Theorem xpsaddlem 16235

Description: Lemma for xpsadd 16236 and xpsmul 16237. (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 6653	. . . . 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 16227	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴𝐹𝐵) = ◡({𝐴} +𝑐 {𝐵}))
6	2, 3, 5	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝐴𝐹𝐵) = ◡({𝐴} +𝑐 {𝐵}))
7	1, 6	syl5eqr 2670	. . . 4 ⊢ (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) = ◡({𝐴} +𝑐 {𝐵}))
8		opelxpi 5148	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
9	2, 3, 8	syl2anc 693	. . . . 5 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
10	4	xpsff1o2 16231	. . . . . . 7 ⊢ 𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹
11		f1of 6137	. . . . . . 7 ⊢ (𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹 → 𝐹:(𝑋 × 𝑌)⟶ran 𝐹)
12	10, 11	ax-mp 5	. . . . . 6 ⊢ 𝐹:(𝑋 × 𝑌)⟶ran 𝐹
13	12	ffvelrni 6358	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ ran 𝐹)
14	9, 13	syl 17	. . . 4 ⊢ (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) ∈ ran 𝐹)
15	7, 14	eqeltrrd 2702	. . 3 ⊢ (𝜑 → ◡({𝐴} +𝑐 {𝐵}) ∈ ran 𝐹)
16		df-ov 6653	. . . . 5 ⊢ (𝐶𝐹𝐷) = (𝐹‘⟨𝐶, 𝐷⟩)
17		xpsadd.5	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
18		xpsadd.6	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
19	4	xpsfval 16227	. . . . . 6 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → (𝐶𝐹𝐷) = ◡({𝐶} +𝑐 {𝐷}))
20	17, 18, 19	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝐶𝐹𝐷) = ◡({𝐶} +𝑐 {𝐷}))
21	16, 20	syl5eqr 2670	. . . 4 ⊢ (𝜑 → (𝐹‘⟨𝐶, 𝐷⟩) = ◡({𝐶} +𝑐 {𝐷}))
22		opelxpi 5148	. . . . . 6 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
23	17, 18, 22	syl2anc 693	. . . . 5 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
24	12	ffvelrni 6358	. . . . 5 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌) → (𝐹‘⟨𝐶, 𝐷⟩) ∈ ran 𝐹)
25	23, 24	syl 17	. . . 4 ⊢ (𝜑 → (𝐹‘⟨𝐶, 𝐷⟩) ∈ ran 𝐹)
26	21, 25	eqeltrrd 2702	. . 3 ⊢ (𝜑 → ◡({𝐶} +𝑐 {𝐷}) ∈ ran 𝐹)
27		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}))))
28	15, 26, 27	mpd3an23 1426	. 2 ⊢ (𝜑 → ((◡𝐹‘◡({𝐴} +𝑐 {𝐵})) ∙ (◡𝐹‘◡({𝐶} +𝑐 {𝐷}))) = (◡𝐹‘(◡({𝐴} +𝑐 {𝐵})(𝐸‘𝑈)◡({𝐶} +𝑐 {𝐷}))))
29		f1ocnvfv 6534	. . . . 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⟩))
30	10, 9, 29	sylancr 695	. . . 4 ⊢ (𝜑 → ((𝐹‘⟨𝐴, 𝐵⟩) = ◡({𝐴} +𝑐 {𝐵}) → (◡𝐹‘◡({𝐴} +𝑐 {𝐵})) = ⟨𝐴, 𝐵⟩))
31	7, 30	mpd 15	. . 3 ⊢ (𝜑 → (◡𝐹‘◡({𝐴} +𝑐 {𝐵})) = ⟨𝐴, 𝐵⟩)
32		f1ocnvfv 6534	. . . . 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⟩))
33	10, 23, 32	sylancr 695	. . . 4 ⊢ (𝜑 → ((𝐹‘⟨𝐶, 𝐷⟩) = ◡({𝐶} +𝑐 {𝐷}) → (◡𝐹‘◡({𝐶} +𝑐 {𝐷})) = ⟨𝐶, 𝐷⟩))
34	21, 33	mpd 15	. . 3 ⊢ (𝜑 → (◡𝐹‘◡({𝐶} +𝑐 {𝐷})) = ⟨𝐶, 𝐷⟩)
35	31, 34	oveq12d 6668	. 2 ⊢ (𝜑 → ((◡𝐹‘◡({𝐴} +𝑐 {𝐵})) ∙ (◡𝐹‘◡({𝐶} +𝑐 {𝐷}))) = (⟨𝐴, 𝐵⟩ ∙ ⟨𝐶, 𝐷⟩))
36		xpsval.1	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
37		xpsval.2	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
38		xpscfn 16219	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜)
39	36, 37, 38	syl2anc 693	. . . . . 6 ⊢ (𝜑 → ◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜)
40		xpsval.t	. . . . . . . 8 ⊢ 𝑇 = (𝑅 ×s 𝑆)
41		xpsval.x	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝑅)
42		xpsval.y	. . . . . . . 8 ⊢ 𝑌 = (Base‘𝑆)
43		eqid 2622	. . . . . . . 8 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
44		xpsaddlem.u	. . . . . . . 8 ⊢ 𝑈 = ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))
45	40, 41, 42, 36, 37, 4, 43, 44	xpslem 16233	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 = (Base‘𝑈))
46	15, 45	eleqtrd 2703	. . . . . 6 ⊢ (𝜑 → ◡({𝐴} +𝑐 {𝐵}) ∈ (Base‘𝑈))
47	26, 45	eleqtrd 2703	. . . . . 6 ⊢ (𝜑 → ◡({𝐶} +𝑐 {𝐷}) ∈ (Base‘𝑈))
48		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))))
49	39, 46, 47, 48	syl3anc 1326	. . . . 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))))
50		xpsadd.7	. . . . . . . 8 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ 𝑋)
51		xpsadd.8	. . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐷) ∈ 𝑌)
52		xpscfn 16219	. . . . . . . 8 ⊢ (((𝐴 · 𝐶) ∈ 𝑋 ∧ (𝐵 × 𝐷) ∈ 𝑌) → ◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}) Fn 2𝑜)
53	50, 51, 52	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → ◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}) Fn 2𝑜)
54		dffn5 6241	. . . . . . 7 ⊢ (◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}) Fn 2𝑜 ↔ ◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}) = (𝑘 ∈ 2𝑜 ↦ (◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)})‘𝑘)))
55	53, 54	sylib 208	. . . . . 6 ⊢ (𝜑 → ◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}) = (𝑘 ∈ 2𝑜 ↦ (◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)})‘𝑘)))
56		iftrue 4092	. . . . . . . . . . . . 13 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑅)
57	56	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = (𝐸‘𝑅))
58		xpsaddlem.m	. . . . . . . . . . . 12 ⊢ · = (𝐸‘𝑅)
59	57, 58	syl6eqr 2674	. . . . . . . . . . 11 ⊢ (𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = · )
60		iftrue 4092	. . . . . . . . . . 11 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐴)
61		iftrue 4092	. . . . . . . . . . 11 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐶, 𝐷) = 𝐶)
62	59, 60, 61	oveq123d 6671	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = (𝐴 · 𝐶))
63		iftrue 4092	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)) = (𝐴 · 𝐶))
64	62, 63	eqtr4d 2659	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
65		iffalse 4095	. . . . . . . . . . . . 13 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑆)
66	65	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = (𝐸‘𝑆))
67		xpsaddlem.n	. . . . . . . . . . . 12 ⊢ × = (𝐸‘𝑆)
68	66, 67	syl6eqr 2674	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = × )
69		iffalse 4095	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐵)
70		iffalse 4095	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, 𝐶, 𝐷) = 𝐷)
71	68, 69, 70	oveq123d 6671	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = (𝐵 × 𝐷))
72		iffalse 4095	. . . . . . . . . 10 ⊢ (¬ 𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)) = (𝐵 × 𝐷))
73	71, 72	eqtr4d 2659	. . . . . . . . 9 ⊢ (¬ 𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
74	64, 73	pm2.61i 176	. . . . . . . 8 ⊢ (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷))
75	36	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → 𝑅 ∈ 𝑉)
76	37	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → 𝑆 ∈ 𝑊)
77		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → 𝑘 ∈ 2𝑜)
78		xpscfv 16222	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝑘 ∈ 2𝑜) → (◡({𝑅} +𝑐 {𝑆})‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
79	75, 76, 77, 78	syl3anc 1326	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → (◡({𝑅} +𝑐 {𝑆})‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
80	79	fveq2d 6195	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → (𝐸‘(◡({𝑅} +𝑐 {𝑆})‘𝑘)) = (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)))
81	2	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → 𝐴 ∈ 𝑋)
82	3	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → 𝐵 ∈ 𝑌)
83		xpscfv 16222	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑘 ∈ 2𝑜) → (◡({𝐴} +𝑐 {𝐵})‘𝑘) = if(𝑘 = ∅, 𝐴, 𝐵))
84	81, 82, 77, 83	syl3anc 1326	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → (◡({𝐴} +𝑐 {𝐵})‘𝑘) = if(𝑘 = ∅, 𝐴, 𝐵))
85	17	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → 𝐶 ∈ 𝑋)
86	18	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → 𝐷 ∈ 𝑌)
87		xpscfv 16222	. . . . . . . . . 10 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌 ∧ 𝑘 ∈ 2𝑜) → (◡({𝐶} +𝑐 {𝐷})‘𝑘) = if(𝑘 = ∅, 𝐶, 𝐷))
88	85, 86, 77, 87	syl3anc 1326	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → (◡({𝐶} +𝑐 {𝐷})‘𝑘) = if(𝑘 = ∅, 𝐶, 𝐷))
89	80, 84, 88	oveq123d 6671	. . . . . . . 8 ⊢ ((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)))
90	50	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → (𝐴 · 𝐶) ∈ 𝑋)
91	51	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → (𝐵 × 𝐷) ∈ 𝑌)
92		xpscfv 16222	. . . . . . . . 9 ⊢ (((𝐴 · 𝐶) ∈ 𝑋 ∧ (𝐵 × 𝐷) ∈ 𝑌 ∧ 𝑘 ∈ 2𝑜) → (◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)})‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
93	90, 91, 77, 92	syl3anc 1326	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → (◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)})‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
94	74, 89, 93	3eqtr4a 2682	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 2𝑜) → ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(𝐸‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘)) = (◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)})‘𝑘))
95	94	mpteq2dva 4744	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 2𝑜 ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(𝐸‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘))) = (𝑘 ∈ 2𝑜 ↦ (◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)})‘𝑘)))
96	55, 95	eqtr4d 2659	. . . . 5 ⊢ (𝜑 → ◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}) = (𝑘 ∈ 2𝑜 ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(𝐸‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘))))
97	49, 96	eqtr4d 2659	. . . 4 ⊢ (𝜑 → (◡({𝐴} +𝑐 {𝐵})(𝐸‘𝑈)◡({𝐶} +𝑐 {𝐷})) = ◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}))
98	97	fveq2d 6195	. . 3 ⊢ (𝜑 → (◡𝐹‘(◡({𝐴} +𝑐 {𝐵})(𝐸‘𝑈)◡({𝐶} +𝑐 {𝐷}))) = (◡𝐹‘◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)})))
99		df-ov 6653	. . . . 5 ⊢ ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = (𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
100	4	xpsfval 16227	. . . . . 6 ⊢ (((𝐴 · 𝐶) ∈ 𝑋 ∧ (𝐵 × 𝐷) ∈ 𝑌) → ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = ◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}))
101	50, 51, 100	syl2anc 693	. . . . 5 ⊢ (𝜑 → ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = ◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}))
102	99, 101	syl5eqr 2670	. . . 4 ⊢ (𝜑 → (𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩) = ◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}))
103		opelxpi 5148	. . . . . 6 ⊢ (((𝐴 · 𝐶) ∈ 𝑋 ∧ (𝐵 × 𝐷) ∈ 𝑌) → ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩ ∈ (𝑋 × 𝑌))
104	50, 51, 103	syl2anc 693	. . . . 5 ⊢ (𝜑 → ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩ ∈ (𝑋 × 𝑌))
105		f1ocnvfv 6534	. . . . 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)⟩))
106	10, 104, 105	sylancr 695	. . . 4 ⊢ (𝜑 → ((𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩) = ◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)}) → (◡𝐹‘◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)})) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩))
107	102, 106	mpd 15	. . 3 ⊢ (𝜑 → (◡𝐹‘◡({(𝐴 · 𝐶)} +𝑐 {(𝐵 × 𝐷)})) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
108	98, 107	eqtrd 2656	. 2 ⊢ (𝜑 → (◡𝐹‘(◡({𝐴} +𝑐 {𝐵})(𝐸‘𝑈)◡({𝐶} +𝑐 {𝐷}))) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
109	28, 35, 108	3eqtr3d 2664	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∙ ⟨𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∅c0 3915  ifcif 4086  {csn 4177  ⟨cop 4183   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113  ran crn 5115   Fn wfn 5883  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  2_𝑜c2o 7554   +_𝑐 ccda 8989  Basecbs 15857  Scalarcsca 15944  X_scprds 16106   ×_s cxps 16166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-prds 16108

This theorem is referenced by:  xpsadd  16236  xpsmul  16237