xpcpropd - Metamath Proof Explorer

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

Theorem xpcpropd 18362

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 2761	. . 3 ⊢ (𝐴 ×_c 𝐶) = (𝐴 ×_c 𝐶)
2		eqid 2761	. . 3 ⊢ (Base‘𝐴) = (Base‘𝐴)
3		eqid 2761	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2761	. . 3 ⊢ (Hom ‘𝐴) = (Hom ‘𝐴)
5		eqid 2761	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2761	. . 3 ⊢ (comp‘𝐴) = (comp‘𝐴)
7		eqid 2761	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
8		xpcpropd.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
9		xpcpropd.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
10		eqidd 2762	. . 3 ⊢ (𝜑 → ((Base‘𝐴) × (Base‘𝐶)) = ((Base‘𝐴) × (Base‘𝐶)))
11	1, 2, 3	xpcbas 18332	. . . . 5 ⊢ ((Base‘𝐴) × (Base‘𝐶)) = (Base‘(𝐴 ×_c 𝐶))
12		eqid 2761	. . . . 5 ⊢ (Hom ‘(𝐴 ×_c 𝐶)) = (Hom ‘(𝐴 ×_c 𝐶))
13	1, 11, 4, 5, 12	xpchomfval 18333	. . . 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 2762	. . 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 18331	. 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 2761	. . 3 ⊢ (𝐵 ×_c 𝐷) = (𝐵 ×_c 𝐷)
18		eqid 2761	. . 3 ⊢ (Base‘𝐵) = (Base‘𝐵)
19		eqid 2761	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
20		eqid 2761	. . 3 ⊢ (Hom ‘𝐵) = (Hom ‘𝐵)
21		eqid 2761	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
22		eqid 2761	. . 3 ⊢ (comp‘𝐵) = (comp‘𝐵)
23		eqid 2761	. . 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 17850	. . . 4 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
28		xpcpropd.3	. . . . 5 ⊢ (𝜑 → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
29	28	homfeqbas 17850	. . . 4 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
30	27, 29	xpeq12d 5682	. . 3 ⊢ (𝜑 → ((Base‘𝐴) × (Base‘𝐶)) = ((Base‘𝐵) × (Base‘𝐷)))
31	26	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (Hom_f ‘𝐴) = (Hom_f ‘𝐵))
32		xp1st 8022	. . . . . . . 8 ⊢ (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1^st ‘𝑢) ∈ (Base‘𝐴))
33	32	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (1^st ‘𝑢) ∈ (Base‘𝐴))
34		xp1st 8022	. . . . . . . 8 ⊢ (𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1^st ‘𝑣) ∈ (Base‘𝐴))
35	34	3ad2ant3 1153	. . . . . . 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 17851	. . . . . 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 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
38		xp2nd 8023	. . . . . . . 8 ⊢ (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2^nd ‘𝑢) ∈ (Base‘𝐶))
39	38	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (2^nd ‘𝑢) ∈ (Base‘𝐶))
40		xp2nd 8023	. . . . . . . 8 ⊢ (𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2^nd ‘𝑣) ∈ (Base‘𝐶))
41	40	3ad2ant3 1153	. . . . . . 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 17851	. . . . . 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 5682	. . . . 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 7489	. . . 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 2808	. . 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 745	. . . . . . . . 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 745	. . . . . . . . 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 796	. . . . . . . . . . 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 8022	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))) → (1^st ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)))
51	49, 50	syl 18	. . . . . . . . . 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 8022	. . . . . . . . . 10 ⊢ ((1^st ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1^st ‘(1^st ‘𝑥)) ∈ (Base‘𝐴))
53	51, 52	syl 18	. . . . . . . . 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 8023	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))) → (2^nd ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)))
55	49, 54	syl 18	. . . . . . . . . 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 8022	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1^st ‘(2^nd ‘𝑥)) ∈ (Base‘𝐴))
57	55, 56	syl 18	. . . . . . . . 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 788	. . . . . . . . . 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 8022	. . . . . . . . . 10 ⊢ (𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1^st ‘𝑦) ∈ (Base‘𝐴))
60	58, 59	syl 18	. . . . . . . . 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 490	. . . . . . . . . . 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 8029	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
63	49, 62	syl 18	. . . . . . . . . . . . . 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 6881	. . . . . . . . . . . . 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 7415	. . . . . . . . . . . . 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 2814	. . . . . . . . . . . 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 18334	. . . . . . . . . . . 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 2796	. . . . . . . . . . 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 2863	. . . . . . . . . 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 8022	. . . . . . . . . 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 18	. . . . . . . . 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 781	. . . . . . . . . . 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 18334	. . . . . . . . . . 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 2863	. . . . . . . . . 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 8022	. . . . . . . . . 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 18	. . . . . . . . 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 17862	. . . . . . . 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 745	. . . . . . . . 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 745	. . . . . . . . 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 8023	. . . . . . . . . 10 ⊢ ((1^st ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2^nd ‘(1^st ‘𝑥)) ∈ (Base‘𝐶))
82	51, 81	syl 18	. . . . . . . . 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 8023	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2^nd ‘(2^nd ‘𝑥)) ∈ (Base‘𝐶))
84	55, 83	syl 18	. . . . . . . . 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 8023	. . . . . . . . . 10 ⊢ (𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2^nd ‘𝑦) ∈ (Base‘𝐶))
86	58, 85	syl 18	. . . . . . . . 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 8023	. . . . . . . . . 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 18	. . . . . . . . 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 8023	. . . . . . . . . 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 18	. . . . . . . . 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 17862	. . . . . . . 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 4841	. . . . . . 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 1127	. . . . . 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 7489	. . . . 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 1127	. . . 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 7489	. . 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 18331	. 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 2799	1 ⊢ (𝜑 → (𝐴 ×_c 𝐶) = (𝐵 ×_c 𝐷))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 {ctp 4588 ⟨cop 4590 × cxp 5649 ‘cfv 6531 (class class class)co 7412 ∈ cmpo 7414 1^st c1st 7988 2^nd c2nd 7989 ndxcnx 17351 Basecbs 17367 Hom chom 17419 compcco 17420 Hom_f chomf 17820 comp_fccomf 17821 ×_c cxpc 18322

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-er 8701 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-z 12675 df-dec 12796 df-uz 12947 df-fz 13621 df-struct 17305 df-slot 17340 df-ndx 17352 df-base 17368 df-hom 17432 df-cco 17433 df-homf 17824 df-comf 17825 df-xpc 18326

This theorem is used by: curfpropd 18387 oppchofcl 18414 1stfpropd 50342 2ndfpropd 50343

W3C validator