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Theorem scaffval 18653
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 6088 . . . . . . . 8 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
3 scaffval.f . . . . . . . 8 𝐹 = (Scalar‘𝑊)
42, 3syl6eqr 2662 . . . . . . 7 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
54fveq2d 6092 . . . . . 6 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
6 scaffval.k . . . . . 6 𝐾 = (Base‘𝐹)
75, 6syl6eqr 2662 . . . . 5 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
8 fveq2 6088 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
9 scaffval.b . . . . . 6 𝐵 = (Base‘𝑊)
108, 9syl6eqr 2662 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
11 fveq2 6088 . . . . . . 7 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
12 scaffval.s . . . . . . 7 · = ( ·𝑠𝑊)
1311, 12syl6eqr 2662 . . . . . 6 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
1413oveqd 6544 . . . . 5 (𝑤 = 𝑊 → (𝑥( ·𝑠𝑤)𝑦) = (𝑥 · 𝑦))
157, 10, 14mpt2eq123dv 6593 . . . 4 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠𝑤)𝑦)) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
16 df-scaf 18638 . . . 4 ·sf = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠𝑤)𝑦)))
17 df-ov 6530 . . . . . . . 8 (𝑥 · 𝑦) = ( · ‘⟨𝑥, 𝑦⟩)
18 fvrn0 6111 . . . . . . . 8 ( · ‘⟨𝑥, 𝑦⟩) ∈ (ran · ∪ {∅})
1917, 18eqeltri 2684 . . . . . . 7 (𝑥 · 𝑦) ∈ (ran · ∪ {∅})
2019rgen2w 2909 . . . . . 6 𝑥𝐾𝑦𝐵 (𝑥 · 𝑦) ∈ (ran · ∪ {∅})
21 eqid 2610 . . . . . . 7 (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
2221fmpt2 7104 . . . . . 6 (∀𝑥𝐾𝑦𝐵 (𝑥 · 𝑦) ∈ (ran · ∪ {∅}) ↔ (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)):(𝐾 × 𝐵)⟶(ran · ∪ {∅}))
2320, 22mpbi 219 . . . . 5 (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)):(𝐾 × 𝐵)⟶(ran · ∪ {∅})
24 fvex 6098 . . . . . . 7 (Base‘𝐹) ∈ V
256, 24eqeltri 2684 . . . . . 6 𝐾 ∈ V
26 fvex 6098 . . . . . . 7 (Base‘𝑊) ∈ V
279, 26eqeltri 2684 . . . . . 6 𝐵 ∈ V
2825, 27xpex 6838 . . . . 5 (𝐾 × 𝐵) ∈ V
29 fvex 6098 . . . . . . . 8 ( ·𝑠𝑊) ∈ V
3012, 29eqeltri 2684 . . . . . . 7 · ∈ V
3130rnex 6970 . . . . . 6 ran · ∈ V
32 p0ex 4774 . . . . . 6 {∅} ∈ V
3331, 32unex 6832 . . . . 5 (ran · ∪ {∅}) ∈ V
34 fex2 6992 . . . . 5 (((𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)):(𝐾 × 𝐵)⟶(ran · ∪ {∅}) ∧ (𝐾 × 𝐵) ∈ V ∧ (ran · ∪ {∅}) ∈ V) → (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) ∈ V)
3523, 28, 33, 34mp3an 1416 . . . 4 (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) ∈ V
3615, 16, 35fvmpt 6176 . . 3 (𝑊 ∈ V → ( ·sf𝑊) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
37 fvprc 6082 . . . . 5 𝑊 ∈ V → ( ·sf𝑊) = ∅)
38 mpt20 6601 . . . . 5 (𝑥 ∈ ∅, 𝑦𝐵 ↦ (𝑥 · 𝑦)) = ∅
3937, 38syl6eqr 2662 . . . 4 𝑊 ∈ V → ( ·sf𝑊) = (𝑥 ∈ ∅, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
40 fvprc 6082 . . . . . . . . 9 𝑊 ∈ V → (Scalar‘𝑊) = ∅)
413, 40syl5eq 2656 . . . . . . . 8 𝑊 ∈ V → 𝐹 = ∅)
4241fveq2d 6092 . . . . . . 7 𝑊 ∈ V → (Base‘𝐹) = (Base‘∅))
436, 42syl5eq 2656 . . . . . 6 𝑊 ∈ V → 𝐾 = (Base‘∅))
44 base0 15689 . . . . . 6 ∅ = (Base‘∅)
4543, 44syl6eqr 2662 . . . . 5 𝑊 ∈ V → 𝐾 = ∅)
46 eqid 2610 . . . . 5 𝐵 = 𝐵
47 mpt2eq12 6591 . . . . 5 ((𝐾 = ∅ ∧ 𝐵 = 𝐵) → (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) = (𝑥 ∈ ∅, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
4845, 46, 47sylancl 693 . . . 4 𝑊 ∈ V → (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)) = (𝑥 ∈ ∅, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
4939, 48eqtr4d 2647 . . 3 𝑊 ∈ V → ( ·sf𝑊) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦)))
5036, 49pm2.61i 175 . 2 ( ·sf𝑊) = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
511, 50eqtri 2632 1 = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  cun 3538  c0 3874  {csn 4125  cop 4131   × cxp 5026  ran crn 5029  wf 5786  cfv 5790  (class class class)co 6527  cmpt2 6529  Basecbs 15644  Scalarcsca 15720   ·𝑠 cvsca 15721   ·sf cscaf 18636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7037  df-2nd 7038  df-slot 15648  df-base 15649  df-scaf 18638
This theorem is referenced by:  scafval  18654  scafeq  18655  scaffn  18656  lmodscaf  18657  rlmscaf  18978
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