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Theorem scaffval 19654
 Description: The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)
Hypotheses
Ref Expression
scaffval.b 𝐵 = (Base‘𝑊)
scaffval.f 𝐹 = (Scalar‘𝑊)
scaffval.k 𝐾 = (Base‘𝐹)
scaffval.a = ( ·sf𝑊)
scaffval.s · = ( ·𝑠𝑊)
Assertion
Ref Expression
scaffval = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥, · ,𝑦   𝑥,𝑊,𝑦
Allowed substitution hints:   (𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem scaffval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 scaffval.a . 2 = ( ·sf𝑊)
2 fveq2 6663 . . . . . . . 8 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
3 scaffval.f . . . . . . . 8 𝐹 = (Scalar‘𝑊)
42, 3eqtr4di 2877 . . . . . . 7 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
54fveq2d 6667 . . . . . 6 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
6 scaffval.k . . . . . 6 𝐾 = (Base‘𝐹)
75, 6eqtr4di 2877 . . . . 5 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
8 fveq2 6663 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
9 scaffval.b . . . . . 6 𝐵 = (Base‘𝑊)
108, 9eqtr4di 2877 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
11 fveq2 6663 . . . . . . 7 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
12 scaffval.s . . . . . . 7 · = ( ·𝑠𝑊)
1311, 12eqtr4di 2877 . . . . . 6 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
1413oveqd 7168 . . . . 5 (𝑤 = 𝑊 → (𝑥( ·𝑠𝑤)𝑦) = (𝑥 · 𝑦))
157, 10, 14mpoeq123dv 7224 . . . 4 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠𝑤)𝑦)) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
16 df-scaf 19639 . . . 4 ·sf = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠𝑤)𝑦)))
176fvexi 6677 . . . . 5 𝐾 ∈ V
189fvexi 6677 . . . . 5 𝐵 ∈ V
1912fvexi 6677 . . . . . . 7 · ∈ V
2019rnex 7614 . . . . . 6 ran · ∈ V
21 p0ex 5273 . . . . . 6 {∅} ∈ V
2220, 21unex 7465 . . . . 5 (ran · ∪ {∅}) ∈ V
23 df-ov 7154 . . . . . . 7 (𝑥 · 𝑦) = ( · ‘⟨𝑥, 𝑦⟩)
24 fvrn0 6691 . . . . . . 7 ( · ‘⟨𝑥, 𝑦⟩) ∈ (ran · ∪ {∅})
2523, 24eqeltri 2912 . . . . . 6 (𝑥 · 𝑦) ∈ (ran · ∪ {∅})
2625rgen2w 3146 . . . . 5 𝑥𝐾𝑦𝐵 (𝑥 · 𝑦) ∈ (ran · ∪ {∅})
2717, 18, 22, 26mpoexw 7774 . . . 4 (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) ∈ V
2815, 16, 27fvmpt 6761 . . 3 (𝑊 ∈ V → ( ·sf𝑊) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
29 fvprc 6656 . . . 4 𝑊 ∈ V → ( ·sf𝑊) = ∅)
30 fvprc 6656 . . . . . . 7 𝑊 ∈ V → (Base‘𝑊) = ∅)
319, 30syl5eq 2871 . . . . . 6 𝑊 ∈ V → 𝐵 = ∅)
3231olcd 871 . . . . 5 𝑊 ∈ V → (𝐾 = ∅ ∨ 𝐵 = ∅))
33 0mpo0 7232 . . . . 5 ((𝐾 = ∅ ∨ 𝐵 = ∅) → (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) = ∅)
3432, 33syl 17 . . . 4 𝑊 ∈ V → (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) = ∅)
3529, 34eqtr4d 2862 . . 3 𝑊 ∈ V → ( ·sf𝑊) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
3628, 35pm2.61i 185 . 2 ( ·sf𝑊) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
371, 36eqtri 2847 1 = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 844   = wceq 1538   ∈ wcel 2115  Vcvv 3480   ∪ cun 3917  ∅c0 4276  {csn 4550  ⟨cop 4556  ran crn 5544  ‘cfv 6345  (class class class)co 7151   ∈ cmpo 7153  Basecbs 16485  Scalarcsca 16570   ·𝑠 cvsca 16571   ·sf cscaf 19637 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7686  df-2nd 7687  df-scaf 19639 This theorem is referenced by:  scafval  19655  scafeq  19656  scaffn  19657  lmodscaf  19658  rlmscaf  19983
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