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Theorem xpcpropd 16617
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 → (Homf𝐴) = (Homf𝐵))
xpcpropd.2 (𝜑 → (compf𝐴) = (compf𝐵))
xpcpropd.3 (𝜑 → (Homf𝐶) = (Homf𝐷))
xpcpropd.4 (𝜑 → (compf𝐶) = (compf𝐷))
xpcpropd.a (𝜑𝐴𝑉)
xpcpropd.b (𝜑𝐵𝑉)
xpcpropd.c (𝜑𝐶𝑉)
xpcpropd.d (𝜑𝐷𝑉)
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.
StepHypRef Expression
1 eqid 2609 . . 3 (𝐴 ×c 𝐶) = (𝐴 ×c 𝐶)
2 eqid 2609 . . 3 (Base‘𝐴) = (Base‘𝐴)
3 eqid 2609 . . 3 (Base‘𝐶) = (Base‘𝐶)
4 eqid 2609 . . 3 (Hom ‘𝐴) = (Hom ‘𝐴)
5 eqid 2609 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2609 . . 3 (comp‘𝐴) = (comp‘𝐴)
7 eqid 2609 . . 3 (comp‘𝐶) = (comp‘𝐶)
8 xpcpropd.a . . 3 (𝜑𝐴𝑉)
9 xpcpropd.c . . 3 (𝜑𝐶𝑉)
10 eqidd 2610 . . 3 (𝜑 → ((Base‘𝐴) × (Base‘𝐶)) = ((Base‘𝐴) × (Base‘𝐶)))
111, 2, 3xpcbas 16587 . . . . 5 ((Base‘𝐴) × (Base‘𝐶)) = (Base‘(𝐴 ×c 𝐶))
12 eqid 2609 . . . . 5 (Hom ‘(𝐴 ×c 𝐶)) = (Hom ‘(𝐴 ×c 𝐶))
131, 11, 4, 5, 12xpchomfval 16588 . . . 4 (Hom ‘(𝐴 ×c 𝐶)) = (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)), 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (((1st𝑢)(Hom ‘𝐴)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐶)(2nd𝑣))))
1413a1i 11 . . 3 (𝜑 → (Hom ‘(𝐴 ×c 𝐶)) = (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)), 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (((1st𝑢)(Hom ‘𝐴)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐶)(2nd𝑣)))))
15 eqidd 2610 . . 3 (𝜑 → (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))), 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐴)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐶)(2nd𝑦))(2nd𝑓))⟩)) = (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))), 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐴)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐶)(2nd𝑦))(2nd𝑓))⟩)))
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15xpcval 16586 . 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 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐴)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐶)(2nd𝑦))(2nd𝑓))⟩))⟩})
17 eqid 2609 . . 3 (𝐵 ×c 𝐷) = (𝐵 ×c 𝐷)
18 eqid 2609 . . 3 (Base‘𝐵) = (Base‘𝐵)
19 eqid 2609 . . 3 (Base‘𝐷) = (Base‘𝐷)
20 eqid 2609 . . 3 (Hom ‘𝐵) = (Hom ‘𝐵)
21 eqid 2609 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
22 eqid 2609 . . 3 (comp‘𝐵) = (comp‘𝐵)
23 eqid 2609 . . 3 (comp‘𝐷) = (comp‘𝐷)
24 xpcpropd.b . . 3 (𝜑𝐵𝑉)
25 xpcpropd.d . . 3 (𝜑𝐷𝑉)
26 xpcpropd.1 . . . . 5 (𝜑 → (Homf𝐴) = (Homf𝐵))
2726homfeqbas 16125 . . . 4 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
28 xpcpropd.3 . . . . 5 (𝜑 → (Homf𝐶) = (Homf𝐷))
2928homfeqbas 16125 . . . 4 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
3027, 29xpeq12d 5054 . . 3 (𝜑 → ((Base‘𝐴) × (Base‘𝐶)) = ((Base‘𝐵) × (Base‘𝐷)))
31263ad2ant1 1074 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (Homf𝐴) = (Homf𝐵))
32 xp1st 7066 . . . . . . . 8 (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1st𝑢) ∈ (Base‘𝐴))
33323ad2ant2 1075 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (1st𝑢) ∈ (Base‘𝐴))
34 xp1st 7066 . . . . . . . 8 (𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1st𝑣) ∈ (Base‘𝐴))
35343ad2ant3 1076 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (1st𝑣) ∈ (Base‘𝐴))
362, 4, 20, 31, 33, 35homfeqval 16126 . . . . . 6 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → ((1st𝑢)(Hom ‘𝐴)(1st𝑣)) = ((1st𝑢)(Hom ‘𝐵)(1st𝑣)))
37283ad2ant1 1074 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (Homf𝐶) = (Homf𝐷))
38 xp2nd 7067 . . . . . . . 8 (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2nd𝑢) ∈ (Base‘𝐶))
39383ad2ant2 1075 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (2nd𝑢) ∈ (Base‘𝐶))
40 xp2nd 7067 . . . . . . . 8 (𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2nd𝑣) ∈ (Base‘𝐶))
41403ad2ant3 1076 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (2nd𝑣) ∈ (Base‘𝐶))
423, 5, 21, 37, 39, 41homfeqval 16126 . . . . . 6 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → ((2nd𝑢)(Hom ‘𝐶)(2nd𝑣)) = ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣)))
4336, 42xpeq12d 5054 . . . . 5 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (((1st𝑢)(Hom ‘𝐴)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐶)(2nd𝑣))) = (((1st𝑢)(Hom ‘𝐵)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣))))
4443mpt2eq3dva 6595 . . . 4 (𝜑 → (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)), 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (((1st𝑢)(Hom ‘𝐴)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐶)(2nd𝑣)))) = (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)), 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (((1st𝑢)(Hom ‘𝐵)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣)))))
4513, 44syl5eq 2655 . . 3 (𝜑 → (Hom ‘(𝐴 ×c 𝐶)) = (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)), 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (((1st𝑢)(Hom ‘𝐵)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣)))))
4626ad4antr 763 . . . . . . . . 9 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (Homf𝐴) = (Homf𝐵))
47 xpcpropd.2 . . . . . . . . . 10 (𝜑 → (compf𝐴) = (compf𝐵))
4847ad4antr 763 . . . . . . . . 9 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (compf𝐴) = (compf𝐵))
49 simp-4r 802 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → 𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))))
50 xp1st 7066 . . . . . . . . . . 11 (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))) → (1st𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)))
5149, 50syl 17 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (1st𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)))
52 xp1st 7066 . . . . . . . . . 10 ((1st𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1st ‘(1st𝑥)) ∈ (Base‘𝐴))
5351, 52syl 17 . . . . . . . . 9 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (1st ‘(1st𝑥)) ∈ (Base‘𝐴))
54 xp2nd 7067 . . . . . . . . . . 11 (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))) → (2nd𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)))
5549, 54syl 17 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (2nd𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)))
56 xp1st 7066 . . . . . . . . . 10 ((2nd𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1st ‘(2nd𝑥)) ∈ (Base‘𝐴))
5755, 56syl 17 . . . . . . . . 9 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (1st ‘(2nd𝑥)) ∈ (Base‘𝐴))
58 simpllr 794 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)))
59 xp1st 7066 . . . . . . . . . 10 (𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1st𝑦) ∈ (Base‘𝐴))
6058, 59syl 17 . . . . . . . . 9 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (1st𝑦) ∈ (Base‘𝐴))
61 simpr 475 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥))
62 1st2nd2 7073 . . . . . . . . . . . . . . 15 (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
6349, 62syl 17 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
6463fveq2d 6092 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) = ((Hom ‘(𝐴 ×c 𝐶))‘⟨(1st𝑥), (2nd𝑥)⟩))
65 df-ov 6530 . . . . . . . . . . . . 13 ((1st𝑥)(Hom ‘(𝐴 ×c 𝐶))(2nd𝑥)) = ((Hom ‘(𝐴 ×c 𝐶))‘⟨(1st𝑥), (2nd𝑥)⟩)
6664, 65syl6eqr 2661 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) = ((1st𝑥)(Hom ‘(𝐴 ×c 𝐶))(2nd𝑥)))
671, 11, 4, 5, 12, 51, 55xpchom 16589 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → ((1st𝑥)(Hom ‘(𝐴 ×c 𝐶))(2nd𝑥)) = (((1st ‘(1st𝑥))(Hom ‘𝐴)(1st ‘(2nd𝑥))) × ((2nd ‘(1st𝑥))(Hom ‘𝐶)(2nd ‘(2nd𝑥)))))
6866, 67eqtrd 2643 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) = (((1st ‘(1st𝑥))(Hom ‘𝐴)(1st ‘(2nd𝑥))) × ((2nd ‘(1st𝑥))(Hom ‘𝐶)(2nd ‘(2nd𝑥)))))
6961, 68eleqtrd 2689 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → 𝑓 ∈ (((1st ‘(1st𝑥))(Hom ‘𝐴)(1st ‘(2nd𝑥))) × ((2nd ‘(1st𝑥))(Hom ‘𝐶)(2nd ‘(2nd𝑥)))))
70 xp1st 7066 . . . . . . . . . 10 (𝑓 ∈ (((1st ‘(1st𝑥))(Hom ‘𝐴)(1st ‘(2nd𝑥))) × ((2nd ‘(1st𝑥))(Hom ‘𝐶)(2nd ‘(2nd𝑥)))) → (1st𝑓) ∈ ((1st ‘(1st𝑥))(Hom ‘𝐴)(1st ‘(2nd𝑥))))
7169, 70syl 17 . . . . . . . . 9 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (1st𝑓) ∈ ((1st ‘(1st𝑥))(Hom ‘𝐴)(1st ‘(2nd𝑥))))
72 simplr 787 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦))
731, 11, 4, 5, 12, 55, 58xpchom 16589 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦) = (((1st ‘(2nd𝑥))(Hom ‘𝐴)(1st𝑦)) × ((2nd ‘(2nd𝑥))(Hom ‘𝐶)(2nd𝑦))))
7472, 73eleqtrd 2689 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → 𝑔 ∈ (((1st ‘(2nd𝑥))(Hom ‘𝐴)(1st𝑦)) × ((2nd ‘(2nd𝑥))(Hom ‘𝐶)(2nd𝑦))))
75 xp1st 7066 . . . . . . . . . 10 (𝑔 ∈ (((1st ‘(2nd𝑥))(Hom ‘𝐴)(1st𝑦)) × ((2nd ‘(2nd𝑥))(Hom ‘𝐶)(2nd𝑦))) → (1st𝑔) ∈ ((1st ‘(2nd𝑥))(Hom ‘𝐴)(1st𝑦)))
7674, 75syl 17 . . . . . . . . 9 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (1st𝑔) ∈ ((1st ‘(2nd𝑥))(Hom ‘𝐴)(1st𝑦)))
772, 4, 6, 22, 46, 48, 53, 57, 60, 71, 76comfeqval 16137 . . . . . . . 8 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐴)(1st𝑦))(1st𝑓)) = ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐵)(1st𝑦))(1st𝑓)))
7828ad4antr 763 . . . . . . . . 9 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (Homf𝐶) = (Homf𝐷))
79 xpcpropd.4 . . . . . . . . . 10 (𝜑 → (compf𝐶) = (compf𝐷))
8079ad4antr 763 . . . . . . . . 9 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (compf𝐶) = (compf𝐷))
81 xp2nd 7067 . . . . . . . . . 10 ((1st𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2nd ‘(1st𝑥)) ∈ (Base‘𝐶))
8251, 81syl 17 . . . . . . . . 9 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (2nd ‘(1st𝑥)) ∈ (Base‘𝐶))
83 xp2nd 7067 . . . . . . . . . 10 ((2nd𝑥) ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2nd ‘(2nd𝑥)) ∈ (Base‘𝐶))
8455, 83syl 17 . . . . . . . . 9 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (2nd ‘(2nd𝑥)) ∈ (Base‘𝐶))
85 xp2nd 7067 . . . . . . . . . 10 (𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2nd𝑦) ∈ (Base‘𝐶))
8658, 85syl 17 . . . . . . . . 9 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (2nd𝑦) ∈ (Base‘𝐶))
87 xp2nd 7067 . . . . . . . . . 10 (𝑓 ∈ (((1st ‘(1st𝑥))(Hom ‘𝐴)(1st ‘(2nd𝑥))) × ((2nd ‘(1st𝑥))(Hom ‘𝐶)(2nd ‘(2nd𝑥)))) → (2nd𝑓) ∈ ((2nd ‘(1st𝑥))(Hom ‘𝐶)(2nd ‘(2nd𝑥))))
8869, 87syl 17 . . . . . . . . 9 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (2nd𝑓) ∈ ((2nd ‘(1st𝑥))(Hom ‘𝐶)(2nd ‘(2nd𝑥))))
89 xp2nd 7067 . . . . . . . . . 10 (𝑔 ∈ (((1st ‘(2nd𝑥))(Hom ‘𝐴)(1st𝑦)) × ((2nd ‘(2nd𝑥))(Hom ‘𝐶)(2nd𝑦))) → (2nd𝑔) ∈ ((2nd ‘(2nd𝑥))(Hom ‘𝐶)(2nd𝑦)))
9074, 89syl 17 . . . . . . . . 9 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → (2nd𝑔) ∈ ((2nd ‘(2nd𝑥))(Hom ‘𝐶)(2nd𝑦)))
913, 5, 7, 23, 78, 80, 82, 84, 86, 88, 90comfeqval 16137 . . . . . . . 8 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐶)(2nd𝑦))(2nd𝑓)) = ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐷)(2nd𝑦))(2nd𝑓)))
9277, 91opeq12d 4342 . . . . . . 7 (((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦)) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐴)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐶)(2nd𝑦))(2nd𝑓))⟩ = ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐵)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐷)(2nd𝑦))(2nd𝑓))⟩)
93923impa 1250 . . . . . 6 ((((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦) ∧ 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥)) → ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐴)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐶)(2nd𝑦))(2nd𝑓))⟩ = ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐵)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐷)(2nd𝑦))(2nd𝑓))⟩)
9493mpt2eq3dva 6595 . . . . 5 (((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶)))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐴)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐶)(2nd𝑦))(2nd𝑓))⟩) = (𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐵)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐷)(2nd𝑦))(2nd𝑓))⟩))
95943impa 1250 . . . 4 ((𝜑𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))) ∧ 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐴)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐶)(2nd𝑦))(2nd𝑓))⟩) = (𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐵)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐷)(2nd𝑦))(2nd𝑓))⟩))
9695mpt2eq3dva 6595 . . 3 (𝜑 → (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))), 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐴)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐶)(2nd𝑦))(2nd𝑓))⟩)) = (𝑥 ∈ (((Base‘𝐴) × (Base‘𝐶)) × ((Base‘𝐴) × (Base‘𝐶))), 𝑦 ∈ ((Base‘𝐴) × (Base‘𝐶)) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐵)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐷)(2nd𝑦))(2nd𝑓))⟩)))
9717, 18, 19, 20, 21, 22, 23, 24, 25, 30, 45, 96xpcval 16586 . 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 ∈ ((2nd𝑥)(Hom ‘(𝐴 ×c 𝐶))𝑦), 𝑓 ∈ ((Hom ‘(𝐴 ×c 𝐶))‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐴)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐶)(2nd𝑦))(2nd𝑓))⟩))⟩})
9816, 97eqtr4d 2646 1 (𝜑 → (𝐴 ×c 𝐶) = (𝐵 ×c 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  {ctp 4128  cop 4130   × cxp 5026  cfv 5790  (class class class)co 6527  cmpt2 6529  1st c1st 7034  2nd c2nd 7035  ndxcnx 15638  Basecbs 15641  Hom chom 15725  compcco 15726  Homf chomf 16096  compfccomf 16097   ×c cxpc 16577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-4 10928  df-5 10929  df-6 10930  df-7 10931  df-8 10932  df-9 10933  df-n0 11140  df-z 11211  df-dec 11326  df-uz 11520  df-fz 12153  df-struct 15643  df-ndx 15644  df-slot 15645  df-base 15646  df-hom 15739  df-cco 15740  df-homf 16100  df-comf 16101  df-xpc 16581
This theorem is referenced by:  curfpropd  16642  oppchofcl  16669
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