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Theorem xpcco 18149

Description: Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)

**Hypotheses**
Ref	Expression
xpccofval.t	⊢ 𝑇 = (𝐶 ×_c 𝐷)
xpccofval.b	⊢ 𝐵 = (Base‘𝑇)
xpccofval.k	⊢ 𝐾 = (Hom ‘𝑇)
xpccofval.o1	⊢ · = (comp‘𝐶)
xpccofval.o2	⊢ ∙ = (comp‘𝐷)
xpccofval.o	⊢ 𝑂 = (comp‘𝑇)
xpcco.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
xpcco.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
xpcco.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
xpcco.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐾𝑌))
xpcco.g	⊢ (𝜑 → 𝐺 ∈ (𝑌𝐾𝑍))

**Assertion**
Ref	Expression
xpcco	⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = ⟨((1^st ‘𝐺)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩ · (1^st ‘𝑍))(1^st ‘𝐹)), ((2^nd ‘𝐺)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩ ∙ (2^nd ‘𝑍))(2^nd ‘𝐹))⟩)

Proof of Theorem xpcco Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpccofval.t	. . 3 ⊢ 𝑇 = (𝐶 ×_c 𝐷)
2		xpccofval.b	. . 3 ⊢ 𝐵 = (Base‘𝑇)
3		xpccofval.k	. . 3 ⊢ 𝐾 = (Hom ‘𝑇)
4		xpccofval.o1	. . 3 ⊢ · = (comp‘𝐶)
5		xpccofval.o2	. . 3 ⊢ ∙ = (comp‘𝐷)
6		xpccofval.o	. . 3 ⊢ 𝑂 = (comp‘𝑇)
7	1, 2, 3, 4, 5, 6	xpccofval 18148	. 2 ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩))
8		xpcco.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		xpcco.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10	8, 9	opelxpd 5670	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
11		xpcco.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
12	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝑋, 𝑌⟩) → 𝑍 ∈ 𝐵)
13		ovex 7400	. . . . 5 ⊢ ((2^nd ‘𝑥)𝐾𝑦) ∈ V
14		fvex 6853	. . . . 5 ⊢ (𝐾‘𝑥) ∈ V
15	13, 14	mpoex 8032	. . . 4 ⊢ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩) ∈ V
16	15	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩) ∈ V)
17		xpcco.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐾𝑍))
18	17	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝐺 ∈ (𝑌𝐾𝑍))
19		simprl 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝑥 = ⟨𝑋, 𝑌⟩)
20	19	fveq2d 6844	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (2^nd ‘𝑥) = (2^nd ‘⟨𝑋, 𝑌⟩))
21		op2ndg 7955	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
22	8, 9, 21	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
23	22	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
24	20, 23	eqtrd 2771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (2^nd ‘𝑥) = 𝑌)
25		simprr 773	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝑦 = 𝑍)
26	24, 25	oveq12d 7385	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → ((2^nd ‘𝑥)𝐾𝑦) = (𝑌𝐾𝑍))
27	18, 26	eleqtrrd 2839	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝐺 ∈ ((2^nd ‘𝑥)𝐾𝑦))
28		xpcco.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐾𝑌))
29	28	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝐹 ∈ (𝑋𝐾𝑌))
30	19	fveq2d 6844	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (𝐾‘𝑥) = (𝐾‘⟨𝑋, 𝑌⟩))
31		df-ov 7370	. . . . . . 7 ⊢ (𝑋𝐾𝑌) = (𝐾‘⟨𝑋, 𝑌⟩)
32	30, 31	eqtr4di 2789	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (𝐾‘𝑥) = (𝑋𝐾𝑌))
33	29, 32	eleqtrrd 2839	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝐹 ∈ (𝐾‘𝑥))
34	33	adantr 480	. . . 4 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ 𝑔 = 𝐺) → 𝐹 ∈ (𝐾‘𝑥))
35		opex 5416	. . . . 5 ⊢ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩ ∈ V
36	35	a1i 11	. . . 4 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩ ∈ V)
37	19	fveq2d 6844	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (1^st ‘𝑥) = (1^st ‘⟨𝑋, 𝑌⟩))
38		op1stg 7954	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
39	8, 9, 38	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
40	39	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
41	37, 40	eqtrd 2771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (1^st ‘𝑥) = 𝑋)
42	41	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (1^st ‘𝑥) = 𝑋)
43	42	fveq2d 6844	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (1^st ‘(1^st ‘𝑥)) = (1^st ‘𝑋))
44	24	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2^nd ‘𝑥) = 𝑌)
45	44	fveq2d 6844	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (1^st ‘(2^nd ‘𝑥)) = (1^st ‘𝑌))
46	43, 45	opeq12d 4824	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ = ⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩)
47		simplrr 778	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑦 = 𝑍)
48	47	fveq2d 6844	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (1^st ‘𝑦) = (1^st ‘𝑍))
49	46, 48	oveq12d 7385	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦)) = (⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩ · (1^st ‘𝑍)))
50		simprl 771	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺)
51	50	fveq2d 6844	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (1^st ‘𝑔) = (1^st ‘𝐺))
52		simprr 773	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
53	52	fveq2d 6844	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (1^st ‘𝑓) = (1^st ‘𝐹))
54	49, 51, 53	oveq123d 7388	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)) = ((1^st ‘𝐺)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩ · (1^st ‘𝑍))(1^st ‘𝐹)))
55	42	fveq2d 6844	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2^nd ‘(1^st ‘𝑥)) = (2^nd ‘𝑋))
56	44	fveq2d 6844	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2^nd ‘(2^nd ‘𝑥)) = (2^nd ‘𝑌))
57	55, 56	opeq12d 4824	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ = ⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩)
58	47	fveq2d 6844	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2^nd ‘𝑦) = (2^nd ‘𝑍))
59	57, 58	oveq12d 7385	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦)) = (⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩ ∙ (2^nd ‘𝑍)))
60	50	fveq2d 6844	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2^nd ‘𝑔) = (2^nd ‘𝐺))
61	52	fveq2d 6844	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2^nd ‘𝑓) = (2^nd ‘𝐹))
62	59, 60, 61	oveq123d 7388	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓)) = ((2^nd ‘𝐺)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩ ∙ (2^nd ‘𝑍))(2^nd ‘𝐹)))
63	54, 62	opeq12d 4824	. . . 4 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩ = ⟨((1^st ‘𝐺)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩ · (1^st ‘𝑍))(1^st ‘𝐹)), ((2^nd ‘𝐺)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩ ∙ (2^nd ‘𝑍))(2^nd ‘𝐹))⟩)
64	27, 34, 36, 63	ovmpodv2 7525	. . 3 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → ((⟨𝑋, 𝑌⟩𝑂𝑍) = (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩) → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = ⟨((1^st ‘𝐺)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩ · (1^st ‘𝑍))(1^st ‘𝐹)), ((2^nd ‘𝐺)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩ ∙ (2^nd ‘𝑍))(2^nd ‘𝐹))⟩))
65	10, 12, 16, 64	ovmpodv 7524	. 2 ⊢ (𝜑 → (𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)) → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = ⟨((1^st ‘𝐺)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩ · (1^st ‘𝑍))(1^st ‘𝐹)), ((2^nd ‘𝐺)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩ ∙ (2^nd ‘𝑍))(2^nd ‘𝐹))⟩))
66	7, 65	mpi 20	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = ⟨((1^st ‘𝐺)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩ · (1^st ‘𝑍))(1^st ‘𝐹)), ((2^nd ‘𝐺)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩ ∙ (2^nd ‘𝑍))(2^nd ‘𝐹))⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3429 ⟨cop 4573 × cxp 5629 ‘cfv 6498 (class class class)co 7367 ∈ cmpo 7369 1^st c1st 7940 2^nd c2nd 7941 Basecbs 17179 Hom chom 17231 compcco 17232 ×_c cxpc 18134

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-fz 13462 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-hom 17244 df-cco 17245 df-xpc 18138

This theorem is referenced by: xpcco1st 18150 xpcco2nd 18151 xpcco2 18153 xpccatid 18154 swapfcoa 49756

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