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Theorem xpcpropd 18265
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 2735 . . 3 (𝐴 ×c 𝐶) = (𝐴 ×c 𝐶)
2 eqid 2735 . . 3 (Base‘𝐴) = (Base‘𝐴)
3 eqid 2735 . . 3 (Base‘𝐶) = (Base‘𝐶)
4 eqid 2735 . . 3 (Hom ‘𝐴) = (Hom ‘𝐴)
5 eqid 2735 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2735 . . 3 (comp‘𝐴) = (comp‘𝐴)
7 eqid 2735 . . 3 (comp‘𝐶) = (comp‘𝐶)
8 xpcpropd.a . . 3 (𝜑𝐴𝑉)
9 xpcpropd.c . . 3 (𝜑𝐶𝑉)
10 eqidd 2736 . . 3 (𝜑 → ((Base‘𝐴) × (Base‘𝐶)) = ((Base‘𝐴) × (Base‘𝐶)))
111, 2, 3xpcbas 18234 . . . . 5 ((Base‘𝐴) × (Base‘𝐶)) = (Base‘(𝐴 ×c 𝐶))
12 eqid 2735 . . . . 5 (Hom ‘(𝐴 ×c 𝐶)) = (Hom ‘(𝐴 ×c 𝐶))
131, 11, 4, 5, 12xpchomfval 18235 . . . 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 2736 . . 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 18233 . 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 2735 . . 3 (𝐵 ×c 𝐷) = (𝐵 ×c 𝐷)
18 eqid 2735 . . 3 (Base‘𝐵) = (Base‘𝐵)
19 eqid 2735 . . 3 (Base‘𝐷) = (Base‘𝐷)
20 eqid 2735 . . 3 (Hom ‘𝐵) = (Hom ‘𝐵)
21 eqid 2735 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
22 eqid 2735 . . 3 (comp‘𝐵) = (comp‘𝐵)
23 eqid 2735 . . 3 (comp‘𝐷) = (comp‘𝐷)
24 xpcpropd.b . . 3 (𝜑𝐵𝑉)
25 xpcpropd.d . . 3 (𝜑𝐷𝑉)
26 xpcpropd.1 . . . . 5 (𝜑 → (Homf𝐴) = (Homf𝐵))
2726homfeqbas 17741 . . . 4 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
28 xpcpropd.3 . . . . 5 (𝜑 → (Homf𝐶) = (Homf𝐷))
2928homfeqbas 17741 . . . 4 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
3027, 29xpeq12d 5720 . . 3 (𝜑 → ((Base‘𝐴) × (Base‘𝐶)) = ((Base‘𝐵) × (Base‘𝐷)))
31263ad2ant1 1132 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (Homf𝐴) = (Homf𝐵))
32 xp1st 8045 . . . . . . . 8 (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1st𝑢) ∈ (Base‘𝐴))
33323ad2ant2 1133 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (1st𝑢) ∈ (Base‘𝐴))
34 xp1st 8045 . . . . . . . 8 (𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1st𝑣) ∈ (Base‘𝐴))
35343ad2ant3 1134 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (1st𝑣) ∈ (Base‘𝐴))
362, 4, 20, 31, 33, 35homfeqval 17742 . . . . . 6 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → ((1st𝑢)(Hom ‘𝐴)(1st𝑣)) = ((1st𝑢)(Hom ‘𝐵)(1st𝑣)))
37283ad2ant1 1132 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (Homf𝐶) = (Homf𝐷))
38 xp2nd 8046 . . . . . . . 8 (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2nd𝑢) ∈ (Base‘𝐶))
39383ad2ant2 1133 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (2nd𝑢) ∈ (Base‘𝐶))
40 xp2nd 8046 . . . . . . . 8 (𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2nd𝑣) ∈ (Base‘𝐶))
41403ad2ant3 1134 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (2nd𝑣) ∈ (Base‘𝐶))
423, 5, 21, 37, 39, 41homfeqval 17742 . . . . . 6 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → ((2nd𝑢)(Hom ‘𝐶)(2nd𝑣)) = ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣)))
4336, 42xpeq12d 5720 . . . . 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))))
4443mpoeq3dva 7510 . . . 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, 44eqtrid 2787 . . 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 732 . . . . . . . . 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 732 . . . . . . . . 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 784 . . . . . . . . . . 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 8045 . . . . . . . . . . 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 8045 . . . . . . . . . 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 8046 . . . . . . . . . . 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 8045 . . . . . . . . . 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 776 . . . . . . . . . 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 8045 . . . . . . . . . 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 484 . . . . . . . . . . 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 8052 . . . . . . . . . . . . . . 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 6911 . . . . . . . . . . . . 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 7434 . . . . . . . . . . . . 13 ((1st𝑥)(Hom ‘(𝐴 ×c 𝐶))(2nd𝑥)) = ((Hom ‘(𝐴 ×c 𝐶))‘⟨(1st𝑥), (2nd𝑥)⟩)
6664, 65eqtr4di 2793 . . . . . . . . . . . 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 18236 . . . . . . . . . . . 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 2775 . . . . . . . . . . 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 2841 . . . . . . . . . 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 8045 . . . . . . . . . 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 769 . . . . . . . . . . 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 18236 . . . . . . . . . . 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 2841 . . . . . . . . . 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 8045 . . . . . . . . . 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 17753 . . . . . . . 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 732 . . . . . . . . 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 732 . . . . . . . . 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 8046 . . . . . . . . . 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 8046 . . . . . . . . . 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 8046 . . . . . . . . . 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 8046 . . . . . . . . . 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 8046 . . . . . . . . . 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 17753 . . . . . . . 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 4886 . . . . . . 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 1109 . . . . . 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))⟩)
9493mpoeq3dva 7510 . . . . 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 1109 . . . 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))⟩))
9695mpoeq3dva 7510 . . 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 18233 . 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 2778 1 (𝜑 → (𝐴 ×c 𝐶) = (𝐵 ×c 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  {ctp 4635  cop 4637   × cxp 5687  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  ndxcnx 17227  Basecbs 17245  Hom chom 17309  compcco 17310  Homf chomf 17711  compfccomf 17712   ×c cxpc 18224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-hom 17322  df-cco 17323  df-homf 17715  df-comf 17716  df-xpc 18228
This theorem is referenced by:  curfpropd  18290  oppchofcl  18317
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