Theorem xpsaddlem 16838

Description: Lemma for xpsadd 16839 and xpsmul 16840. (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 7138	. . . . 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 16831	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴𝐹𝐵) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
6	2, 3, 5	syl2anc 587	. . . . 5 ⊢ (𝜑 → (𝐴𝐹𝐵) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
7	1, 6	syl5eqr 2847	. . . 4 ⊢ (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
8	2, 3	opelxpd 5557	. . . . 5 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
9	4	xpsff1o2 16834	. . . . . . 7 ⊢ 𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹
10		f1of 6590	. . . . . . 7 ⊢ (𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹 → 𝐹:(𝑋 × 𝑌)⟶ran 𝐹)
11	9, 10	ax-mp 5	. . . . . 6 ⊢ 𝐹:(𝑋 × 𝑌)⟶ran 𝐹
12	11	ffvelrni 6827	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ ran 𝐹)
13	8, 12	syl 17	. . . 4 ⊢ (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) ∈ ran 𝐹)
14	7, 13	eqeltrrd 2891	. . 3 ⊢ (𝜑 → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ ran 𝐹)
15		df-ov 7138	. . . . 5 ⊢ (𝐶𝐹𝐷) = (𝐹‘⟨𝐶, 𝐷⟩)
16		xpsadd.5	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
17		xpsadd.6	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
18	4	xpsfval 16831	. . . . . 6 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → (𝐶𝐹𝐷) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})
19	16, 17, 18	syl2anc 587	. . . . 5 ⊢ (𝜑 → (𝐶𝐹𝐷) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})
20	15, 19	syl5eqr 2847	. . . 4 ⊢ (𝜑 → (𝐹‘⟨𝐶, 𝐷⟩) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})
21	16, 17	opelxpd 5557	. . . . 5 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
22	11	ffvelrni 6827	. . . . 5 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌) → (𝐹‘⟨𝐶, 𝐷⟩) ∈ ran 𝐹)
23	21, 22	syl 17	. . . 4 ⊢ (𝜑 → (𝐹‘⟨𝐶, 𝐷⟩) ∈ ran 𝐹)
24	20, 23	eqeltrrd 2891	. . 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 1460	. 2 ⊢ (𝜑 → ((◡𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) ∙ (◡𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = (◡𝐹‘({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})))
27		f1ocnvfv 7013	. . . . 5 ⊢ ((𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌)) → ((𝐹‘⟨𝐴, 𝐵⟩) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (◡𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) = ⟨𝐴, 𝐵⟩))
28	9, 8, 27	sylancr 590	. . . 4 ⊢ (𝜑 → ((𝐹‘⟨𝐴, 𝐵⟩) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (◡𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) = ⟨𝐴, 𝐵⟩))
29	7, 28	mpd 15	. . 3 ⊢ (𝜑 → (◡𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) = ⟨𝐴, 𝐵⟩)
30		f1ocnvfv 7013	. . . . 5 ⊢ ((𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹 ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌)) → ((𝐹‘⟨𝐶, 𝐷⟩) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} → (◡𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = ⟨𝐶, 𝐷⟩))
31	9, 21, 30	sylancr 590	. . . 4 ⊢ (𝜑 → ((𝐹‘⟨𝐶, 𝐷⟩) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} → (◡𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = ⟨𝐶, 𝐷⟩))
32	20, 31	mpd 15	. . 3 ⊢ (𝜑 → (◡𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = ⟨𝐶, 𝐷⟩)
33	29, 32	oveq12d 7153	. 2 ⊢ (𝜑 → ((◡𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) ∙ (◡𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = (⟨𝐴, 𝐵⟩ ∙ ⟨𝐶, 𝐷⟩))
34		iftrue 4431	. . . . . . . . . . . 12 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑅)
35	34	fveq2d 6649	. . . . . . . . . . 11 ⊢ (𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = (𝐸‘𝑅))
36		xpsaddlem.m	. . . . . . . . . . 11 ⊢ · = (𝐸‘𝑅)
37	35, 36	eqtr4di 2851	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = · )
38		iftrue 4431	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐴)
39		iftrue 4431	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐶, 𝐷) = 𝐶)
40	37, 38, 39	oveq123d 7156	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = (𝐴 · 𝐶))
41		iftrue 4431	. . . . . . . . 9 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)) = (𝐴 · 𝐶))
42	40, 41	eqtr4d 2836	. . . . . . . 8 ⊢ (𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
43		iffalse 4434	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑆)
44	43	fveq2d 6649	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = (𝐸‘𝑆))
45		xpsaddlem.n	. . . . . . . . . . 11 ⊢ × = (𝐸‘𝑆)
46	44, 45	eqtr4di 2851	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = × )
47		iffalse 4434	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐵)
48		iffalse 4434	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝐶, 𝐷) = 𝐷)
49	46, 47, 48	oveq123d 7156	. . . . . . . . 9 ⊢ (¬ 𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = (𝐵 × 𝐷))
50		iffalse 4434	. . . . . . . . 9 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)) = (𝐵 × 𝐷))
51	49, 50	eqtr4d 2836	. . . . . . . 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 185	. . . . . . 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 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝑅 ∈ 𝑉)
55		xpsval.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
56	55	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝑆 ∈ 𝑊)
57		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝑘 ∈ 2o)
58		fvprif 16826	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
59	54, 56, 57, 58	syl3anc 1368	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
60	59	fveq2d 6649	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (𝐸‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘)) = (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)))
61	2	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝐴 ∈ 𝑋)
62	3	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝐵 ∈ 𝑌)
63		fvprif 16826	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘) = if(𝑘 = ∅, 𝐴, 𝐵))
64	61, 62, 57, 63	syl3anc 1368	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘) = if(𝑘 = ∅, 𝐴, 𝐵))
65	16	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝐶 ∈ 𝑋)
66	17	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝐷 ∈ 𝑌)
67		fvprif 16826	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘) = if(𝑘 = ∅, 𝐶, 𝐷))
68	65, 66, 57, 67	syl3anc 1368	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘) = if(𝑘 = ∅, 𝐶, 𝐷))
69	60, 64, 68	oveq123d 7156	. . . . . . 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 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (𝐴 · 𝐶) ∈ 𝑋)
72		xpsadd.8	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 × 𝐷) ∈ 𝑌)
73	72	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (𝐵 × 𝐷) ∈ 𝑌)
74		fvprif 16826	. . . . . . . 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 1368	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
76	52, 69, 75	3eqtr4a 2859	. . . . . 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 5125	. . . . 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 16822	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o)
79	53, 55, 78	syl2anc 587	. . . . . 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 2798	. . . . . . . 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 16836	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 = (Base‘𝑈))
86	14, 85	eleqtrd 2892	. . . . . 6 ⊢ (𝜑 → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ (Base‘𝑈))
87	24, 85	eleqtrd 2892	. . . . . 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 1368	. . . . 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 16822	. . . . . . 7 ⊢ (((𝐴 · 𝐶) ∈ 𝑋 ∧ (𝐵 × 𝐷) ∈ 𝑌) → {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩} Fn 2o)
91	70, 72, 90	syl2anc 587	. . . . . 6 ⊢ (𝜑 → {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩} Fn 2o)
92		dffn5 6699	. . . . . 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 2843	. . . 4 ⊢ (𝜑 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩})
95	94	fveq2d 6649	. . 3 ⊢ (𝜑 → (◡𝐹‘({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = (◡𝐹‘{⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}))
96		df-ov 7138	. . . . 5 ⊢ ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = (𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
97	4	xpsfval 16831	. . . . . 6 ⊢ (((𝐴 · 𝐶) ∈ 𝑋 ∧ (𝐵 × 𝐷) ∈ 𝑌) → ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩})
98	70, 72, 97	syl2anc 587	. . . . 5 ⊢ (𝜑 → ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩})
99	96, 98	syl5eqr 2847	. . . 4 ⊢ (𝜑 → (𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩})
100	70, 72	opelxpd 5557	. . . . 5 ⊢ (𝜑 → ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩ ∈ (𝑋 × 𝑌))
101		f1ocnvfv 7013	. . . . 5 ⊢ ((𝐹:(𝑋 × 𝑌)–1-1-onto→ran 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 590	. . . 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 15	. . 3 ⊢ (𝜑 → (◡𝐹‘{⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
104	95, 103	eqtrd 2833	. 2 ⊢ (𝜑 → (◡𝐹‘({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
105	26, 33, 104	3eqtr3d 2841	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∙ ⟨𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∅c0 4243  ifcif 4425  {cpr 4527  ⟨cop 4531   ↦ cmpt 5110   × cxp 5517  ^◡ccnv 5518  ran crn 5520   Fn wfn 6319  ⟶wf 6320  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  1_oc1o 8078  2_oc2o 8079  Basecbs 16475  Scalarcsca 16560  X_scprds 16711   ×_s cxps 16771

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12886  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-hom 16581  df-cco 16582  df-prds 16713

This theorem is referenced by:  xpsadd  16839  xpsmul  16840