Theorem scaffval 20875

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 6840	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
3		scaffval.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
4	2, 3	eqtr4di 2789	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
5	4	fveq2d 6844	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
6		scaffval.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝐹)
7	5, 6	eqtr4di 2789	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
8		fveq2 6840	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
9		scaffval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
10	8, 9	eqtr4di 2789	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
11		fveq2 6840	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
12		scaffval.s	. . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊)
13	11, 12	eqtr4di 2789	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
14	13	oveqd 7384	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠 ‘𝑤)𝑦) = (𝑥 · 𝑦))
15	7, 10, 14	mpoeq123dv 7442	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠 ‘𝑤)𝑦)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
16		df-scaf 20858	. . . 4 ⊢ ·sf = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠 ‘𝑤)𝑦)))
17	6	fvexi 6854	. . . . 5 ⊢ 𝐾 ∈ V
18	9	fvexi 6854	. . . . 5 ⊢ 𝐵 ∈ V
19	12	fvexi 6854	. . . . . . 7 ⊢ · ∈ V
20	19	rnex 7861	. . . . . 6 ⊢ ran · ∈ V
21		p0ex 5326	. . . . . 6 ⊢ {∅} ∈ V
22	20, 21	unex 7698	. . . . 5 ⊢ (ran · ∪ {∅}) ∈ V
23		df-ov 7370	. . . . . . 7 ⊢ (𝑥 · 𝑦) = ( · ‘⟨𝑥, 𝑦⟩)
24		fvrn0 6868	. . . . . . 7 ⊢ ( · ‘⟨𝑥, 𝑦⟩) ∈ (ran · ∪ {∅})
25	23, 24	eqeltri 2832	. . . . . 6 ⊢ (𝑥 · 𝑦) ∈ (ran · ∪ {∅})
26	25	rgen2w 3056	. . . . 5 ⊢ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥 · 𝑦) ∈ (ran · ∪ {∅})
27	17, 18, 22, 26	mpoexw 8031	. . . 4 ⊢ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) ∈ V
28	15, 16, 27	fvmpt 6947	. . 3 ⊢ (𝑊 ∈ V → ( ·sf ‘𝑊) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
29		fvprc 6832	. . . 4 ⊢ (¬ 𝑊 ∈ V → ( ·sf ‘𝑊) = ∅)
30		fvprc 6832	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
31	9, 30	eqtrid 2783	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → 𝐵 = ∅)
32	31	olcd 875	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐾 = ∅ ∨ 𝐵 = ∅))
33		0mpo0 7450	. . . . 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 2774	. . 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 2759	1 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦))

Syntax hints:  ¬ wn 3   ∨ wo 848   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ∪ cun 3887  ∅c0 4273  {csn 4567  ⟨cop 4573  ran crn 5632  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  Basecbs 17179  Scalarcsca 17223   ·_𝑠 cvsca 17224   ·_sf cscaf 20856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-scaf 20858

This theorem is referenced by:  scafval  20876  scafeq  20877  scaffn  20878  lmodscaf  20879  rlmscaf  21202