Theorem xpsvsca 17523

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 7412	. . . . 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 2733	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
6	5	xpsfval 17512	. . . . . 6 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌) → (𝐵(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})𝐶) = {⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩})
7	3, 4, 6	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝐵(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})𝐶) = {⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩})
8	2, 7	eqtr3id 2787	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐵, 𝐶⟩) = {⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩})
9	3, 4	opelxpd 5716	. . . . 5 ⊢ (𝜑 → ⟨𝐵, 𝐶⟩ ∈ (𝑋 × 𝑌))
10	5	xpsff1o2 17515	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
11		f1of 6834	. . . . . . 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 7086	. . . . 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 17	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐵, 𝐶⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
15	8, 14	eqeltrrd 2835	. . 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 2733	. . . . 5 ⊢ (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}) = (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})
23	16, 17, 18, 19, 20, 5, 21, 22	xpsval 17516	. . . 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 17517	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = (Base‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})))
25		f1ocnv 6846	. . . . . 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 13	. . . . 5 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})–1-1-onto→(𝑋 × 𝑌))
27		f1ofo 6841	. . . . 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 17	. . . 4 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})–onto→(𝑋 × 𝑌))
29		ovexd 7444	. . . 4 ⊢ (𝜑 → (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}) ∈ V)
30	21	fvexi 6906	. . . . . . 7 ⊢ 𝐺 ∈ V
31	30	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐺 ∈ V)
32		prex 5433	. . . . . . 7 ⊢ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} ∈ V
33	32	a1i 11	. . . . . 6 ⊢ (⊤ → {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} ∈ V)
34	22, 31, 33	prdssca 17402	. . . . 5 ⊢ (⊤ → 𝐺 = (Scalar‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})))
35	34	mptru 1549	. . . 4 ⊢ 𝐺 = (Scalar‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}))
36		xpsvsca.k	. . . 4 ⊢ 𝐾 = (Base‘𝐺)
37		eqid 2733	. . . 4 ⊢ ( ·𝑠 ‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) = ( ·𝑠 ‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}))
38		xpsvsca.p	. . . 4 ⊢ ∙ = ( ·𝑠 ‘𝑇)
39	26	f1ovscpbl 17472	. . . 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 17484	. . 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 1464	. 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 7276	. . . . 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 588	. . . 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 15	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩}) = ⟨𝐵, 𝐶⟩)
45	44	oveq2d 7425	. 2 ⊢ (𝜑 → (𝐴 ∙ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩})) = (𝐴 ∙ ⟨𝐵, 𝐶⟩))
46		iftrue 4535	. . . . . . . . . . . 12 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑅)
47	46	fveq2d 6896	. . . . . . . . . . 11 ⊢ (𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = ( ·𝑠 ‘𝑅))
48		xpsvsca.m	. . . . . . . . . . 11 ⊢ · = ( ·𝑠 ‘𝑅)
49	47, 48	eqtr4di 2791	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = · )
50		eqidd 2734	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → 𝐴 = 𝐴)
51		iftrue 4535	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐵, 𝐶) = 𝐵)
52	49, 50, 51	oveq123d 7430	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = (𝐴 · 𝐵))
53		iftrue 4535	. . . . . . . . 9 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)) = (𝐴 · 𝐵))
54	52, 53	eqtr4d 2776	. . . . . . . 8 ⊢ (𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
55		iffalse 4538	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑆)
56	55	fveq2d 6896	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = ( ·𝑠 ‘𝑆))
57		xpsvsca.n	. . . . . . . . . . 11 ⊢ × = ( ·𝑠 ‘𝑆)
58	56, 57	eqtr4di 2791	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)) = × )
59		eqidd 2734	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → 𝐴 = 𝐴)
60		iffalse 4538	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝐵, 𝐶) = 𝐶)
61	58, 59, 60	oveq123d 7430	. . . . . . . . 9 ⊢ (¬ 𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = (𝐴 × 𝐶))
62		iffalse 4538	. . . . . . . . 9 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)) = (𝐴 × 𝐶))
63	61, 62	eqtr4d 2776	. . . . . . . 8 ⊢ (¬ 𝑘 = ∅ → (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
64	54, 63	pm2.61i 182	. . . . . . 7 ⊢ (𝐴( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐵, 𝐶)) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶))
65	19	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝑅 ∈ 𝑉)
66	20	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝑆 ∈ 𝑊)
67		simpr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝑘 ∈ 2o)
68		fvprif 17507	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
69	65, 66, 67, 68	syl3anc 1372	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
70	69	fveq2d 6896	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ( ·𝑠 ‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘)) = ( ·𝑠 ‘if(𝑘 = ∅, 𝑅, 𝑆)))
71		eqidd 2734	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝐴 = 𝐴)
72	3	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝐵 ∈ 𝑋)
73	4	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝐶 ∈ 𝑌)
74		fvprif 17507	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩}‘𝑘) = if(𝑘 = ∅, 𝐵, 𝐶))
75	72, 73, 67, 74	syl3anc 1372	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩}‘𝑘) = if(𝑘 = ∅, 𝐵, 𝐶))
76	70, 71, 75	oveq123d 7430	. . . . . . 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 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (𝐴 · 𝐵) ∈ 𝑋)
79		xpsvsca.7	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ 𝑌)
80	79	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (𝐴 × 𝐶) ∈ 𝑌)
81		fvprif 17507	. . . . . . . 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 1372	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ({⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩}‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐵), (𝐴 × 𝐶)))
83	64, 76, 82	3eqtr4a 2799	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (𝐴( ·𝑠 ‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩}‘𝑘)) = ({⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩}‘𝑘))
84	83	mpteq2dva 5249	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 2o ↦ (𝐴( ·𝑠 ‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩}‘𝑘))) = (𝑘 ∈ 2o ↦ ({⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩}‘𝑘)))
85		eqid 2733	. . . . . 6 ⊢ (Base‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) = (Base‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}))
86	30	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ V)
87		2on 8480	. . . . . . 7 ⊢ 2o ∈ On
88	87	a1i 11	. . . . . 6 ⊢ (𝜑 → 2o ∈ On)
89		fnpr2o 17503	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o)
90	19, 20, 89	syl2anc 585	. . . . . 6 ⊢ (𝜑 → {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o)
91	15, 24	eleqtrd 2836	. . . . . 6 ⊢ (𝜑 → {⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩} ∈ (Base‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})))
92	22, 85, 37, 36, 86, 88, 90, 1, 91	prdsvscaval 17425	. . . . 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 17503	. . . . . . 7 ⊢ (((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐴 × 𝐶) ∈ 𝑌) → {⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩} Fn 2o)
94	77, 79, 93	syl2anc 585	. . . . . 6 ⊢ (𝜑 → {⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩} Fn 2o)
95		dffn5 6951	. . . . . 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 217	. . . . 5 ⊢ (𝜑 → {⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩} = (𝑘 ∈ 2o ↦ ({⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩}‘𝑘)))
97	84, 92, 96	3eqtr4d 2783	. . . 4 ⊢ (𝜑 → (𝐴( ·𝑠 ‘(𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})){⟨∅, 𝐵⟩, ⟨1o, 𝐶⟩}) = {⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩})
98	97	fveq2d 6896	. . 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 7412	. . . . 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 17512	. . . . . 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 585	. . . . 5 ⊢ (𝜑 → ((𝐴 · 𝐵)(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})(𝐴 × 𝐶)) = {⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩})
102	99, 101	eqtr3id 2787	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩) = {⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩})
103	77, 79	opelxpd 5716	. . . . 5 ⊢ (𝜑 → ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩ ∈ (𝑋 × 𝑌))
104		f1ocnvfv 7276	. . . . 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 588	. . . 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 15	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, (𝐴 · 𝐵)⟩, ⟨1o, (𝐴 × 𝐶)⟩}) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)
107	98, 106	eqtrd 2773	. 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 2781	1 ⊢ (𝜑 → (𝐴 ∙ ⟨𝐵, 𝐶⟩) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1542  ⊤wtru 1543   ∈ wcel 2107  Vcvv 3475  ∅c0 4323  ifcif 4529  {cpr 4631  ⟨cop 4635   ↦ cmpt 5232   × cxp 5675  ^◡ccnv 5676  ran crn 5678  Oncon0 6365   Fn wfn 6539  ⟶wf 6540  –onto→wfo 6542  –1-1-onto→wf1o 6543  ‘cfv 6544  (class class class)co 7409   ∈ cmpo 7411  1_oc1o 8459  2_oc2o 8460  Basecbs 17144  Scalarcsca 17200   ·_𝑠 cvsca 17201  X_scprds 17391   ×_s cxps 17452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-fz 13485  df-struct 17080  df-slot 17115  df-ndx 17127  df-base 17145  df-plusg 17210  df-mulr 17211  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ple 17217  df-ds 17219  df-hom 17221  df-cco 17222  df-prds 17393  df-imas 17454  df-xps 17456

This theorem is referenced by: (None)