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Theorem xpsaddlem 17516

Description: Lemma for xpsadd 17517 and xpsmul 17518. (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⟩, ⟨1_o, 𝑦⟩})
xpsaddlem.u	⊢ 𝑈 = ((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})
xpsaddlem.1	⊢ ((𝜑 ∧ {⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} ∈ ran 𝐹 ∧ {⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩} ∈ ran 𝐹) → ((^◡𝐹‘{⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}) ∙ (^◡𝐹‘{⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩})) = (^◡𝐹‘({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩})))
xpsaddlem.2	⊢ (({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩} Fn 2_o ∧ {⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} ∈ (Base‘𝑈) ∧ {⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩} ∈ (Base‘𝑈)) → ({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}) = (𝑘 ∈ 2_o ↦ (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(𝐸‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 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 7409	. . . . 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⟩, ⟨1_o, 𝑦⟩})
5	4	xpsfval 17509	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴𝐹𝐵) = {⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩})
6	2, 3, 5	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝐴𝐹𝐵) = {⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩})
7	1, 6	eqtr3id 2787	. . . 4 ⊢ (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) = {⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩})
8	2, 3	opelxpd 5714	. . . . 5 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
9	4	xpsff1o2 17512	. . . . . . 7 ⊢ 𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹
10		f1of 6831	. . . . . . 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 7083	. . . . 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 2835	. . 3 ⊢ (𝜑 → {⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} ∈ ran 𝐹)
15		df-ov 7409	. . . . 5 ⊢ (𝐶𝐹𝐷) = (𝐹‘⟨𝐶, 𝐷⟩)
16		xpsadd.5	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
17		xpsadd.6	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
18	4	xpsfval 17509	. . . . . 6 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → (𝐶𝐹𝐷) = {⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩})
19	16, 17, 18	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝐶𝐹𝐷) = {⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩})
20	15, 19	eqtr3id 2787	. . . 4 ⊢ (𝜑 → (𝐹‘⟨𝐶, 𝐷⟩) = {⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩})
21	16, 17	opelxpd 5714	. . . . 5 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
22	11	ffvelcdmi 7083	. . . . 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 2835	. . 3 ⊢ (𝜑 → {⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩} ∈ ran 𝐹)
25		xpsaddlem.1	. . 3 ⊢ ((𝜑 ∧ {⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} ∈ ran 𝐹 ∧ {⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩} ∈ ran 𝐹) → ((^◡𝐹‘{⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}) ∙ (^◡𝐹‘{⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩})) = (^◡𝐹‘({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩})))
26	14, 24, 25	mpd3an23 1464	. 2 ⊢ (𝜑 → ((^◡𝐹‘{⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}) ∙ (^◡𝐹‘{⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩})) = (^◡𝐹‘({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩})))
27		f1ocnvfv 7273	. . . . 5 ⊢ ((𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌)) → ((𝐹‘⟨𝐴, 𝐵⟩) = {⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} → (^◡𝐹‘{⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}) = ⟨𝐴, 𝐵⟩))
28	9, 8, 27	sylancr 588	. . . 4 ⊢ (𝜑 → ((𝐹‘⟨𝐴, 𝐵⟩) = {⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} → (^◡𝐹‘{⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}) = ⟨𝐴, 𝐵⟩))
29	7, 28	mpd 15	. . 3 ⊢ (𝜑 → (^◡𝐹‘{⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}) = ⟨𝐴, 𝐵⟩)
30		f1ocnvfv 7273	. . . . 5 ⊢ ((𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹 ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌)) → ((𝐹‘⟨𝐶, 𝐷⟩) = {⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩} → (^◡𝐹‘{⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}) = ⟨𝐶, 𝐷⟩))
31	9, 21, 30	sylancr 588	. . . 4 ⊢ (𝜑 → ((𝐹‘⟨𝐶, 𝐷⟩) = {⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩} → (^◡𝐹‘{⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}) = ⟨𝐶, 𝐷⟩))
32	20, 31	mpd 15	. . 3 ⊢ (𝜑 → (^◡𝐹‘{⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}) = ⟨𝐶, 𝐷⟩)
33	29, 32	oveq12d 7424	. 2 ⊢ (𝜑 → ((^◡𝐹‘{⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}) ∙ (^◡𝐹‘{⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩})) = (⟨𝐴, 𝐵⟩ ∙ ⟨𝐶, 𝐷⟩))
34		iftrue 4534	. . . . . . . . . . . 12 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑅)
35	34	fveq2d 6893	. . . . . . . . . . 11 ⊢ (𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = (𝐸‘𝑅))
36		xpsaddlem.m	. . . . . . . . . . 11 ⊢ · = (𝐸‘𝑅)
37	35, 36	eqtr4di 2791	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = · )
38		iftrue 4534	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐴)
39		iftrue 4534	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐶, 𝐷) = 𝐶)
40	37, 38, 39	oveq123d 7427	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = (𝐴 · 𝐶))
41		iftrue 4534	. . . . . . . . 9 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)) = (𝐴 · 𝐶))
42	40, 41	eqtr4d 2776	. . . . . . . 8 ⊢ (𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
43		iffalse 4537	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑆)
44	43	fveq2d 6893	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = (𝐸‘𝑆))
45		xpsaddlem.n	. . . . . . . . . . 11 ⊢ × = (𝐸‘𝑆)
46	44, 45	eqtr4di 2791	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = × )
47		iffalse 4537	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐵)
48		iffalse 4537	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝐶, 𝐷) = 𝐷)
49	46, 47, 48	oveq123d 7427	. . . . . . . . 9 ⊢ (¬ 𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = (𝐵 × 𝐷))
50		iffalse 4537	. . . . . . . . 9 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)) = (𝐵 × 𝐷))
51	49, 50	eqtr4d 2776	. . . . . . . 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 182	. . . . . . 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 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2_o) → 𝑅 ∈ 𝑉)
55		xpsval.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
56	55	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2_o) → 𝑆 ∈ 𝑊)
57		simpr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2_o) → 𝑘 ∈ 2_o)
58		fvprif 17504	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝑘 ∈ 2_o) → ({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
59	54, 56, 57, 58	syl3anc 1372	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2_o) → ({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
60	59	fveq2d 6893	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2_o) → (𝐸‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘)) = (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)))
61	2	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2_o) → 𝐴 ∈ 𝑋)
62	3	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2_o) → 𝐵 ∈ 𝑌)
63		fvprif 17504	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑘 ∈ 2_o) → ({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘) = if(𝑘 = ∅, 𝐴, 𝐵))
64	61, 62, 57, 63	syl3anc 1372	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2_o) → ({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘) = if(𝑘 = ∅, 𝐴, 𝐵))
65	16	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2_o) → 𝐶 ∈ 𝑋)
66	17	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2_o) → 𝐷 ∈ 𝑌)
67		fvprif 17504	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌 ∧ 𝑘 ∈ 2_o) → ({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘) = if(𝑘 = ∅, 𝐶, 𝐷))
68	65, 66, 57, 67	syl3anc 1372	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2_o) → ({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘) = if(𝑘 = ∅, 𝐶, 𝐷))
69	60, 64, 68	oveq123d 7427	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 2_o) → (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(𝐸‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 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 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2_o) → (𝐴 · 𝐶) ∈ 𝑋)
72		xpsadd.8	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 × 𝐷) ∈ 𝑌)
73	72	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2_o) → (𝐵 × 𝐷) ∈ 𝑌)
74		fvprif 17504	. . . . . . . 8 ⊢ (((𝐴 · 𝐶) ∈ 𝑋 ∧ (𝐵 × 𝐷) ∈ 𝑌 ∧ 𝑘 ∈ 2_o) → ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩}‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
75	71, 73, 57, 74	syl3anc 1372	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 2_o) → ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩}‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
76	52, 69, 75	3eqtr4a 2799	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 2_o) → (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(𝐸‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘)) = ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩}‘𝑘))
77	76	mpteq2dva 5248	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 2_o ↦ (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(𝐸‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘))) = (𝑘 ∈ 2_o ↦ ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩}‘𝑘)))
78		fnpr2o 17500	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩} Fn 2_o)
79	53, 55, 78	syl2anc 585	. . . . . 6 ⊢ (𝜑 → {⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩} Fn 2_o)
80		xpsval.t	. . . . . . . 8 ⊢ 𝑇 = (𝑅 ×_s 𝑆)
81		xpsval.x	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝑅)
82		xpsval.y	. . . . . . . 8 ⊢ 𝑌 = (Base‘𝑆)
83		eqid 2733	. . . . . . . 8 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
84		xpsaddlem.u	. . . . . . . 8 ⊢ 𝑈 = ((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})
85	80, 81, 82, 53, 55, 4, 83, 84	xpsrnbas 17514	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 = (Base‘𝑈))
86	14, 85	eleqtrd 2836	. . . . . 6 ⊢ (𝜑 → {⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} ∈ (Base‘𝑈))
87	24, 85	eleqtrd 2836	. . . . . 6 ⊢ (𝜑 → {⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩} ∈ (Base‘𝑈))
88		xpsaddlem.2	. . . . . 6 ⊢ (({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩} Fn 2_o ∧ {⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} ∈ (Base‘𝑈) ∧ {⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩} ∈ (Base‘𝑈)) → ({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}) = (𝑘 ∈ 2_o ↦ (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(𝐸‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘))))
89	79, 86, 87, 88	syl3anc 1372	. . . . 5 ⊢ (𝜑 → ({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}) = (𝑘 ∈ 2_o ↦ (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(𝐸‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘))))
90		fnpr2o 17500	. . . . . . 7 ⊢ (((𝐴 · 𝐶) ∈ 𝑋 ∧ (𝐵 × 𝐷) ∈ 𝑌) → {⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩} Fn 2_o)
91	70, 72, 90	syl2anc 585	. . . . . 6 ⊢ (𝜑 → {⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩} Fn 2_o)
92		dffn5 6948	. . . . . 6 ⊢ ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩} Fn 2_o ↔ {⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩} = (𝑘 ∈ 2_o ↦ ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩}‘𝑘)))
93	91, 92	sylib 217	. . . . 5 ⊢ (𝜑 → {⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩} = (𝑘 ∈ 2_o ↦ ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩}‘𝑘)))
94	77, 89, 93	3eqtr4d 2783	. . . 4 ⊢ (𝜑 → ({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩})
95	94	fveq2d 6893	. . 3 ⊢ (𝜑 → (^◡𝐹‘({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩})) = (^◡𝐹‘{⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩}))
96		df-ov 7409	. . . . 5 ⊢ ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = (𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
97	4	xpsfval 17509	. . . . . 6 ⊢ (((𝐴 · 𝐶) ∈ 𝑋 ∧ (𝐵 × 𝐷) ∈ 𝑌) → ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩})
98	70, 72, 97	syl2anc 585	. . . . 5 ⊢ (𝜑 → ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩})
99	96, 98	eqtr3id 2787	. . . 4 ⊢ (𝜑 → (𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩})
100	70, 72	opelxpd 5714	. . . . 5 ⊢ (𝜑 → ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩ ∈ (𝑋 × 𝑌))
101		f1ocnvfv 7273	. . . . 5 ⊢ ((𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹 ∧ ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩ ∈ (𝑋 × 𝑌)) → ((𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩} → (^◡𝐹‘{⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩}) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩))
102	9, 100, 101	sylancr 588	. . . 4 ⊢ (𝜑 → ((𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩} → (^◡𝐹‘{⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩}) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩))
103	99, 102	mpd 15	. . 3 ⊢ (𝜑 → (^◡𝐹‘{⟨∅, (𝐴 · 𝐶)⟩, ⟨1_o, (𝐵 × 𝐷)⟩}) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
104	95, 103	eqtrd 2773	. 2 ⊢ (𝜑 → (^◡𝐹‘({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩})) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
105	26, 33, 104	3eqtr3d 2781	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∙ ⟨𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∅c0 4322 ifcif 4528 {cpr 4630 ⟨cop 4634 ↦ cmpt 5231 × cxp 5674 ^◡ccnv 5675 ran crn 5677 Fn wfn 6536 ⟶wf 6537 –1-1-onto→wf1o 6540 ‘cfv 6541 (class class class)co 7406 ∈ cmpo 7408 1_oc1o 8456 2_oc2o 8457 Basecbs 17141 Scalarcsca 17197 X_scprds 17388 ×_s cxps 17449

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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-2o 8464 df-er 8700 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-struct 17077 df-slot 17112 df-ndx 17124 df-base 17142 df-plusg 17207 df-mulr 17208 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-hom 17218 df-cco 17219 df-prds 17390

This theorem is referenced by: xpsadd 17517 xpsmul 17518

W3C validator