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Theorem dsmmacl 20004
 Description: The finite hull is closed under addition. (Contributed by Stefan O'Rear, 11-Jan-2015.)
Hypotheses
Ref Expression
dsmmcl.p 𝑃 = (𝑆Xs𝑅)
dsmmcl.h 𝐻 = (Base‘(𝑆m 𝑅))
dsmmcl.i (𝜑𝐼𝑊)
dsmmcl.s (𝜑𝑆𝑉)
dsmmcl.r (𝜑𝑅:𝐼⟶Mnd)
dsmmacl.j (𝜑𝐽𝐻)
dsmmacl.k (𝜑𝐾𝐻)
dsmmacl.a + = (+g𝑃)
Assertion
Ref Expression
dsmmacl (𝜑 → (𝐽 + 𝐾) ∈ 𝐻)

Proof of Theorem dsmmacl
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 dsmmcl.p . . 3 𝑃 = (𝑆Xs𝑅)
2 eqid 2621 . . 3 (Base‘𝑃) = (Base‘𝑃)
3 dsmmacl.a . . 3 + = (+g𝑃)
4 dsmmcl.s . . 3 (𝜑𝑆𝑉)
5 dsmmcl.i . . 3 (𝜑𝐼𝑊)
6 dsmmcl.r . . 3 (𝜑𝑅:𝐼⟶Mnd)
7 dsmmacl.j . . . . 5 (𝜑𝐽𝐻)
8 eqid 2621 . . . . . 6 (𝑆m 𝑅) = (𝑆m 𝑅)
9 dsmmcl.h . . . . . 6 𝐻 = (Base‘(𝑆m 𝑅))
10 ffn 6002 . . . . . . 7 (𝑅:𝐼⟶Mnd → 𝑅 Fn 𝐼)
116, 10syl 17 . . . . . 6 (𝜑𝑅 Fn 𝐼)
121, 8, 2, 9, 5, 11dsmmelbas 20002 . . . . 5 (𝜑 → (𝐽𝐻 ↔ (𝐽 ∈ (Base‘𝑃) ∧ {𝑎𝐼 ∣ (𝐽𝑎) ≠ (0g‘(𝑅𝑎))} ∈ Fin)))
137, 12mpbid 222 . . . 4 (𝜑 → (𝐽 ∈ (Base‘𝑃) ∧ {𝑎𝐼 ∣ (𝐽𝑎) ≠ (0g‘(𝑅𝑎))} ∈ Fin))
1413simpld 475 . . 3 (𝜑𝐽 ∈ (Base‘𝑃))
15 dsmmacl.k . . . . 5 (𝜑𝐾𝐻)
161, 8, 2, 9, 5, 11dsmmelbas 20002 . . . . 5 (𝜑 → (𝐾𝐻 ↔ (𝐾 ∈ (Base‘𝑃) ∧ {𝑎𝐼 ∣ (𝐾𝑎) ≠ (0g‘(𝑅𝑎))} ∈ Fin)))
1715, 16mpbid 222 . . . 4 (𝜑 → (𝐾 ∈ (Base‘𝑃) ∧ {𝑎𝐼 ∣ (𝐾𝑎) ≠ (0g‘(𝑅𝑎))} ∈ Fin))
1817simpld 475 . . 3 (𝜑𝐾 ∈ (Base‘𝑃))
191, 2, 3, 4, 5, 6, 14, 18prdsplusgcl 17242 . 2 (𝜑 → (𝐽 + 𝐾) ∈ (Base‘𝑃))
204adantr 481 . . . . . 6 ((𝜑𝑎𝐼) → 𝑆𝑉)
215adantr 481 . . . . . 6 ((𝜑𝑎𝐼) → 𝐼𝑊)
2211adantr 481 . . . . . 6 ((𝜑𝑎𝐼) → 𝑅 Fn 𝐼)
2314adantr 481 . . . . . 6 ((𝜑𝑎𝐼) → 𝐽 ∈ (Base‘𝑃))
2418adantr 481 . . . . . 6 ((𝜑𝑎𝐼) → 𝐾 ∈ (Base‘𝑃))
25 simpr 477 . . . . . 6 ((𝜑𝑎𝐼) → 𝑎𝐼)
261, 2, 20, 21, 22, 23, 24, 3, 25prdsplusgfval 16055 . . . . 5 ((𝜑𝑎𝐼) → ((𝐽 + 𝐾)‘𝑎) = ((𝐽𝑎)(+g‘(𝑅𝑎))(𝐾𝑎)))
2726neeq1d 2849 . . . 4 ((𝜑𝑎𝐼) → (((𝐽 + 𝐾)‘𝑎) ≠ (0g‘(𝑅𝑎)) ↔ ((𝐽𝑎)(+g‘(𝑅𝑎))(𝐾𝑎)) ≠ (0g‘(𝑅𝑎))))
2827rabbidva 3176 . . 3 (𝜑 → {𝑎𝐼 ∣ ((𝐽 + 𝐾)‘𝑎) ≠ (0g‘(𝑅𝑎))} = {𝑎𝐼 ∣ ((𝐽𝑎)(+g‘(𝑅𝑎))(𝐾𝑎)) ≠ (0g‘(𝑅𝑎))})
2913simprd 479 . . . . 5 (𝜑 → {𝑎𝐼 ∣ (𝐽𝑎) ≠ (0g‘(𝑅𝑎))} ∈ Fin)
3017simprd 479 . . . . 5 (𝜑 → {𝑎𝐼 ∣ (𝐾𝑎) ≠ (0g‘(𝑅𝑎))} ∈ Fin)
31 unfi 8171 . . . . 5 (({𝑎𝐼 ∣ (𝐽𝑎) ≠ (0g‘(𝑅𝑎))} ∈ Fin ∧ {𝑎𝐼 ∣ (𝐾𝑎) ≠ (0g‘(𝑅𝑎))} ∈ Fin) → ({𝑎𝐼 ∣ (𝐽𝑎) ≠ (0g‘(𝑅𝑎))} ∪ {𝑎𝐼 ∣ (𝐾𝑎) ≠ (0g‘(𝑅𝑎))}) ∈ Fin)
3229, 30, 31syl2anc 692 . . . 4 (𝜑 → ({𝑎𝐼 ∣ (𝐽𝑎) ≠ (0g‘(𝑅𝑎))} ∪ {𝑎𝐼 ∣ (𝐾𝑎) ≠ (0g‘(𝑅𝑎))}) ∈ Fin)
33 neorian 2884 . . . . . . . . . 10 (((𝐽𝑎) ≠ (0g‘(𝑅𝑎)) ∨ (𝐾𝑎) ≠ (0g‘(𝑅𝑎))) ↔ ¬ ((𝐽𝑎) = (0g‘(𝑅𝑎)) ∧ (𝐾𝑎) = (0g‘(𝑅𝑎))))
3433bicomi 214 . . . . . . . . 9 (¬ ((𝐽𝑎) = (0g‘(𝑅𝑎)) ∧ (𝐾𝑎) = (0g‘(𝑅𝑎))) ↔ ((𝐽𝑎) ≠ (0g‘(𝑅𝑎)) ∨ (𝐾𝑎) ≠ (0g‘(𝑅𝑎))))
3534con1bii 346 . . . . . . . 8 (¬ ((𝐽𝑎) ≠ (0g‘(𝑅𝑎)) ∨ (𝐾𝑎) ≠ (0g‘(𝑅𝑎))) ↔ ((𝐽𝑎) = (0g‘(𝑅𝑎)) ∧ (𝐾𝑎) = (0g‘(𝑅𝑎))))
366ffvelrnda 6315 . . . . . . . . . 10 ((𝜑𝑎𝐼) → (𝑅𝑎) ∈ Mnd)
37 eqid 2621 . . . . . . . . . . . 12 (Base‘(𝑅𝑎)) = (Base‘(𝑅𝑎))
38 eqid 2621 . . . . . . . . . . . 12 (0g‘(𝑅𝑎)) = (0g‘(𝑅𝑎))
3937, 38mndidcl 17229 . . . . . . . . . . 11 ((𝑅𝑎) ∈ Mnd → (0g‘(𝑅𝑎)) ∈ (Base‘(𝑅𝑎)))
4036, 39syl 17 . . . . . . . . . 10 ((𝜑𝑎𝐼) → (0g‘(𝑅𝑎)) ∈ (Base‘(𝑅𝑎)))
41 eqid 2621 . . . . . . . . . . 11 (+g‘(𝑅𝑎)) = (+g‘(𝑅𝑎))
4237, 41, 38mndlid 17232 . . . . . . . . . 10 (((𝑅𝑎) ∈ Mnd ∧ (0g‘(𝑅𝑎)) ∈ (Base‘(𝑅𝑎))) → ((0g‘(𝑅𝑎))(+g‘(𝑅𝑎))(0g‘(𝑅𝑎))) = (0g‘(𝑅𝑎)))
4336, 40, 42syl2anc 692 . . . . . . . . 9 ((𝜑𝑎𝐼) → ((0g‘(𝑅𝑎))(+g‘(𝑅𝑎))(0g‘(𝑅𝑎))) = (0g‘(𝑅𝑎)))
44 oveq12 6613 . . . . . . . . . 10 (((𝐽𝑎) = (0g‘(𝑅𝑎)) ∧ (𝐾𝑎) = (0g‘(𝑅𝑎))) → ((𝐽𝑎)(+g‘(𝑅𝑎))(𝐾𝑎)) = ((0g‘(𝑅𝑎))(+g‘(𝑅𝑎))(0g‘(𝑅𝑎))))
4544eqeq1d 2623 . . . . . . . . 9 (((𝐽𝑎) = (0g‘(𝑅𝑎)) ∧ (𝐾𝑎) = (0g‘(𝑅𝑎))) → (((𝐽𝑎)(+g‘(𝑅𝑎))(𝐾𝑎)) = (0g‘(𝑅𝑎)) ↔ ((0g‘(𝑅𝑎))(+g‘(𝑅𝑎))(0g‘(𝑅𝑎))) = (0g‘(𝑅𝑎))))
4643, 45syl5ibrcom 237 . . . . . . . 8 ((𝜑𝑎𝐼) → (((𝐽𝑎) = (0g‘(𝑅𝑎)) ∧ (𝐾𝑎) = (0g‘(𝑅𝑎))) → ((𝐽𝑎)(+g‘(𝑅𝑎))(𝐾𝑎)) = (0g‘(𝑅𝑎))))
4735, 46syl5bi 232 . . . . . . 7 ((𝜑𝑎𝐼) → (¬ ((𝐽𝑎) ≠ (0g‘(𝑅𝑎)) ∨ (𝐾𝑎) ≠ (0g‘(𝑅𝑎))) → ((𝐽𝑎)(+g‘(𝑅𝑎))(𝐾𝑎)) = (0g‘(𝑅𝑎))))
4847necon1ad 2807 . . . . . 6 ((𝜑𝑎𝐼) → (((𝐽𝑎)(+g‘(𝑅𝑎))(𝐾𝑎)) ≠ (0g‘(𝑅𝑎)) → ((𝐽𝑎) ≠ (0g‘(𝑅𝑎)) ∨ (𝐾𝑎) ≠ (0g‘(𝑅𝑎)))))
4948ss2rabdv 3662 . . . . 5 (𝜑 → {𝑎𝐼 ∣ ((𝐽𝑎)(+g‘(𝑅𝑎))(𝐾𝑎)) ≠ (0g‘(𝑅𝑎))} ⊆ {𝑎𝐼 ∣ ((𝐽𝑎) ≠ (0g‘(𝑅𝑎)) ∨ (𝐾𝑎) ≠ (0g‘(𝑅𝑎)))})
50 unrab 3874 . . . . 5 ({𝑎𝐼 ∣ (𝐽𝑎) ≠ (0g‘(𝑅𝑎))} ∪ {𝑎𝐼 ∣ (𝐾𝑎) ≠ (0g‘(𝑅𝑎))}) = {𝑎𝐼 ∣ ((𝐽𝑎) ≠ (0g‘(𝑅𝑎)) ∨ (𝐾𝑎) ≠ (0g‘(𝑅𝑎)))}
5149, 50syl6sseqr 3631 . . . 4 (𝜑 → {𝑎𝐼 ∣ ((𝐽𝑎)(+g‘(𝑅𝑎))(𝐾𝑎)) ≠ (0g‘(𝑅𝑎))} ⊆ ({𝑎𝐼 ∣ (𝐽𝑎) ≠ (0g‘(𝑅𝑎))} ∪ {𝑎𝐼 ∣ (𝐾𝑎) ≠ (0g‘(𝑅𝑎))}))
52 ssfi 8124 . . . 4 ((({𝑎𝐼 ∣ (𝐽𝑎) ≠ (0g‘(𝑅𝑎))} ∪ {𝑎𝐼 ∣ (𝐾𝑎) ≠ (0g‘(𝑅𝑎))}) ∈ Fin ∧ {𝑎𝐼 ∣ ((𝐽𝑎)(+g‘(𝑅𝑎))(𝐾𝑎)) ≠ (0g‘(𝑅𝑎))} ⊆ ({𝑎𝐼 ∣ (𝐽𝑎) ≠ (0g‘(𝑅𝑎))} ∪ {𝑎𝐼 ∣ (𝐾𝑎) ≠ (0g‘(𝑅𝑎))})) → {𝑎𝐼 ∣ ((𝐽𝑎)(+g‘(𝑅𝑎))(𝐾𝑎)) ≠ (0g‘(𝑅𝑎))} ∈ Fin)
5332, 51, 52syl2anc 692 . . 3 (𝜑 → {𝑎𝐼 ∣ ((𝐽𝑎)(+g‘(𝑅𝑎))(𝐾𝑎)) ≠ (0g‘(𝑅𝑎))} ∈ Fin)
5428, 53eqeltrd 2698 . 2 (𝜑 → {𝑎𝐼 ∣ ((𝐽 + 𝐾)‘𝑎) ≠ (0g‘(𝑅𝑎))} ∈ Fin)
551, 8, 2, 9, 5, 11dsmmelbas 20002 . 2 (𝜑 → ((𝐽 + 𝐾) ∈ 𝐻 ↔ ((𝐽 + 𝐾) ∈ (Base‘𝑃) ∧ {𝑎𝐼 ∣ ((𝐽 + 𝐾)‘𝑎) ≠ (0g‘(𝑅𝑎))} ∈ Fin)))
5619, 54, 55mpbir2and 956 1 (𝜑 → (𝐽 + 𝐾) ∈ 𝐻)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  {crab 2911   ∪ cun 3553   ⊆ wss 3555   Fn wfn 5842  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604  Fincfn 7899  Basecbs 15781  +gcplusg 15862  0gc0g 16021  Xscprds 16027  Mndcmnd 17215   ⊕m cdsmm 19994 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-hom 15887  df-cco 15888  df-0g 16023  df-prds 16029  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-dsmm 19995 This theorem is referenced by:  dsmmsubg  20006
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