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Theorem scaffval 19368
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 6493 . . . . . . . 8 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
3 scaffval.f . . . . . . . 8 𝐹 = (Scalar‘𝑊)
42, 3syl6eqr 2826 . . . . . . 7 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
54fveq2d 6497 . . . . . 6 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
6 scaffval.k . . . . . 6 𝐾 = (Base‘𝐹)
75, 6syl6eqr 2826 . . . . 5 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
8 fveq2 6493 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
9 scaffval.b . . . . . 6 𝐵 = (Base‘𝑊)
108, 9syl6eqr 2826 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
11 fveq2 6493 . . . . . . 7 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
12 scaffval.s . . . . . . 7 · = ( ·𝑠𝑊)
1311, 12syl6eqr 2826 . . . . . 6 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
1413oveqd 6987 . . . . 5 (𝑤 = 𝑊 → (𝑥( ·𝑠𝑤)𝑦) = (𝑥 · 𝑦))
157, 10, 14mpoeq123dv 7041 . . . 4 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠𝑤)𝑦)) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
16 df-scaf 19353 . . . 4 ·sf = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠𝑤)𝑦)))
17 df-ov 6973 . . . . . . . 8 (𝑥 · 𝑦) = ( · ‘⟨𝑥, 𝑦⟩)
18 fvrn0 6521 . . . . . . . 8 ( · ‘⟨𝑥, 𝑦⟩) ∈ (ran · ∪ {∅})
1917, 18eqeltri 2856 . . . . . . 7 (𝑥 · 𝑦) ∈ (ran · ∪ {∅})
2019rgen2w 3095 . . . . . 6 𝑥𝐾𝑦𝐵 (𝑥 · 𝑦) ∈ (ran · ∪ {∅})
21 eqid 2772 . . . . . . 7 (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
2221fmpo 7568 . . . . . 6 (∀𝑥𝐾𝑦𝐵 (𝑥 · 𝑦) ∈ (ran · ∪ {∅}) ↔ (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)):(𝐾 × 𝐵)⟶(ran · ∪ {∅}))
2320, 22mpbi 222 . . . . 5 (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)):(𝐾 × 𝐵)⟶(ran · ∪ {∅})
246fvexi 6507 . . . . . 6 𝐾 ∈ V
259fvexi 6507 . . . . . 6 𝐵 ∈ V
2624, 25xpex 7287 . . . . 5 (𝐾 × 𝐵) ∈ V
2712fvexi 6507 . . . . . . 7 · ∈ V
2827rnex 7426 . . . . . 6 ran · ∈ V
29 p0ex 5131 . . . . . 6 {∅} ∈ V
3028, 29unex 7280 . . . . 5 (ran · ∪ {∅}) ∈ V
31 fex2 7447 . . . . 5 (((𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)):(𝐾 × 𝐵)⟶(ran · ∪ {∅}) ∧ (𝐾 × 𝐵) ∈ V ∧ (ran · ∪ {∅}) ∈ V) → (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) ∈ V)
3223, 26, 30, 31mp3an 1440 . . . 4 (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) ∈ V
3315, 16, 32fvmpt 6589 . . 3 (𝑊 ∈ V → ( ·sf𝑊) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
34 fvprc 6486 . . . . 5 𝑊 ∈ V → ( ·sf𝑊) = ∅)
35 mpo0 7049 . . . . 5 (𝑥 ∈ ∅, 𝑦𝐵 ↦ (𝑥 · 𝑦)) = ∅
3634, 35syl6eqr 2826 . . . 4 𝑊 ∈ V → ( ·sf𝑊) = (𝑥 ∈ ∅, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
37 fvprc 6486 . . . . . . . . 9 𝑊 ∈ V → (Scalar‘𝑊) = ∅)
383, 37syl5eq 2820 . . . . . . . 8 𝑊 ∈ V → 𝐹 = ∅)
3938fveq2d 6497 . . . . . . 7 𝑊 ∈ V → (Base‘𝐹) = (Base‘∅))
406, 39syl5eq 2820 . . . . . 6 𝑊 ∈ V → 𝐾 = (Base‘∅))
41 base0 16386 . . . . . 6 ∅ = (Base‘∅)
4240, 41syl6eqr 2826 . . . . 5 𝑊 ∈ V → 𝐾 = ∅)
43 eqid 2772 . . . . 5 𝐵 = 𝐵
44 mpoeq12 7039 . . . . 5 ((𝐾 = ∅ ∧ 𝐵 = 𝐵) → (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) = (𝑥 ∈ ∅, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
4542, 43, 44sylancl 577 . . . 4 𝑊 ∈ V → (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) = (𝑥 ∈ ∅, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
4636, 45eqtr4d 2811 . . 3 𝑊 ∈ V → ( ·sf𝑊) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
4733, 46pm2.61i 177 . 2 ( ·sf𝑊) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
481, 47eqtri 2796 1 = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1507  wcel 2050  wral 3082  Vcvv 3409  cun 3821  c0 4172  {csn 4435  cop 4441   × cxp 5399  ran crn 5402  wf 6178  cfv 6182  (class class class)co 6970  cmpo 6972  Basecbs 16333  Scalarcsca 16418   ·𝑠 cvsca 16419   ·sf cscaf 19351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-1st 7495  df-2nd 7496  df-slot 16337  df-base 16339  df-scaf 19353
This theorem is referenced by:  scafval  19369  scafeq  19370  scaffn  19371  lmodscaf  19372  rlmscaf  19696
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