xpcpropd - Metamath Proof Explorer

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Theorem xpcpropd 18143

Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same product category. (Contributed by Mario Carneiro, 17-Jan-2017.)

**Hypotheses**
Ref	Expression
xpcpropd.1	⊢ (𝜑 → (Hom_f ‘𝐴) = (Hom_f ‘𝐵))
xpcpropd.2	⊢ (𝜑 → (comp_f‘𝐴) = (comp_f‘𝐵))
xpcpropd.3	⊢ (𝜑 → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
xpcpropd.4	⊢ (𝜑 → (comp_f‘𝐶) = (comp_f‘𝐷))
xpcpropd.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
xpcpropd.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
xpcpropd.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
xpcpropd.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)

**Assertion**
Ref	Expression
xpcpropd	⊢ (𝜑 → (𝐴 ×_c 𝐶) = (𝐵 ×_c 𝐷))

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

Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ (𝐴 ×_c 𝐶) = (𝐴 ×_c 𝐶)
2		eqid 2737	. . 3 ⊢ (Base‘𝐴) = (Base‘𝐴)
3		eqid 2737	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2737	. . 3 ⊢ (Hom ‘𝐴) = (Hom ‘𝐴)
5		eqid 2737	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2737	. . 3 ⊢ (comp‘𝐴) = (comp‘𝐴)
7		eqid 2737	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
8		xpcpropd.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
9		xpcpropd.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
10		eqidd 2738	. . 3 ⊢ (𝜑 → ((Base‘𝐴) × (Base‘𝐶)) = ((Base‘𝐴) × (Base‘𝐶)))
11	1, 2, 3	xpcbas 18113	. . . . 5 ⊢ ((Base‘𝐴) × (Base‘𝐶)) = (Base‘(𝐴 ×_c 𝐶))
12		eqid 2737	. . . . 5 ⊢ (Hom ‘(𝐴 ×_c 𝐶)) = (Hom ‘(𝐴 ×_c 𝐶))
13	1, 11, 4, 5, 12	xpchomfval 18114	. . . 4 ⊢ (Hom ‘(𝐴 ×_c 𝐶)) = (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)), 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (((1^st ‘𝑢)(Hom ‘𝐴)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐶)(2^nd ‘𝑣))))
14	13	a1i 11	. . 3 ⊢ (𝜑 → (Hom ‘(𝐴 ×_c 𝐶)) = (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)), 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (((1^st ‘𝑢)(Hom ‘𝐴)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐶)(2^nd ‘𝑣)))))
15		eqidd 2738	. . 3 ⊢ (𝜑 → (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))), 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐴)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐶)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩)) = (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))), 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐴)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐶)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩)))
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15	xpcval 18112	. 2 ⊢ (𝜑 → (𝐴 ×_c 𝐶) = {⟨(Base‘ndx), ((Base‘𝐴) × (Base‘𝐶))⟩, ⟨(Hom ‘ndx), (Hom ‘(𝐴 ×_c 𝐶))⟩, ⟨(comp‘ndx), (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))), 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐴)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐶)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩})
17		eqid 2737	. . 3 ⊢ (𝐵 ×_c 𝐷) = (𝐵 ×_c 𝐷)
18		eqid 2737	. . 3 ⊢ (Base‘𝐵) = (Base‘𝐵)
19		eqid 2737	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
20		eqid 2737	. . 3 ⊢ (Hom ‘𝐵) = (Hom ‘𝐵)
21		eqid 2737	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
22		eqid 2737	. . 3 ⊢ (comp‘𝐵) = (comp‘𝐵)
23		eqid 2737	. . 3 ⊢ (comp‘𝐷) = (comp‘𝐷)
24		xpcpropd.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
25		xpcpropd.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
26		xpcpropd.1	. . . . 5 ⊢ (𝜑 → (Hom_f ‘𝐴) = (Hom_f ‘𝐵))
27	26	homfeqbas 17631	. . . 4 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
28		xpcpropd.3	. . . . 5 ⊢ (𝜑 → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
29	28	homfeqbas 17631	. . . 4 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
30	27, 29	xpeq12d 5663	. . 3 ⊢ (𝜑 → ((Base‘𝐴) × (Base‘𝐶)) = ((Base‘𝐵) × (Base‘𝐷)))
31	26	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (Hom_f ‘𝐴) = (Hom_f ‘𝐵))
32		xp1st 7975	. . . . . . . 8 ⊢ (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1^st ‘𝑢) ∈ (Base‘𝐴))
33	32	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (1^st ‘𝑢) ∈ (Base‘𝐴))
34		xp1st 7975	. . . . . . . 8 ⊢ (𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1^st ‘𝑣) ∈ (Base‘𝐴))
35	34	3ad2ant3 1136	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (1^st ‘𝑣) ∈ (Base‘𝐴))
36	2, 4, 20, 31, 33, 35	homfeqval 17632	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → ((1^st ‘𝑢)(Hom ‘𝐴)(1^st ‘𝑣)) = ((1^st ‘𝑢)(Hom ‘𝐵)(1^st ‘𝑣)))
37	28	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
38		xp2nd 7976	. . . . . . . 8 ⊢ (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2^nd ‘𝑢) ∈ (Base‘𝐶))
39	38	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (2^nd ‘𝑢) ∈ (Base‘𝐶))
40		xp2nd 7976	. . . . . . . 8 ⊢ (𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2^nd ‘𝑣) ∈ (Base‘𝐶))
41	40	3ad2ant3 1136	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (2^nd ‘𝑣) ∈ (Base‘𝐶))
42	3, 5, 21, 37, 39, 41	homfeqval 17632	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → ((2^nd ‘𝑢)(Hom ‘𝐶)(2^nd ‘𝑣)) = ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣)))
43	36, 42	xpeq12d 5663	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (((1^st ‘𝑢)(Hom ‘𝐴)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐶)(2^nd ‘𝑣))) = (((1^st ‘𝑢)(Hom ‘𝐵)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))))
44	43	mpoeq3dva 7445	. . . 4 ⊢ (𝜑 → (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)), 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (((1^st ‘𝑢)(Hom ‘𝐴)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐶)(2^nd ‘𝑣)))) = (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)), 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (((1^st ‘𝑢)(Hom ‘𝐵)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣)))))
45	13, 44	eqtrid 2784	. . 3 ⊢ (𝜑 → (Hom ‘(𝐴 ×_c 𝐶)) = (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)), 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (((1^st ‘𝑢)(Hom ‘𝐵)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣)))))
46	26	ad4antr 733	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → (Hom_f ‘𝐴) = (Hom_f ‘𝐵))
47		xpcpropd.2	. . . . . . . . . 10 ⊢ (𝜑 → (comp_f‘𝐴) = (comp_f‘𝐵))
48	47	ad4antr 733	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → (comp_f‘𝐴) = (comp_f‘𝐵))
49		simp-4r 784	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))))
50		xp1st 7975	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))) → (1^st ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)))
51	49, 50	syl 17	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → (1^st ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)))
52		xp1st 7975	. . . . . . . . . 10 ⊢ ((1^st ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1^st ‘(1^st ‘𝑥)) ∈ (Base‘𝐴))
53	51, 52	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → (1^st ‘(1^st ‘𝑥)) ∈ (Base‘𝐴))
54		xp2nd 7976	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))) → (2^nd ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)))
55	49, 54	syl 17	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → (2^nd ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)))
56		xp1st 7975	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1^st ‘(2^nd ‘𝑥)) ∈ (Base‘𝐴))
57	55, 56	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → (1^st ‘(2^nd ‘𝑥)) ∈ (Base‘𝐴))
58		simpllr 776	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)))
59		xp1st 7975	. . . . . . . . . 10 ⊢ (𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1^st ‘𝑦) ∈ (Base‘𝐴))
60	58, 59	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → (1^st ‘𝑦) ∈ (Base‘𝐴))
61		simpr 484	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥))
62		1st2nd2 7982	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
63	49, 62	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
64	63	fveq2d 6846	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥) = ((Hom ‘(𝐴 ×_c 𝐶))‘⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩))
65		df-ov 7371	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))(2^nd ‘𝑥)) = ((Hom ‘(𝐴 ×_c 𝐶))‘⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
66	64, 65	eqtr4di 2790	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥) = ((1^st ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))(2^nd ‘𝑥)))
67	1, 11, 4, 5, 12, 51, 55	xpchom 18115	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → ((1^st ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))(2^nd ‘𝑥)) = (((1^st ‘(1^st ‘𝑥))(Hom ‘𝐴)(1^st ‘(2^nd ‘𝑥))) × ((2^nd ‘(1^st ‘𝑥))(Hom ‘𝐶)(2^nd ‘(2^nd ‘𝑥)))))
68	66, 67	eqtrd 2772	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥) = (((1^st ‘(1^st ‘𝑥))(Hom ‘𝐴)(1^st ‘(2^nd ‘𝑥))) × ((2^nd ‘(1^st ‘𝑥))(Hom ‘𝐶)(2^nd ‘(2^nd ‘𝑥)))))
69	61, 68	eleqtrd 2839	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → 𝑓 ∈ (((1^st ‘(1^st ‘𝑥))(Hom ‘𝐴)(1^st ‘(2^nd ‘𝑥))) × ((2^nd ‘(1^st ‘𝑥))(Hom ‘𝐶)(2^nd ‘(2^nd ‘𝑥)))))
70		xp1st 7975	. . . . . . . . . 10 ⊢ (𝑓 ∈ (((1^st ‘(1^st ‘𝑥))(Hom ‘𝐴)(1^st ‘(2^nd ‘𝑥))) × ((2^nd ‘(1^st ‘𝑥))(Hom ‘𝐶)(2^nd ‘(2^nd ‘𝑥)))) → (1^st ‘𝑓) ∈ ((1^st ‘(1^st ‘𝑥))(Hom ‘𝐴)(1^st ‘(2^nd ‘𝑥))))
71	69, 70	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → (1^st ‘𝑓) ∈ ((1^st ‘(1^st ‘𝑥))(Hom ‘𝐴)(1^st ‘(2^nd ‘𝑥))))
72		simplr 769	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦))
73	1, 11, 4, 5, 12, 55, 58	xpchom 18115	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦) = (((1^st ‘(2^nd ‘𝑥))(Hom ‘𝐴)(1^st ‘𝑦)) × ((2^nd ‘(2^nd ‘𝑥))(Hom ‘𝐶)(2^nd ‘𝑦))))
74	72, 73	eleqtrd 2839	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → 𝑔 ∈ (((1^st ‘(2^nd ‘𝑥))(Hom ‘𝐴)(1^st ‘𝑦)) × ((2^nd ‘(2^nd ‘𝑥))(Hom ‘𝐶)(2^nd ‘𝑦))))
75		xp1st 7975	. . . . . . . . . 10 ⊢ (𝑔 ∈ (((1^st ‘(2^nd ‘𝑥))(Hom ‘𝐴)(1^st ‘𝑦)) × ((2^nd ‘(2^nd ‘𝑥))(Hom ‘𝐶)(2^nd ‘𝑦))) → (1^st ‘𝑔) ∈ ((1^st ‘(2^nd ‘𝑥))(Hom ‘𝐴)(1^st ‘𝑦)))
76	74, 75	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → (1^st ‘𝑔) ∈ ((1^st ‘(2^nd ‘𝑥))(Hom ‘𝐴)(1^st ‘𝑦)))
77	2, 4, 6, 22, 46, 48, 53, 57, 60, 71, 76	comfeqval 17643	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → ((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐴)(1^st ‘𝑦))(1^st ‘𝑓)) = ((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐵)(1^st ‘𝑦))(1^st ‘𝑓)))
78	28	ad4antr 733	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
79		xpcpropd.4	. . . . . . . . . 10 ⊢ (𝜑 → (comp_f‘𝐶) = (comp_f‘𝐷))
80	79	ad4antr 733	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → (comp_f‘𝐶) = (comp_f‘𝐷))
81		xp2nd 7976	. . . . . . . . . 10 ⊢ ((1^st ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2^nd ‘(1^st ‘𝑥)) ∈ (Base‘𝐶))
82	51, 81	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → (2^nd ‘(1^st ‘𝑥)) ∈ (Base‘𝐶))
83		xp2nd 7976	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2^nd ‘(2^nd ‘𝑥)) ∈ (Base‘𝐶))
84	55, 83	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → (2^nd ‘(2^nd ‘𝑥)) ∈ (Base‘𝐶))
85		xp2nd 7976	. . . . . . . . . 10 ⊢ (𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2^nd ‘𝑦) ∈ (Base‘𝐶))
86	58, 85	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → (2^nd ‘𝑦) ∈ (Base‘𝐶))
87		xp2nd 7976	. . . . . . . . . 10 ⊢ (𝑓 ∈ (((1^st ‘(1^st ‘𝑥))(Hom ‘𝐴)(1^st ‘(2^nd ‘𝑥))) × ((2^nd ‘(1^st ‘𝑥))(Hom ‘𝐶)(2^nd ‘(2^nd ‘𝑥)))) → (2^nd ‘𝑓) ∈ ((2^nd ‘(1^st ‘𝑥))(Hom ‘𝐶)(2^nd ‘(2^nd ‘𝑥))))
88	69, 87	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → (2^nd ‘𝑓) ∈ ((2^nd ‘(1^st ‘𝑥))(Hom ‘𝐶)(2^nd ‘(2^nd ‘𝑥))))
89		xp2nd 7976	. . . . . . . . . 10 ⊢ (𝑔 ∈ (((1^st ‘(2^nd ‘𝑥))(Hom ‘𝐴)(1^st ‘𝑦)) × ((2^nd ‘(2^nd ‘𝑥))(Hom ‘𝐶)(2^nd ‘𝑦))) → (2^nd ‘𝑔) ∈ ((2^nd ‘(2^nd ‘𝑥))(Hom ‘𝐶)(2^nd ‘𝑦)))
90	74, 89	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → (2^nd ‘𝑔) ∈ ((2^nd ‘(2^nd ‘𝑥))(Hom ‘𝐶)(2^nd ‘𝑦)))
91	3, 5, 7, 23, 78, 80, 82, 84, 86, 88, 90	comfeqval 17643	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐶)(2^nd ‘𝑦))(2^nd ‘𝑓)) = ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐷)(2^nd ‘𝑦))(2^nd ‘𝑓)))
92	77, 91	opeq12d 4839	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐴)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐶)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩ = ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐵)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐷)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩)
93	92	3impa 1110	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥)) → ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐴)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐶)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩ = ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐵)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐷)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩)
94	93	mpoeq3dva 7445	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐴)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐶)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩) = (𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐵)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐷)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))
95	94	3impa 1110	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐴)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐶)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩) = (𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐵)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐷)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))
96	95	mpoeq3dva 7445	. . 3 ⊢ (𝜑 → (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))), 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐴)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐶)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩)) = (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))), 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐵)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐷)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩)))
97	17, 18, 19, 20, 21, 22, 23, 24, 25, 30, 45, 96	xpcval 18112	. 2 ⊢ (𝜑 → (𝐵 ×_c 𝐷) = {⟨(Base‘ndx), ((Base‘𝐴) × (Base‘𝐶))⟩, ⟨(Hom ‘ndx), (Hom ‘(𝐴 ×_c 𝐶))⟩, ⟨(comp‘ndx), (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))), 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘(𝐴 ×_c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×_c 𝐶))‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐴)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐶)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩})
98	16, 97	eqtr4d 2775	1 ⊢ (𝜑 → (𝐴 ×_c 𝐶) = (𝐵 ×_c 𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 {ctp 4586 ⟨cop 4588 × cxp 5630 ‘cfv 6500 (class class class)co 7368 ∈ cmpo 7370 1^st c1st 7941 2^nd c2nd 7942 ndxcnx 17132 Basecbs 17148 Hom chom 17200 compcco 17201 Hom_f chomf 17601 comp_fccomf 17602 ×_c cxpc 18103

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 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-fz 13436 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17149 df-hom 17213 df-cco 17214 df-homf 17605 df-comf 17606 df-xpc 18107

This theorem is referenced by: curfpropd 18168 oppchofcl 18195 1stfpropd 49643 2ndfpropd 49644

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