Theorem xpsaddlem 17603

Description: Lemma for xpsadd 17604 and xpsmul 17605. (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 7399	. . . . 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 17596	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴𝐹𝐵) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
6	2, 3, 5	syl2anc 593	. . . . 5 ⊢ (𝜑 → (𝐴𝐹𝐵) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
7	1, 6	eqtr3id 2811	. . . 4 ⊢ (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
8	2, 3	opelxpd 5686	. . . . 5 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
9	4	xpsff1o2 17599	. . . . . . 7 ⊢ 𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹
10		f1of 6806	. . . . . . 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	ffvelcdmi 7064	. . . . 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 2863	. . 3 ⊢ (𝜑 → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ ran 𝐹)
15		df-ov 7399	. . . . 5 ⊢ (𝐶𝐹𝐷) = (𝐹‘⟨𝐶, 𝐷⟩)
16		xpsadd.5	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
17		xpsadd.6	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
18	4	xpsfval 17596	. . . . . 6 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → (𝐶𝐹𝐷) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})
19	16, 17, 18	syl2anc 593	. . . . 5 ⊢ (𝜑 → (𝐶𝐹𝐷) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})
20	15, 19	eqtr3id 2811	. . . 4 ⊢ (𝜑 → (𝐹‘⟨𝐶, 𝐷⟩) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})
21	16, 17	opelxpd 5686	. . . . 5 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
22	11	ffvelcdmi 7064	. . . . 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 2863	. . 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 1484	. 2 ⊢ (𝜑 → ((◡𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) ∙ (◡𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = (◡𝐹‘({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})))
27		f1ocnvfv 7262	. . . . 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 596	. . . 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 7262	. . . . 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 596	. . . 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 7414	. 2 ⊢ (𝜑 → ((◡𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) ∙ (◡𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = (⟨𝐴, 𝐵⟩ ∙ ⟨𝐶, 𝐷⟩))
34		iftrue 4486	. . . . . . . . . . . 12 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑅)
35	34	fveq2d 6871	. . . . . . . . . . 11 ⊢ (𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = (𝐸‘𝑅))
36		xpsaddlem.m	. . . . . . . . . . 11 ⊢ · = (𝐸‘𝑅)
37	35, 36	eqtr4di 2815	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = · )
38		iftrue 4486	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐴)
39		iftrue 4486	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐶, 𝐷) = 𝐶)
40	37, 38, 39	oveq123d 7417	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = (𝐴 · 𝐶))
41		iftrue 4486	. . . . . . . . 9 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)) = (𝐴 · 𝐶))
42	40, 41	eqtr4d 2800	. . . . . . . 8 ⊢ (𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
43		iffalse 4489	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑆)
44	43	fveq2d 6871	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = (𝐸‘𝑆))
45		xpsaddlem.n	. . . . . . . . . . 11 ⊢ × = (𝐸‘𝑆)
46	44, 45	eqtr4di 2815	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = × )
47		iffalse 4489	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐵)
48		iffalse 4489	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝐶, 𝐷) = 𝐷)
49	46, 47, 48	oveq123d 7417	. . . . . . . . 9 ⊢ (¬ 𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = (𝐵 × 𝐷))
50		iffalse 4489	. . . . . . . . 9 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)) = (𝐵 × 𝐷))
51	49, 50	eqtr4d 2800	. . . . . . . 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 183	. . . . . . 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 17591	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
59	54, 56, 57, 58	syl3anc 1390	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
60	59	fveq2d 6871	. . . . . . . 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 17591	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘) = if(𝑘 = ∅, 𝐴, 𝐵))
64	61, 62, 57, 63	syl3anc 1390	. . . . . . . 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 17591	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘) = if(𝑘 = ∅, 𝐶, 𝐷))
68	65, 66, 57, 67	syl3anc 1390	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘) = if(𝑘 = ∅, 𝐶, 𝐷))
69	60, 64, 68	oveq123d 7417	. . . . . . 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 17591	. . . . . . . 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 1390	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
76	52, 69, 75	3eqtr4a 2823	. . . . . 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 5193	. . . . 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 17587	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o)
79	53, 55, 78	syl2anc 593	. . . . . 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 2762	. . . . . . . 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 17601	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 = (Base‘𝑈))
86	14, 85	eleqtrd 2864	. . . . . 6 ⊢ (𝜑 → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ (Base‘𝑈))
87	24, 85	eleqtrd 2864	. . . . . 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 1390	. . . . 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 17587	. . . . . . 7 ⊢ (((𝐴 · 𝐶) ∈ 𝑋 ∧ (𝐵 × 𝐷) ∈ 𝑌) → {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩} Fn 2o)
91	70, 72, 90	syl2anc 593	. . . . . 6 ⊢ (𝜑 → {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩} Fn 2o)
92		dffn5 6925	. . . . . 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 220	. . . . 5 ⊢ (𝜑 → {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩} = (𝑘 ∈ 2o ↦ ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}‘𝑘)))
94	77, 89, 93	3eqtr4d 2807	. . . 4 ⊢ (𝜑 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩})
95	94	fveq2d 6871	. . 3 ⊢ (𝜑 → (◡𝐹‘({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = (◡𝐹‘{⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}))
96		df-ov 7399	. . . . 5 ⊢ ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = (𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
97	4	xpsfval 17596	. . . . . 6 ⊢ (((𝐴 · 𝐶) ∈ 𝑋 ∧ (𝐵 × 𝐷) ∈ 𝑌) → ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩})
98	70, 72, 97	syl2anc 593	. . . . 5 ⊢ (𝜑 → ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩})
99	96, 98	eqtr3id 2811	. . . 4 ⊢ (𝜑 → (𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩})
100	70, 72	opelxpd 5686	. . . . 5 ⊢ (𝜑 → ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩ ∈ (𝑋 × 𝑌))
101		f1ocnvfv 7262	. . . . 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 596	. . . 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 2797	. 2 ⊢ (𝜑 → (◡𝐹‘({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
105	26, 33, 104	3eqtr3d 2805	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∙ ⟨𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∅c0 4285  ifcif 4480  {cpr 4584  ⟨cop 4588   ↦ cmpt 5181   × cxp 5645  ^◡ccnv 5646  ran crn 5648   Fn wfn 6516  ⟶wf 6517  –1-1-onto→wf1o 6520  ‘cfv 6521  (class class class)co 7396   ∈ cmpo 7398  1_oc1o 8430  2_oc2o 8431  Basecbs 17245  Scalarcsca 17289  X_scprds 17474   ×_s cxps 17536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8678  df-map 8810  df-ixp 8880  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-sup 9388  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-nn 12211  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12482  df-z 12569  df-dec 12689  df-uz 12840  df-fz 13513  df-struct 17183  df-slot 17218  df-ndx 17230  df-base 17246  df-plusg 17299  df-mulr 17300  df-sca 17302  df-vsca 17303  df-ip 17304  df-tset 17305  df-ple 17306  df-ds 17308  df-hom 17310  df-cco 17311  df-prds 17476

This theorem is referenced by:  xpsadd  17604  xpsmul  17605