Theorem scaffval 20752

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	⊢ ∙ = ( ·sf ‘𝑊)
scaffval.s	⊢ · = ( ·𝑠 ‘𝑊)

Assertion

Ref	Expression
scaffval	⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (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.

Step	Hyp	Ref	Expression
1		scaffval.a	. 2 ⊢ ∙ = ( ·sf ‘𝑊)
2		fveq2 6891	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
3		scaffval.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
4	2, 3	eqtr4di 2785	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
5	4	fveq2d 6895	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
6		scaffval.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝐹)
7	5, 6	eqtr4di 2785	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
8		fveq2 6891	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
9		scaffval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
10	8, 9	eqtr4di 2785	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
11		fveq2 6891	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
12		scaffval.s	. . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊)
13	11, 12	eqtr4di 2785	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
14	13	oveqd 7431	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠 ‘𝑤)𝑦) = (𝑥 · 𝑦))
15	7, 10, 14	mpoeq123dv 7489	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠 ‘𝑤)𝑦)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
16		df-scaf 20735	. . . 4 ⊢ ·sf = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠 ‘𝑤)𝑦)))
17	6	fvexi 6905	. . . . 5 ⊢ 𝐾 ∈ V
18	9	fvexi 6905	. . . . 5 ⊢ 𝐵 ∈ V
19	12	fvexi 6905	. . . . . . 7 ⊢ · ∈ V
20	19	rnex 7912	. . . . . 6 ⊢ ran · ∈ V
21		p0ex 5378	. . . . . 6 ⊢ {∅} ∈ V
22	20, 21	unex 7742	. . . . 5 ⊢ (ran · ∪ {∅}) ∈ V
23		df-ov 7417	. . . . . . 7 ⊢ (𝑥 · 𝑦) = ( · ‘⟨𝑥, 𝑦⟩)
24		fvrn0 6921	. . . . . . 7 ⊢ ( · ‘⟨𝑥, 𝑦⟩) ∈ (ran · ∪ {∅})
25	23, 24	eqeltri 2824	. . . . . 6 ⊢ (𝑥 · 𝑦) ∈ (ran · ∪ {∅})
26	25	rgen2w 3061	. . . . 5 ⊢ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥 · 𝑦) ∈ (ran · ∪ {∅})
27	17, 18, 22, 26	mpoexw 8077	. . . 4 ⊢ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) ∈ V
28	15, 16, 27	fvmpt 6999	. . 3 ⊢ (𝑊 ∈ V → ( ·sf ‘𝑊) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
29		fvprc 6883	. . . 4 ⊢ (¬ 𝑊 ∈ V → ( ·sf ‘𝑊) = ∅)
30		fvprc 6883	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
31	9, 30	eqtrid 2779	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → 𝐵 = ∅)
32	31	olcd 873	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐾 = ∅ ∨ 𝐵 = ∅))
33		0mpo0 7497	. . . . 5 ⊢ ((𝐾 = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) = ∅)
34	32, 33	syl 17	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) = ∅)
35	29, 34	eqtr4d 2770	. . 3 ⊢ (¬ 𝑊 ∈ V → ( ·sf ‘𝑊) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
36	28, 35	pm2.61i 182	. 2 ⊢ ( ·sf ‘𝑊) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦))
37	1, 36	eqtri 2755	1 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦))

Syntax hints:  ¬ wn 3   ∨ wo 846   = wceq 1534   ∈ wcel 2099  Vcvv 3469   ∪ cun 3942  ∅c0 4318  {csn 4624  ⟨cop 4630  ran crn 5673  ‘cfv 6542  (class class class)co 7414   ∈ cmpo 7416  Basecbs 17171  Scalarcsca 17227   ·_𝑠 cvsca 17228   ·_sf cscaf 20733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-scaf 20735

This theorem is referenced by:  scafval  20753  scafeq  20754  scaffn  20755  lmodscaf  20756  rlmscaf  21089