Theorem xpsvsca 17685

Description: Value of the scalar multiplication function in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)

Hypotheses

Ref	Expression
xpssca.t	⊢ 𝑇 = (𝑅 ×s 𝑆)
xpssca.g	⊢ 𝐺 = (Scalar‘𝑅)
xpssca.1	⊢ (𝜑 → 𝑅 ∈ 𝑉)
xpssca.2	⊢ (𝜑 → 𝑆 ∈ 𝑊)
xpsvsca.x	⊢ 𝑋 = (Base‘𝑅)
xpsvsca.y	⊢ 𝑌 = (Base‘𝑆)
xpsvsca.k	⊢ 𝐾 = (Base‘𝐺)
xpsvsca.m	⊢ · = ( ·𝑠 ‘𝑅)
xpsvsca.n	⊢ × = ( ·𝑠 ‘𝑆)
xpsvsca.p	⊢ ∙ = ( ·𝑠 ‘𝑇)
xpsvsca.3	⊢ (𝜑 → 𝐴 ∈ 𝐾)
xpsvsca.4	⊢ (𝜑 → 𝐵 ∈ 𝑋)
xpsvsca.5	⊢ (𝜑 → 𝐶 ∈ 𝑌)
xpsvsca.6	⊢ (𝜑 → (𝐴 · 𝐵) ∈ 𝑋)
xpsvsca.7	⊢ (𝜑 → (𝐴 × 𝐶) ∈ 𝑌)

Assertion

Ref	Expression
xpsvsca	⊢ (𝜑 → (𝐴 ∙ ⟨𝐵, 𝐶⟩) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)

Proof of Theorem xpsvsca

Dummy variables 𝑘 𝑎 𝑥 𝑦 𝑐 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpsvsca.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐾)
2		df-ov 7418	. . . . 5 ⊢ (𝐵(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})𝐶) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐵, 𝐶⟩)
3		xpsvsca.4	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑋)
4		xpsvsca.5	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑌)
5		eqid 2760	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
6	5	xpsfval 17674	. . . . . 6 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌) → (𝐵(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})𝐶) = {⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩})
7	3, 4, 6	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝐵(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})𝐶) = {⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩})
8	2, 7	eqtr3id 2809	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐵, 𝐶⟩) = {⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩})
9	3, 4	opelxpd 5694	. . . . 5 ⊢ (𝜑 → ⟨𝐵, 𝐶⟩ ∈ (𝑋 × 𝑌))
10	5	xpsff1o2 17677	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
11		f1of 6819	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):(𝑋 × 𝑌)⟶ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
12	10, 11	ax-mp 5	. . . . . 6 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):(𝑋 × 𝑌)⟶ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
13	12	ffvelcdmi 7078	. . . . 5 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝑋 × 𝑌) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐵, 𝐶⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
14	9, 13	syl 18	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐵, 𝐶⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
15	8, 14	eqeltrrd 2861	. . 3 ⊢ (𝜑 → {⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩} ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
16		xpssca.t	. . . . 5 ⊢ 𝑇 = (𝑅 ×s 𝑆)
17		xpsvsca.x	. . . . 5 ⊢ 𝑋 = (Base‘𝑅)
18		xpsvsca.y	. . . . 5 ⊢ 𝑌 = (Base‘𝑆)
19		xpssca.1	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
20		xpssca.2	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
21		xpssca.g	. . . . 5 ⊢ 𝐺 = (Scalar‘𝑅)
22		eqid 2760	. . . . 5 ⊢ (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}) = (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})
23	16, 17, 18, 19, 20, 5, 21, 22	xpsval 17678	. . . 4 ⊢ (𝜑 → 𝑇 = (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) “s (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})))
24	16, 17, 18, 19, 20, 5, 21, 22	xpsrnbas 17679	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = (Base‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})))
25		f1ocnv 6832	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})–1-1-onto→(𝑋 × 𝑌))
26	10, 25	mp1i 14	. . . . 5 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})–1-1-onto→(𝑋 × 𝑌))
27		f1ofo 6827	. . . . 5 ⊢ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})–1-1-onto→(𝑋 × 𝑌) → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})–onto→(𝑋 × 𝑌))
28	26, 27	syl 18	. . . 4 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})–onto→(𝑋 × 𝑌))
29		ovexd 7450	. . . 4 ⊢ (𝜑 → (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}) ∈ V)
30	21	fvexi 6894	. . . . . . 7 ⊢ 𝐺 ∈ V
31	30	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐺 ∈ V)
32		prex 5403	. . . . . . 7 ⊢ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} ∈ V
33	32	a1i 11	. . . . . 6 ⊢ (⊤ → {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} ∈ V)
34	22, 31, 33	prdssca 17563	. . . . 5 ⊢ (⊤ → 𝐺 = (Scalar‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})))
35	34	mptru 1577	. . . 4 ⊢ 𝐺 = (Scalar‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}))
36		xpsvsca.k	. . . 4 ⊢ 𝐾 = (Base‘𝐺)
37		eqid 2760	. . . 4 ⊢ ( ·𝑠 ‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) = ( ·𝑠 ‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}))
38		xpsvsca.p	. . . 4 ⊢ ∙ = ( ·𝑠 ‘𝑇)
39	26	f1ovscpbl 17634	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐾 ∧ 𝑏 ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) ∧ 𝑐 ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))) → ((◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘𝑏) = (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘𝑐) → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘(𝑎( ·𝑠 ‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}))𝑏)) = (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘(𝑎( ·𝑠 ‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}))𝑐))))
40	23, 24, 28, 29, 35, 36, 37, 38, 39	imasvscaval 17646	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐾 ∧ {⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩} ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})) → (𝐴 ∙ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩})) = (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘(𝐴( ·𝑠 ‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})){⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩})))
41	1, 15, 40	mpd3an23 1492	. 2 ⊢ (𝜑 → (𝐴 ∙ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩})) = (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘(𝐴( ·𝑠 ‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})){⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩})))
42		f1ocnvfv 7281	. . . . 5 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) ∧ ⟨𝐵, 𝐶⟩ ∈ (𝑋 × 𝑌)) → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐵, 𝐶⟩) = {⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩} → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩}) = ⟨𝐵, 𝐶⟩))
43	10, 9, 42	sylancr 599	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐵, 𝐶⟩) = {⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩} → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩}) = ⟨𝐵, 𝐶⟩))
44	8, 43	mpd 16	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩}) = ⟨𝐵, 𝐶⟩)
45	44	oveq2d 7431	. 2 ⊢ (𝜑 → (𝐴 ∙ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩})) = (𝐴 ∙ ⟨𝐵, 𝐶⟩))
46		iftrue 4488	. . . . . . . . . . . 12 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑅)
47	46	fveq2d 6884	. . . . . . . . . . 11 ⊢ (𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = ( ·𝑠 ‘𝑅))
48		xpsvsca.m	. . . . . . . . . . 11 ⊢ · = ( ·𝑠 ‘𝑅)
49	47, 48	eqtr4di 2813	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = · )
50		eqidd 2761	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → 𝐴 = 𝐴)
51		iftrue 4488	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐵, 𝐶) = 𝐵)
52	49, 50, 51	oveq123d 7436	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = (𝐴 · 𝐵))
53		iftrue 4488	. . . . . . . . 9 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)) = (𝐴 · 𝐵))
54	52, 53	eqtr4d 2798	. . . . . . . 8 ⊢ (𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
55		iffalse 4491	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑆)
56	55	fveq2d 6884	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = ( ·𝑠 ‘𝑆))
57		xpsvsca.n	. . . . . . . . . . 11 ⊢ × = ( ·𝑠 ‘𝑆)
58	56, 57	eqtr4di 2813	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = × )
59		eqidd 2761	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → 𝐴 = 𝐴)
60		iffalse 4491	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝐵, 𝐶) = 𝐶)
61	58, 59, 60	oveq123d 7436	. . . . . . . . 9 ⊢ (¬ 𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = (𝐴 × 𝐶))
62		iffalse 4491	. . . . . . . . 9 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)) = (𝐴 × 𝐶))
63	61, 62	eqtr4d 2798	. . . . . . . 8 ⊢ (¬ 𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
64	54, 63	pm2.61i 184	. . . . . . 7 ⊢ (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶))
65	19	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝑅 ∈ 𝑉)
66	20	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝑆 ∈ 𝑊)
67		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝑘 ∈ 2o)
68		fvprif 17669	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
69	65, 66, 67, 68	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
70	69	fveq2d 6884	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ( ·𝑠 ‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘)) = ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)))
71		eqidd 2761	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝐴 = 𝐴)
72	3	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝐵 ∈ 𝑋)
73	4	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝐶 ∈ 𝑌)
74		fvprif 17669	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩}‘𝑘) = if(𝑘 = ∅, 𝐵, 𝐶))
75	72, 73, 67, 74	syl3anc 1398	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩}‘𝑘) = if(𝑘 = ∅, 𝐵, 𝐶))
76	70, 71, 75	oveq123d 7436	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (𝐴( ·𝑠 ‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩}‘𝑘)) = (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)))
77		xpsvsca.6	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ 𝑋)
78	77	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (𝐴 · 𝐵) ∈ 𝑋)
79		xpsvsca.7	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ 𝑌)
80	79	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (𝐴 × 𝐶) ∈ 𝑌)
81		fvprif 17669	. . . . . . . 8 ⊢ (((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐴 × 𝐶) ∈ 𝑌 ∧ 𝑘 ∈ 2o) → ({⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩}‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
82	78, 80, 67, 81	syl3anc 1398	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ({⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩}‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
83	64, 76, 82	3eqtr4a 2821	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (𝐴( ·𝑠 ‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩}‘𝑘)) = ({⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩}‘𝑘))
84	83	mpteq2dva 5198	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 2o ↦ (𝐴( ·𝑠 ‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩}‘𝑘))) = (𝑘 ∈ 2o ↦ ({⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩}‘𝑘)))
85		eqid 2760	. . . . . 6 ⊢ (Base‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) = (Base‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}))
86	30	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ V)
87		2on 8473	. . . . . . 7 ⊢ 2o ∈ On
88	87	a1i 11	. . . . . 6 ⊢ (𝜑 → 2o ∈ On)
89		fnpr2o 17665	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o)
90	19, 20, 89	syl2anc 596	. . . . . 6 ⊢ (𝜑 → {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o)
91	15, 24	eleqtrd 2862	. . . . . 6 ⊢ (𝜑 → {⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩} ∈ (Base‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})))
92	22, 85, 37, 36, 86, 88, 90, 1, 91	prdsvscaval 17586	. . . . 5 ⊢ (𝜑 → (𝐴( ·𝑠 ‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})){⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩}) = (𝑘 ∈ 2o ↦ (𝐴( ·𝑠 ‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩}‘𝑘))))
93		fnpr2o 17665	. . . . . . 7 ⊢ (((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐴 × 𝐶) ∈ 𝑌) → {⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩} Fn 2o)
94	77, 79, 93	syl2anc 596	. . . . . 6 ⊢ (𝜑 → {⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩} Fn 2o)
95		dffn5 6938	. . . . . 6 ⊢ ({⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩} Fn 2o ↔ {⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩} = (𝑘 ∈ 2o ↦ ({⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩}‘𝑘)))
96	94, 95	sylib 221	. . . . 5 ⊢ (𝜑 → {⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩} = (𝑘 ∈ 2o ↦ ({⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩}‘𝑘)))
97	84, 92, 96	3eqtr4d 2805	. . . 4 ⊢ (𝜑 → (𝐴( ·𝑠 ‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})){⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩}) = {⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩})
98	97	fveq2d 6884	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘(𝐴( ·𝑠 ‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})){⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩})) = (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩}))
99		df-ov 7418	. . . . 5 ⊢ ((𝐴 · 𝐵)(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})(𝐴 × 𝐶)) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)
100	5	xpsfval 17674	. . . . . 6 ⊢ (((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐴 × 𝐶) ∈ 𝑌) → ((𝐴 · 𝐵)(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})(𝐴 × 𝐶)) = {⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩})
101	77, 79, 100	syl2anc 596	. . . . 5 ⊢ (𝜑 → ((𝐴 · 𝐵)(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})(𝐴 × 𝐶)) = {⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩})
102	99, 101	eqtr3id 2809	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩) = {⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩})
103	77, 79	opelxpd 5694	. . . . 5 ⊢ (𝜑 → ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩ ∈ (𝑋 × 𝑌))
104		f1ocnvfv 7281	. . . . 5 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) ∧ ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩ ∈ (𝑋 × 𝑌)) → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩) = {⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩} → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩}) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩))
105	10, 103, 104	sylancr 599	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩) = {⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩} → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩}) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩))
106	102, 105	mpd 16	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩}) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)
107	98, 106	eqtrd 2795	. 2 ⊢ (𝜑 → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘(𝐴( ·𝑠 ‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})){⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩})) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)
108	41, 45, 107	3eqtr3d 2803	1 ⊢ (𝜑 → (𝐴 ∙ ⟨𝐵, 𝐶⟩) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570  ⊤wtru 1571   ∈ wcel 2145  Vcvv 3450  ∅c0 4279  ifcif 4482  {cpr 4586  ⟨cop 4590   ↦ cmpt 5186   × cxp 5653  ^◡ccnv 5654  ran crn 5656  Oncon0 6358   Fn wfn 6529  ⟶wf 6530  –onto→wfo 6532  –1-1-onto→wf1o 6533  ‘cfv 6534  (class class class)co 7415   ∈ cmpo 7417  1_oc1o 8452  2_oc2o 8453  Basecbs 17323  Scalarcsca 17367   ·_𝑠 cvsca 17368  X_scprds 17552   ×_s cxps 17614

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7738  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7372  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8282  df-wrecs 8313  df-recs 8362  df-rdg 8401  df-1o 8459  df-2o 8460  df-er 8700  df-map 8832  df-ixp 8909  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-sup 9416  df-inf 9417  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11489  df-neg 11490  df-nn 12280  df-2 12349  df-3 12350  df-4 12351  df-5 12352  df-6 12353  df-7 12354  df-8 12355  df-9 12356  df-n0 12551  df-z 12638  df-dec 12759  df-uz 12910  df-fz 13584  df-struct 17261  df-slot 17296  df-ndx 17308  df-base 17324  df-plusg 17377  df-mulr 17378  df-sca 17380  df-vsca 17381  df-ip 17382  df-tset 17383  df-ple 17384  df-ds 17386  df-hom 17388  df-cco 17389  df-prds 17554  df-imas 17616  df-xps 17618

This theorem is used by: (None)