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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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