MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  scaffval Structured version   Visualization version   GIF version

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
  Copyright terms: Public domain W3C validator