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Theorem scaffval 19091
Description: The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
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 6333 . . . . . . . 8 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
3 scaffval.f . . . . . . . 8 𝐹 = (Scalar‘𝑊)
42, 3syl6eqr 2823 . . . . . . 7 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
54fveq2d 6337 . . . . . 6 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
6 scaffval.k . . . . . 6 𝐾 = (Base‘𝐹)
75, 6syl6eqr 2823 . . . . 5 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
8 fveq2 6333 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
9 scaffval.b . . . . . 6 𝐵 = (Base‘𝑊)
108, 9syl6eqr 2823 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
11 fveq2 6333 . . . . . . 7 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
12 scaffval.s . . . . . . 7 · = ( ·𝑠𝑊)
1311, 12syl6eqr 2823 . . . . . 6 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
1413oveqd 6813 . . . . 5 (𝑤 = 𝑊 → (𝑥( ·𝑠𝑤)𝑦) = (𝑥 · 𝑦))
157, 10, 14mpt2eq123dv 6868 . . . 4 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠𝑤)𝑦)) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
16 df-scaf 19076 . . . 4 ·sf = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠𝑤)𝑦)))
17 df-ov 6799 . . . . . . . 8 (𝑥 · 𝑦) = ( · ‘⟨𝑥, 𝑦⟩)
18 fvrn0 6359 . . . . . . . 8 ( · ‘⟨𝑥, 𝑦⟩) ∈ (ran · ∪ {∅})
1917, 18eqeltri 2846 . . . . . . 7 (𝑥 · 𝑦) ∈ (ran · ∪ {∅})
2019rgen2w 3074 . . . . . 6 𝑥𝐾𝑦𝐵 (𝑥 · 𝑦) ∈ (ran · ∪ {∅})
21 eqid 2771 . . . . . . 7 (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
2221fmpt2 7391 . . . . . 6 (∀𝑥𝐾𝑦𝐵 (𝑥 · 𝑦) ∈ (ran · ∪ {∅}) ↔ (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)):(𝐾 × 𝐵)⟶(ran · ∪ {∅}))
2320, 22mpbi 220 . . . . 5 (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)):(𝐾 × 𝐵)⟶(ran · ∪ {∅})
246fvexi 6345 . . . . . 6 𝐾 ∈ V
259fvexi 6345 . . . . . 6 𝐵 ∈ V
2624, 25xpex 7113 . . . . 5 (𝐾 × 𝐵) ∈ V
2712fvexi 6345 . . . . . . 7 · ∈ V
2827rnex 7251 . . . . . 6 ran · ∈ V
29 p0ex 4985 . . . . . 6 {∅} ∈ V
3028, 29unex 7107 . . . . 5 (ran · ∪ {∅}) ∈ V
31 fex2 7272 . . . . 5 (((𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)):(𝐾 × 𝐵)⟶(ran · ∪ {∅}) ∧ (𝐾 × 𝐵) ∈ V ∧ (ran · ∪ {∅}) ∈ V) → (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) ∈ V)
3223, 26, 30, 31mp3an 1572 . . . 4 (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) ∈ V
3315, 16, 32fvmpt 6426 . . 3 (𝑊 ∈ V → ( ·sf𝑊) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
34 fvprc 6327 . . . . 5 𝑊 ∈ V → ( ·sf𝑊) = ∅)
35 mpt20 6876 . . . . 5 (𝑥 ∈ ∅, 𝑦𝐵 ↦ (𝑥 · 𝑦)) = ∅
3634, 35syl6eqr 2823 . . . 4 𝑊 ∈ V → ( ·sf𝑊) = (𝑥 ∈ ∅, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
37 fvprc 6327 . . . . . . . . 9 𝑊 ∈ V → (Scalar‘𝑊) = ∅)
383, 37syl5eq 2817 . . . . . . . 8 𝑊 ∈ V → 𝐹 = ∅)
3938fveq2d 6337 . . . . . . 7 𝑊 ∈ V → (Base‘𝐹) = (Base‘∅))
406, 39syl5eq 2817 . . . . . 6 𝑊 ∈ V → 𝐾 = (Base‘∅))
41 base0 16119 . . . . . 6 ∅ = (Base‘∅)
4240, 41syl6eqr 2823 . . . . 5 𝑊 ∈ V → 𝐾 = ∅)
43 eqid 2771 . . . . 5 𝐵 = 𝐵
44 mpt2eq12 6866 . . . . 5 ((𝐾 = ∅ ∧ 𝐵 = 𝐵) → (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) = (𝑥 ∈ ∅, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
4542, 43, 44sylancl 574 . . . 4 𝑊 ∈ V → (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) = (𝑥 ∈ ∅, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
4636, 45eqtr4d 2808 . . 3 𝑊 ∈ V → ( ·sf𝑊) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
4733, 46pm2.61i 176 . 2 ( ·sf𝑊) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
481, 47eqtri 2793 1 = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  cun 3721  c0 4063  {csn 4317  cop 4323   × cxp 5248  ran crn 5251  wf 6026  cfv 6030  (class class class)co 6796  cmpt2 6798  Basecbs 16064  Scalarcsca 16152   ·𝑠 cvsca 16153   ·sf cscaf 19074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-slot 16068  df-base 16070  df-scaf 19076
This theorem is referenced by:  scafval  19092  scafeq  19093  scaffn  19094  lmodscaf  19095  rlmscaf  19423
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