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Theorem scaffval 20340
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 6842 . . . . . . . 8 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
3 scaffval.f . . . . . . . 8 𝐹 = (Scalar‘𝑊)
42, 3eqtr4di 2794 . . . . . . 7 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
54fveq2d 6846 . . . . . 6 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
6 scaffval.k . . . . . 6 𝐾 = (Base‘𝐹)
75, 6eqtr4di 2794 . . . . 5 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
8 fveq2 6842 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
9 scaffval.b . . . . . 6 𝐵 = (Base‘𝑊)
108, 9eqtr4di 2794 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
11 fveq2 6842 . . . . . . 7 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
12 scaffval.s . . . . . . 7 · = ( ·𝑠𝑊)
1311, 12eqtr4di 2794 . . . . . 6 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
1413oveqd 7374 . . . . 5 (𝑤 = 𝑊 → (𝑥( ·𝑠𝑤)𝑦) = (𝑥 · 𝑦))
157, 10, 14mpoeq123dv 7432 . . . 4 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠𝑤)𝑦)) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
16 df-scaf 20325 . . . 4 ·sf = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠𝑤)𝑦)))
176fvexi 6856 . . . . 5 𝐾 ∈ V
189fvexi 6856 . . . . 5 𝐵 ∈ V
1912fvexi 6856 . . . . . . 7 · ∈ V
2019rnex 7849 . . . . . 6 ran · ∈ V
21 p0ex 5339 . . . . . 6 {∅} ∈ V
2220, 21unex 7680 . . . . 5 (ran · ∪ {∅}) ∈ V
23 df-ov 7360 . . . . . . 7 (𝑥 · 𝑦) = ( · ‘⟨𝑥, 𝑦⟩)
24 fvrn0 6872 . . . . . . 7 ( · ‘⟨𝑥, 𝑦⟩) ∈ (ran · ∪ {∅})
2523, 24eqeltri 2834 . . . . . 6 (𝑥 · 𝑦) ∈ (ran · ∪ {∅})
2625rgen2w 3069 . . . . 5 𝑥𝐾𝑦𝐵 (𝑥 · 𝑦) ∈ (ran · ∪ {∅})
2717, 18, 22, 26mpoexw 8011 . . . 4 (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) ∈ V
2815, 16, 27fvmpt 6948 . . 3 (𝑊 ∈ V → ( ·sf𝑊) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
29 fvprc 6834 . . . 4 𝑊 ∈ V → ( ·sf𝑊) = ∅)
30 fvprc 6834 . . . . . . 7 𝑊 ∈ V → (Base‘𝑊) = ∅)
319, 30eqtrid 2788 . . . . . 6 𝑊 ∈ V → 𝐵 = ∅)
3231olcd 872 . . . . 5 𝑊 ∈ V → (𝐾 = ∅ ∨ 𝐵 = ∅))
33 0mpo0 7440 . . . . 5 ((𝐾 = ∅ ∨ 𝐵 = ∅) → (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) = ∅)
3432, 33syl 17 . . . 4 𝑊 ∈ V → (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) = ∅)
3529, 34eqtr4d 2779 . . 3 𝑊 ∈ V → ( ·sf𝑊) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
3628, 35pm2.61i 182 . 2 ( ·sf𝑊) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
371, 36eqtri 2764 1 = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 845   = wceq 1541  wcel 2106  Vcvv 3445  cun 3908  c0 4282  {csn 4586  cop 4592  ran crn 5634  cfv 6496  (class class class)co 7357  cmpo 7359  Basecbs 17083  Scalarcsca 17136   ·𝑠 cvsca 17137   ·sf cscaf 20323
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-scaf 20325
This theorem is referenced by:  scafval  20341  scafeq  20342  scaffn  20343  lmodscaf  20344  rlmscaf  20678
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