Theorem xpsaddlem 17795

Description: Lemma for xpsadd 17796 and xpsmul 17797. (Contributed by Mario Carneiro, 15-Aug-2015.)

Hypotheses

Ref	Expression
xpsval.t	⊢ 𝑇 = (𝑅 ×s 𝑆)
xpsval.x	⊢ 𝑋 = (Base‘𝑅)
xpsval.y	⊢ 𝑌 = (Base‘𝑆)
xpsval.1	⊢ (𝜑 → 𝑅 ∈ 𝑉)
xpsval.2	⊢ (𝜑 → 𝑆 ∈ 𝑊)
xpsadd.3	⊢ (𝜑 → 𝐴 ∈ 𝑋)
xpsadd.4	⊢ (𝜑 → 𝐵 ∈ 𝑌)
xpsadd.5	⊢ (𝜑 → 𝐶 ∈ 𝑋)
xpsadd.6	⊢ (𝜑 → 𝐷 ∈ 𝑌)
xpsadd.7	⊢ (𝜑 → (𝐴 · 𝐶) ∈ 𝑋)
xpsadd.8	⊢ (𝜑 → (𝐵 × 𝐷) ∈ 𝑌)
xpsaddlem.m	⊢ · = (𝐸‘𝑅)
xpsaddlem.n	⊢ × = (𝐸‘𝑆)
xpsaddlem.p	⊢ ∙ = (𝐸‘𝑇)
xpsaddlem.f	⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
xpsaddlem.u	⊢ 𝑈 = ((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})
xpsaddlem.1	⊢ ((𝜑 ∧ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ ran 𝐹 ∧ {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} ∈ ran 𝐹) → ((◡𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) ∙ (◡𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = (◡𝐹‘({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})))
xpsaddlem.2	⊢ (({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o ∧ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ (Base‘𝑈) ∧ {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} ∈ (Base‘𝑈)) → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(𝐸‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))))

Assertion

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

Distinct variable groups:   𝑥,𝑘,𝑦,𝐴   𝐵,𝑘,𝑥,𝑦   𝐶,𝑘,𝑥,𝑦   𝐷,𝑘,𝑥,𝑦   𝑆,𝑘   𝑈,𝑘   𝑥,𝑊   𝜑,𝑘   · ,𝑘,𝑥,𝑦   × ,𝑘,𝑥,𝑦   𝑘,𝑋,𝑥,𝑦   𝑅,𝑘,𝑥   𝑘,𝑌,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝑅(𝑦)   𝑆(𝑥, 𝑦)   ∙ (𝑥, 𝑦, 𝑘)   𝑇(𝑥, 𝑦, 𝑘)   𝑈(𝑥, 𝑦)   𝐸(𝑥, 𝑦, 𝑘)   𝐹(𝑥, 𝑦, 𝑘)   𝑉(𝑥, 𝑦, 𝑘)   𝑊(𝑦, 𝑘)

Proof of Theorem xpsaddlem

Step	Hyp	Ref	Expression
1		df-ov 7454	. . . . 5 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2		xpsadd.3	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
3		xpsadd.4	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑌)
4		xpsaddlem.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
5	4	xpsfval 17788	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴𝐹𝐵) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
6	2, 3, 5	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝐴𝐹𝐵) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
7	1, 6	eqtr3id 2812	. . . 4 ⊢ (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
8	2, 3	opelxpd 5697	. . . . 5 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
9	4	xpsff1o2 17791	. . . . . . 7 ⊢ 𝐹:(𝑋 × 𝑌)⟶1-1ontoran 𝐹
10		f1of 6854	. . . . . . 7 ⊢ (𝐹:(𝑋 × 𝑌)⟶1-1ontoran 𝐹 → 𝐹:(𝑋 × 𝑌)⟶ran 𝐹)
11	9, 10	ax-mp 5	. . . . . 6 ⊢ 𝐹:(𝑋 × 𝑌)⟶ran 𝐹
12	11	ffvelcdmi 7115	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ ran 𝐹)
13	8, 12	syl 18	. . . 4 ⊢ (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) ∈ ran 𝐹)
14	7, 13	eqeltrrd 2865	. . 3 ⊢ (𝜑 → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ ran 𝐹)
15		df-ov 7454	. . . . 5 ⊢ (𝐶𝐹𝐷) = (𝐹‘⟨𝐶, 𝐷⟩)
16		xpsadd.5	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
17		xpsadd.6	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
18	4	xpsfval 17788	. . . . . 6 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → (𝐶𝐹𝐷) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})
19	16, 17, 18	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝐶𝐹𝐷) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})
20	15, 19	eqtr3id 2812	. . . 4 ⊢ (𝜑 → (𝐹‘⟨𝐶, 𝐷⟩) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})
21	16, 17	opelxpd 5697	. . . . 5 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
22	11	ffvelcdmi 7115	. . . . 5 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌) → (𝐹‘⟨𝐶, 𝐷⟩) ∈ ran 𝐹)
23	21, 22	syl 18	. . . 4 ⊢ (𝜑 → (𝐹‘⟨𝐶, 𝐷⟩) ∈ ran 𝐹)
24	20, 23	eqeltrrd 2865	. . 3 ⊢ (𝜑 → {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} ∈ ran 𝐹)
25		xpsaddlem.1	. . 3 ⊢ ((𝜑 ∧ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ ran 𝐹 ∧ {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} ∈ ran 𝐹) → ((◡𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) ∙ (◡𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = (◡𝐹‘({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})))
26	14, 24, 25	mpd3an23 1493	. 2 ⊢ (𝜑 → ((◡𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) ∙ (◡𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = (◡𝐹‘({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})))
27		f1ocnvfv 7317	. . . . 5 ⊢ ((𝐹:(𝑋 × 𝑌)⟶1-1ontoran 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌)) → ((𝐹‘⟨𝐴, 𝐵⟩) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (◡𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) = ⟨𝐴, 𝐵⟩))
28	9, 8, 27	sylancr 599	. . . 4 ⊢ (𝜑 → ((𝐹‘⟨𝐴, 𝐵⟩) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (◡𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) = ⟨𝐴, 𝐵⟩))
29	7, 28	mpd 16	. . 3 ⊢ (𝜑 → (◡𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) = ⟨𝐴, 𝐵⟩)
30		f1ocnvfv 7317	. . . . 5 ⊢ ((𝐹:(𝑋 × 𝑌)⟶1-1ontoran 𝐹 ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌)) → ((𝐹‘⟨𝐶, 𝐷⟩) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} → (◡𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = ⟨𝐶, 𝐷⟩))
31	9, 21, 30	sylancr 599	. . . 4 ⊢ (𝜑 → ((𝐹‘⟨𝐶, 𝐷⟩) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} → (◡𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = ⟨𝐶, 𝐷⟩))
32	20, 31	mpd 16	. . 3 ⊢ (𝜑 → (◡𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = ⟨𝐶, 𝐷⟩)
33	29, 32	oveq12d 7469	. 2 ⊢ (𝜑 → ((◡𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) ∙ (◡𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = (⟨𝐴, 𝐵⟩ ∙ ⟨𝐶, 𝐷⟩))
34		iftrue 4491	. . . . . . . . . . . 12 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑅)
35	34	fveq2d 6920	. . . . . . . . . . 11 ⊢ (𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = (𝐸‘𝑅))
36		xpsaddlem.m	. . . . . . . . . . 11 ⊢ · = (𝐸‘𝑅)
37	35, 36	eqtr4di 2816	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = · )
38		iftrue 4491	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐴)
39		iftrue 4491	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐶, 𝐷) = 𝐶)
40	37, 38, 39	oveq123d 7472	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = (𝐴 · 𝐶))
41		iftrue 4491	. . . . . . . . 9 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)) = (𝐴 · 𝐶))
42	40, 41	eqtr4d 2801	. . . . . . . 8 ⊢ (𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
43		iffalse 4494	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑆)
44	43	fveq2d 6920	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = (𝐸‘𝑆))
45		xpsaddlem.n	. . . . . . . . . . 11 ⊢ × = (𝐸‘𝑆)
46	44, 45	eqtr4di 2816	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = × )
47		iffalse 4494	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐵)
48		iffalse 4494	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝐶, 𝐷) = 𝐷)
49	46, 47, 48	oveq123d 7472	. . . . . . . . 9 ⊢ (¬ 𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = (𝐵 × 𝐷))
50		iffalse 4494	. . . . . . . . 9 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)) = (𝐵 × 𝐷))
51	49, 50	eqtr4d 2801	. . . . . . . 8 ⊢ (¬ 𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
52	42, 51	pm2.61i 184	. . . . . . 7 ⊢ (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷))
53		xpsval.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
54	53	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝑅 ∈ 𝑉)
55		xpsval.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
56	55	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝑆 ∈ 𝑊)
57		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝑘 ∈ 2o)
58		fvprif 17783	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
59	54, 56, 57, 58	syl3anc 1399	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
60	59	fveq2d 6920	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (𝐸‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘)) = (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)))
61	2	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝐴 ∈ 𝑋)
62	3	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝐵 ∈ 𝑌)
63		fvprif 17783	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘) = if(𝑘 = ∅, 𝐴, 𝐵))
64	61, 62, 57, 63	syl3anc 1399	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘) = if(𝑘 = ∅, 𝐴, 𝐵))
65	16	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝐶 ∈ 𝑋)
66	17	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝐷 ∈ 𝑌)
67		fvprif 17783	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘) = if(𝑘 = ∅, 𝐶, 𝐷))
68	65, 66, 57, 67	syl3anc 1399	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘) = if(𝑘 = ∅, 𝐶, 𝐷))
69	60, 64, 68	oveq123d 7472	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(𝐸‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘)) = (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)))
70		xpsadd.7	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ 𝑋)
71	70	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (𝐴 · 𝐶) ∈ 𝑋)
72		xpsadd.8	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 × 𝐷) ∈ 𝑌)
73	72	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (𝐵 × 𝐷) ∈ 𝑌)
74		fvprif 17783	. . . . . . . 8 ⊢ (((𝐴 · 𝐶) ∈ 𝑋 ∧ (𝐵 × 𝐷) ∈ 𝑌 ∧ 𝑘 ∈ 2o) → ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
75	71, 73, 57, 74	syl3anc 1399	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
76	52, 69, 75	3eqtr4a 2824	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(𝐸‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘)) = ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}‘𝑘))
77	76	mpteq2dva 5204	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(𝐸‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))) = (𝑘 ∈ 2o ↦ ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}‘𝑘)))
78		fnpr2o 17779	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o)
79	53, 55, 78	syl2anc 596	. . . . . 6 ⊢ (𝜑 → {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o)
80		xpsval.t	. . . . . . . 8 ⊢ 𝑇 = (𝑅 ×s 𝑆)
81		xpsval.x	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝑅)
82		xpsval.y	. . . . . . . 8 ⊢ 𝑌 = (Base‘𝑆)
83		eqid 2763	. . . . . . . 8 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
84		xpsaddlem.u	. . . . . . . 8 ⊢ 𝑈 = ((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})
85	80, 81, 82, 53, 55, 4, 83, 84	xpsrnbas 17793	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 = (Base‘𝑈))
86	14, 85	eleqtrd 2866	. . . . . 6 ⊢ (𝜑 → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ (Base‘𝑈))
87	24, 85	eleqtrd 2866	. . . . . 6 ⊢ (𝜑 → {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} ∈ (Base‘𝑈))
88		xpsaddlem.2	. . . . . 6 ⊢ (({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o ∧ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ (Base‘𝑈) ∧ {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} ∈ (Base‘𝑈)) → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(𝐸‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))))
89	79, 86, 87, 88	syl3anc 1399	. . . . 5 ⊢ (𝜑 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(𝐸‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))))
90		fnpr2o 17779	. . . . . . 7 ⊢ (((𝐴 · 𝐶) ∈ 𝑋 ∧ (𝐵 × 𝐷) ∈ 𝑌) → {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩} Fn 2o)
91	70, 72, 90	syl2anc 596	. . . . . 6 ⊢ (𝜑 → {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩} Fn 2o)
92		dffn5 6974	. . . . . 6 ⊢ ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩} Fn 2o ↔ {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩} = (𝑘 ∈ 2o ↦ ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}‘𝑘)))
93	91, 92	sylib 221	. . . . 5 ⊢ (𝜑 → {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩} = (𝑘 ∈ 2o ↦ ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}‘𝑘)))
94	77, 89, 93	3eqtr4d 2808	. . . 4 ⊢ (𝜑 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩})
95	94	fveq2d 6920	. . 3 ⊢ (𝜑 → (◡𝐹‘({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = (◡𝐹‘{⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}))
96		df-ov 7454	. . . . 5 ⊢ ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = (𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
97	4	xpsfval 17788	. . . . . 6 ⊢ (((𝐴 · 𝐶) ∈ 𝑋 ∧ (𝐵 × 𝐷) ∈ 𝑌) → ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩})
98	70, 72, 97	syl2anc 596	. . . . 5 ⊢ (𝜑 → ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩})
99	96, 98	eqtr3id 2812	. . . 4 ⊢ (𝜑 → (𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩})
100	70, 72	opelxpd 5697	. . . . 5 ⊢ (𝜑 → ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩ ∈ (𝑋 × 𝑌))
101		f1ocnvfv 7317	. . . . 5 ⊢ ((𝐹:(𝑋 × 𝑌)⟶1-1ontoran 𝐹 ∧ ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩ ∈ (𝑋 × 𝑌)) → ((𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩} → (◡𝐹‘{⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩))
102	9, 100, 101	sylancr 599	. . . 4 ⊢ (𝜑 → ((𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩} → (◡𝐹‘{⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩))
103	99, 102	mpd 16	. . 3 ⊢ (𝜑 → (◡𝐹‘{⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
104	95, 103	eqtrd 2798	. 2 ⊢ (𝜑 → (◡𝐹‘({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
105	26, 33, 104	3eqtr3d 2806	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∙ ⟨𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∅c0 4281  ifcif 4485  {cpr 4590  ⟨cop 4840   ↦ cmpt 5192   × cxp 5656  ^◡ccnv 5657  ran crn 5659   Fn wfn 6561  ⟶wf 6562  ⟶1-1ontowf1o 6565  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  1_oc1o 8507  2_oc2o 8508  Basecbs 17431  Scalarcsca 17475  X_scprds 17666   ×_s cxps 17728

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-er 8755  df-map 8887  df-ixp 8964  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-sup 9471  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-z 12734  df-dec 12855  df-uz 13006  df-fz 13680  df-struct 17369  df-slot 17404  df-ndx 17416  df-base 17432  df-plusg 17486  df-mulr 17487  df-sca 17489  df-vsca 17490  df-ip 17491  df-tset 17492  df-ple 17493  df-ds 17495  df-hom 17497  df-cco 17498  df-prds 17668

This theorem is used by:  xpsadd  17796  xpsmul  17797