Theorem xpsaddlem 16621

Description: Lemma for xpsadd 16622 and xpsmul 16623. (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} +𝑐 {𝑦}))
xpsaddlem.u	⊢ 𝑈 = ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))
xpsaddlem.1	⊢ ((𝜑 ∧ ◡({𝐴} +𝑐 {𝐵}) ∈ ran 𝐹 ∧ ◡({𝐶} +𝑐 {𝐷}) ∈ ran 𝐹) → ((◡𝐹‘◡({𝐴} +𝑐 {𝐵})) ∙ (◡𝐹‘◡({𝐶} +𝑐 {𝐷}))) = (◡𝐹‘(◡({𝐴} +𝑐 {𝐵})(𝐸‘𝑈)◡({𝐶} +𝑐 {𝐷}))))
xpsaddlem.2	⊢ ((◡({𝑅} +𝑐 {𝑆}) Fn 2o ∧ ◡({𝐴} +𝑐 {𝐵}) ∈ (Base‘𝑈) ∧ ◡({𝐶} +𝑐 {𝐷}) ∈ (Base‘𝑈)) → (◡({𝐴} +𝑐 {𝐵})(𝐸‘𝑈)◡({𝐶} +𝑐 {𝐷})) = (𝑘 ∈ 2o ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(𝐸‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {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 6925	. . . . 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} +𝑐 {𝑦}))
5	4	xpsfval 16613	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴𝐹𝐵) = ◡({𝐴} +𝑐 {𝐵}))
6	2, 3, 5	syl2anc 579	. . . . 5 ⊢ (𝜑 → (𝐴𝐹𝐵) = ◡({𝐴} +𝑐 {𝐵}))
7	1, 6	syl5eqr 2827	. . . 4 ⊢ (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) = ◡({𝐴} +𝑐 {𝐵}))
8	2, 3	opelxpd 5393	. . . . 5 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
9	4	xpsff1o2 16617	. . . . . . 7 ⊢ 𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹
10		f1of 6391	. . . . . . 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 6622	. . . . 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 2859	. . 3 ⊢ (𝜑 → ◡({𝐴} +𝑐 {𝐵}) ∈ ran 𝐹)
15		df-ov 6925	. . . . 5 ⊢ (𝐶𝐹𝐷) = (𝐹‘⟨𝐶, 𝐷⟩)
16		xpsadd.5	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
17		xpsadd.6	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
18	4	xpsfval 16613	. . . . . 6 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → (𝐶𝐹𝐷) = ◡({𝐶} +𝑐 {𝐷}))
19	16, 17, 18	syl2anc 579	. . . . 5 ⊢ (𝜑 → (𝐶𝐹𝐷) = ◡({𝐶} +𝑐 {𝐷}))
20	15, 19	syl5eqr 2827	. . . 4 ⊢ (𝜑 → (𝐹‘⟨𝐶, 𝐷⟩) = ◡({𝐶} +𝑐 {𝐷}))
21	16, 17	opelxpd 5393	. . . . 5 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
22	11	ffvelrni 6622	. . . . 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 2859	. . 3 ⊢ (𝜑 → ◡({𝐶} +𝑐 {𝐷}) ∈ ran 𝐹)
25		xpsaddlem.1	. . 3 ⊢ ((𝜑 ∧ ◡({𝐴} +𝑐 {𝐵}) ∈ ran 𝐹 ∧ ◡({𝐶} +𝑐 {𝐷}) ∈ ran 𝐹) → ((◡𝐹‘◡({𝐴} +𝑐 {𝐵})) ∙ (◡𝐹‘◡({𝐶} +𝑐 {𝐷}))) = (◡𝐹‘(◡({𝐴} +𝑐 {𝐵})(𝐸‘𝑈)◡({𝐶} +𝑐 {𝐷}))))
26	14, 24, 25	mpd3an23 1536	. 2 ⊢ (𝜑 → ((◡𝐹‘◡({𝐴} +𝑐 {𝐵})) ∙ (◡𝐹‘◡({𝐶} +𝑐 {𝐷}))) = (◡𝐹‘(◡({𝐴} +𝑐 {𝐵})(𝐸‘𝑈)◡({𝐶} +𝑐 {𝐷}))))
27		f1ocnvfv 6806	. . . . 5 ⊢ ((𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌)) → ((𝐹‘⟨𝐴, 𝐵⟩) = ◡({𝐴} +𝑐 {𝐵}) → (◡𝐹‘◡({𝐴} +𝑐 {𝐵})) = ⟨𝐴, 𝐵⟩))
28	9, 8, 27	sylancr 581	. . . 4 ⊢ (𝜑 → ((𝐹‘⟨𝐴, 𝐵⟩) = ◡({𝐴} +𝑐 {𝐵}) → (◡𝐹‘◡({𝐴} +𝑐 {𝐵})) = ⟨𝐴, 𝐵⟩))
29	7, 28	mpd 15	. . 3 ⊢ (𝜑 → (◡𝐹‘◡({𝐴} +𝑐 {𝐵})) = ⟨𝐴, 𝐵⟩)
30		f1ocnvfv 6806	. . . . 5 ⊢ ((𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹 ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌)) → ((𝐹‘⟨𝐶, 𝐷⟩) = ◡({𝐶} +𝑐 {𝐷}) → (◡𝐹‘◡({𝐶} +𝑐 {𝐷})) = ⟨𝐶, 𝐷⟩))
31	9, 21, 30	sylancr 581	. . . 4 ⊢ (𝜑 → ((𝐹‘⟨𝐶, 𝐷⟩) = ◡({𝐶} +𝑐 {𝐷}) → (◡𝐹‘◡({𝐶} +𝑐 {𝐷})) = ⟨𝐶, 𝐷⟩))
32	20, 31	mpd 15	. . 3 ⊢ (𝜑 → (◡𝐹‘◡({𝐶} +𝑐 {𝐷})) = ⟨𝐶, 𝐷⟩)
33	29, 32	oveq12d 6940	. 2 ⊢ (𝜑 → ((◡𝐹‘◡({𝐴} +𝑐 {𝐵})) ∙ (◡𝐹‘◡({𝐶} +𝑐 {𝐷}))) = (⟨𝐴, 𝐵⟩ ∙ ⟨𝐶, 𝐷⟩))
34		iftrue 4312	. . . . . . . . . . . 12 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑅)
35	34	fveq2d 6450	. . . . . . . . . . 11 ⊢ (𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = (𝐸‘𝑅))
36		xpsaddlem.m	. . . . . . . . . . 11 ⊢ · = (𝐸‘𝑅)
37	35, 36	syl6eqr 2831	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = · )
38		iftrue 4312	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐴)
39		iftrue 4312	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐶, 𝐷) = 𝐶)
40	37, 38, 39	oveq123d 6943	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = (𝐴 · 𝐶))
41		iftrue 4312	. . . . . . . . 9 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)) = (𝐴 · 𝐶))
42	40, 41	eqtr4d 2816	. . . . . . . 8 ⊢ (𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
43		iffalse 4315	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑆)
44	43	fveq2d 6450	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = (𝐸‘𝑆))
45		xpsaddlem.n	. . . . . . . . . . 11 ⊢ × = (𝐸‘𝑆)
46	44, 45	syl6eqr 2831	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = × )
47		iffalse 4315	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐵)
48		iffalse 4315	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝐶, 𝐷) = 𝐷)
49	46, 47, 48	oveq123d 6943	. . . . . . . . 9 ⊢ (¬ 𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = (𝐵 × 𝐷))
50		iffalse 4315	. . . . . . . . 9 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)) = (𝐵 × 𝐷))
51	49, 50	eqtr4d 2816	. . . . . . . 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 177	. . . . . . 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 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝑅 ∈ 𝑉)
55		xpsval.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
56	55	adantr 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝑆 ∈ 𝑊)
57		simpr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝑘 ∈ 2o)
58		xpscfv 16608	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝑘 ∈ 2o) → (◡({𝑅} +𝑐 {𝑆})‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
59	54, 56, 57, 58	syl3anc 1439	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (◡({𝑅} +𝑐 {𝑆})‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
60	59	fveq2d 6450	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (𝐸‘(◡({𝑅} +𝑐 {𝑆})‘𝑘)) = (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)))
61	2	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝐴 ∈ 𝑋)
62	3	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝐵 ∈ 𝑌)
63		xpscfv 16608	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑘 ∈ 2o) → (◡({𝐴} +𝑐 {𝐵})‘𝑘) = if(𝑘 = ∅, 𝐴, 𝐵))
64	61, 62, 57, 63	syl3anc 1439	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (◡({𝐴} +𝑐 {𝐵})‘𝑘) = if(𝑘 = ∅, 𝐴, 𝐵))
65	16	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝐶 ∈ 𝑋)
66	17	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → 𝐷 ∈ 𝑌)
67		xpscfv 16608	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌 ∧ 𝑘 ∈ 2o) → (◡({𝐶} +𝑐 {𝐷})‘𝑘) = if(𝑘 = ∅, 𝐶, 𝐷))
68	65, 66, 57, 67	syl3anc 1439	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (◡({𝐶} +𝑐 {𝐷})‘𝑘) = if(𝑘 = ∅, 𝐶, 𝐷))
69	60, 64, 68	oveq123d 6943	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(𝐸‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {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 474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (𝐴 · 𝐶) ∈ 𝑋)
72		xpsadd.8	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 × 𝐷) ∈ 𝑌)
73	72	adantr 474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (𝐵 × 𝐷) ∈ 𝑌)
74		xpscfv 16608	. . . . . . . 8 ⊢ (((𝐴 · 𝐶) ∈ 𝑋 ∧ (𝐵 × 𝐷) ∈ 𝑌 ∧ 𝑘 ∈ 2o) → (◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)})‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
75	71, 73, 57, 74	syl3anc 1439	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → (◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)})‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
76	52, 69, 75	3eqtr4a 2839	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 2o) → ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(𝐸‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘)) = (◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)})‘𝑘))
77	76	mpteq2dva 4979	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 2o ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(𝐸‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘))) = (𝑘 ∈ 2o ↦ (◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)})‘𝑘)))
78		xpscfn 16605	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ◡({𝑅} +𝑐 {𝑆}) Fn 2o)
79	53, 55, 78	syl2anc 579	. . . . . 6 ⊢ (𝜑 → ◡({𝑅} +𝑐 {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 2777	. . . . . . . 8 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
84		xpsaddlem.u	. . . . . . . 8 ⊢ 𝑈 = ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))
85	80, 81, 82, 53, 55, 4, 83, 84	xpslem 16619	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 = (Base‘𝑈))
86	14, 85	eleqtrd 2860	. . . . . 6 ⊢ (𝜑 → ◡({𝐴} +𝑐 {𝐵}) ∈ (Base‘𝑈))
87	24, 85	eleqtrd 2860	. . . . . 6 ⊢ (𝜑 → ◡({𝐶} +𝑐 {𝐷}) ∈ (Base‘𝑈))
88		xpsaddlem.2	. . . . . 6 ⊢ ((◡({𝑅} +𝑐 {𝑆}) Fn 2o ∧ ◡({𝐴} +𝑐 {𝐵}) ∈ (Base‘𝑈) ∧ ◡({𝐶} +𝑐 {𝐷}) ∈ (Base‘𝑈)) → (◡({𝐴} +𝑐 {𝐵})(𝐸‘𝑈)◡({𝐶} +𝑐 {𝐷})) = (𝑘 ∈ 2o ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(𝐸‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘))))
89	79, 86, 87, 88	syl3anc 1439	. . . . 5 ⊢ (𝜑 → (◡({𝐴} +𝑐 {𝐵})(𝐸‘𝑈)◡({𝐶} +𝑐 {𝐷})) = (𝑘 ∈ 2o ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(𝐸‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘))))
90		xpscfn 16605	. . . . . . 7 ⊢ (((𝐴 · 𝐶) ∈ 𝑋 ∧ (𝐵 × 𝐷) ∈ 𝑌) → ◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}) Fn 2o)
91	70, 72, 90	syl2anc 579	. . . . . 6 ⊢ (𝜑 → ◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}) Fn 2o)
92		dffn5 6501	. . . . . 6 ⊢ (◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}) Fn 2o ↔ ◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}) = (𝑘 ∈ 2o ↦ (◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)})‘𝑘)))
93	91, 92	sylib 210	. . . . 5 ⊢ (𝜑 → ◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}) = (𝑘 ∈ 2o ↦ (◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)})‘𝑘)))
94	77, 89, 93	3eqtr4d 2823	. . . 4 ⊢ (𝜑 → (◡({𝐴} +𝑐 {𝐵})(𝐸‘𝑈)◡({𝐶} +𝑐 {𝐷})) = ◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}))
95	94	fveq2d 6450	. . 3 ⊢ (𝜑 → (◡𝐹‘(◡({𝐴} +𝑐 {𝐵})(𝐸‘𝑈)◡({𝐶} +𝑐 {𝐷}))) = (◡𝐹‘◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)})))
96		df-ov 6925	. . . . 5 ⊢ ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = (𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
97	4	xpsfval 16613	. . . . . 6 ⊢ (((𝐴 · 𝐶) ∈ 𝑋 ∧ (𝐵 × 𝐷) ∈ 𝑌) → ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = ◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}))
98	70, 72, 97	syl2anc 579	. . . . 5 ⊢ (𝜑 → ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = ◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}))
99	96, 98	syl5eqr 2827	. . . 4 ⊢ (𝜑 → (𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩) = ◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}))
100	70, 72	opelxpd 5393	. . . . 5 ⊢ (𝜑 → ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩ ∈ (𝑋 × 𝑌))
101		f1ocnvfv 6806	. . . . 5 ⊢ ((𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹 ∧ ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩ ∈ (𝑋 × 𝑌)) → ((𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩) = ◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}) → (◡𝐹‘◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)})) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩))
102	9, 100, 101	sylancr 581	. . . 4 ⊢ (𝜑 → ((𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩) = ◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}) → (◡𝐹‘◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)})) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩))
103	99, 102	mpd 15	. . 3 ⊢ (𝜑 → (◡𝐹‘◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)})) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
104	95, 103	eqtrd 2813	. 2 ⊢ (𝜑 → (◡𝐹‘(◡({𝐴} +𝑐 {𝐵})(𝐸‘𝑈)◡({𝐶} +𝑐 {𝐷}))) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
105	26, 33, 104	3eqtr3d 2821	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∙ ⟨𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2106  ∅c0 4140  ifcif 4306  {csn 4397  ⟨cop 4403   ↦ cmpt 4965   × cxp 5353  ^◡ccnv 5354  ran crn 5356   Fn wfn 6130  ⟶wf 6131  –1-1-onto→wf1o 6134  ‘cfv 6135  (class class class)co 6922   ↦ cmpt2 6924  2_oc2o 7837   +_𝑐 ccda 9324  Basecbs 16255  Scalarcsca 16341  X_scprds 16492   ×_s cxps 16552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-map 8142  df-ixp 8195  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-sup 8636  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-z 11729  df-dec 11846  df-uz 11993  df-fz 12644  df-struct 16257  df-ndx 16258  df-slot 16259  df-base 16261  df-plusg 16351  df-mulr 16352  df-sca 16354  df-vsca 16355  df-ip 16356  df-tset 16357  df-ple 16358  df-ds 16360  df-hom 16362  df-cco 16363  df-prds 16494

This theorem is referenced by:  xpsadd  16622  xpsmul  16623