xpcco - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > xpcco		Structured version Visualization version GIF version

Theorem xpcco 18084

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 18083	. 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 5650	. . 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 7374	. . . . 5 ⊢ ((2^nd ‘𝑥)𝐾𝑦) ∈ V
14		fvex 6830	. . . . 5 ⊢ (𝐾‘𝑥) ∈ V
15	13, 14	mpoex 8006	. . . 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 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝑥 = ⟨𝑋, 𝑌⟩)
20	19	fveq2d 6821	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (2^nd ‘𝑥) = (2^nd ‘⟨𝑋, 𝑌⟩))
21		op2ndg 7929	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
22	8, 9, 21	syl2anc 584	. . . . . . . 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 2766	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (2^nd ‘𝑥) = 𝑌)
25		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝑦 = 𝑍)
26	24, 25	oveq12d 7359	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → ((2^nd ‘𝑥)𝐾𝑦) = (𝑌𝐾𝑍))
27	18, 26	eleqtrrd 2834	. . . 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 6821	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (𝐾‘𝑥) = (𝐾‘⟨𝑋, 𝑌⟩))
31		df-ov 7344	. . . . . . 7 ⊢ (𝑋𝐾𝑌) = (𝐾‘⟨𝑋, 𝑌⟩)
32	30, 31	eqtr4di 2784	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (𝐾‘𝑥) = (𝑋𝐾𝑌))
33	29, 32	eleqtrrd 2834	. . . . 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 5399	. . . . 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 6821	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (1^st ‘𝑥) = (1^st ‘⟨𝑋, 𝑌⟩))
38		op1stg 7928	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
39	8, 9, 38	syl2anc 584	. . . . . . . . . . . 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 2766	. . . . . . . . . 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 6821	. . . . . . . 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 6821	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (1^st ‘(2^nd ‘𝑥)) = (1^st ‘𝑌))
46	43, 45	opeq12d 4828	. . . . . . 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 777	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑦 = 𝑍)
48	47	fveq2d 6821	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (1^st ‘𝑦) = (1^st ‘𝑍))
49	46, 48	oveq12d 7359	. . . . . 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 770	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺)
51	50	fveq2d 6821	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (1^st ‘𝑔) = (1^st ‘𝐺))
52		simprr 772	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
53	52	fveq2d 6821	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (1^st ‘𝑓) = (1^st ‘𝐹))
54	49, 51, 53	oveq123d 7362	. . . . 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 6821	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2^nd ‘(1^st ‘𝑥)) = (2^nd ‘𝑋))
56	44	fveq2d 6821	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2^nd ‘(2^nd ‘𝑥)) = (2^nd ‘𝑌))
57	55, 56	opeq12d 4828	. . . . . . 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 6821	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2^nd ‘𝑦) = (2^nd ‘𝑍))
59	57, 58	oveq12d 7359	. . . . . 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 6821	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2^nd ‘𝑔) = (2^nd ‘𝐺))
61	52	fveq2d 6821	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2^nd ‘𝑓) = (2^nd ‘𝐹))
62	59, 60, 61	oveq123d 7362	. . . . 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 4828	. . . 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 7499	. . 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 7498	. 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 1541 ∈ wcel 2111 Vcvv 3436 ⟨cop 4577 × cxp 5609 ‘cfv 6476 (class class class)co 7341 ∈ cmpo 7343 1^st c1st 7914 2^nd c2nd 7915 Basecbs 17115 Hom chom 17167 compcco 17168 ×_c cxpc 18069

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-er 8617 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-z 12464 df-dec 12584 df-uz 12728 df-fz 13403 df-struct 17053 df-slot 17088 df-ndx 17100 df-base 17116 df-hom 17180 df-cco 17181 df-xpc 18073

This theorem is referenced by: xpcco1st 18085 xpcco2nd 18086 xpcco2 18088 xpccatid 18089 swapfcoa 49313

W3C validator