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Theorem xpcpropd 16895
 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 2651 . . 3 (𝐴 ×c 𝐶) = (𝐴 ×c 𝐶)
2 eqid 2651 . . 3 (Base‘𝐴) = (Base‘𝐴)
3 eqid 2651 . . 3 (Base‘𝐶) = (Base‘𝐶)
4 eqid 2651 . . 3 (Hom ‘𝐴) = (Hom ‘𝐴)
5 eqid 2651 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2651 . . 3 (comp‘𝐴) = (comp‘𝐴)
7 eqid 2651 . . 3 (comp‘𝐶) = (comp‘𝐶)
8 xpcpropd.a . . 3 (𝜑𝐴𝑉)
9 xpcpropd.c . . 3 (𝜑𝐶𝑉)
10 eqidd 2652 . . 3 (𝜑 → ((Base‘𝐴) × (Base‘𝐶)) = ((Base‘𝐴) × (Base‘𝐶)))
111, 2, 3xpcbas 16865 . . . . 5 ((Base‘𝐴) × (Base‘𝐶)) = (Base‘(𝐴 ×c 𝐶))
12 eqid 2651 . . . . 5 (Hom ‘(𝐴 ×c 𝐶)) = (Hom ‘(𝐴 ×c 𝐶))
131, 11, 4, 5, 12xpchomfval 16866 . . . 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 2652 . . 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 16864 . 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 2651 . . 3 (𝐵 ×c 𝐷) = (𝐵 ×c 𝐷)
18 eqid 2651 . . 3 (Base‘𝐵) = (Base‘𝐵)
19 eqid 2651 . . 3 (Base‘𝐷) = (Base‘𝐷)
20 eqid 2651 . . 3 (Hom ‘𝐵) = (Hom ‘𝐵)
21 eqid 2651 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
22 eqid 2651 . . 3 (comp‘𝐵) = (comp‘𝐵)
23 eqid 2651 . . 3 (comp‘𝐷) = (comp‘𝐷)
24 xpcpropd.b . . 3 (𝜑𝐵𝑉)
25 xpcpropd.d . . 3 (𝜑𝐷𝑉)
26 xpcpropd.1 . . . . 5 (𝜑 → (Homf𝐴) = (Homf𝐵))
2726homfeqbas 16403 . . . 4 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
28 xpcpropd.3 . . . . 5 (𝜑 → (Homf𝐶) = (Homf𝐷))
2928homfeqbas 16403 . . . 4 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
3027, 29xpeq12d 5174 . . 3 (𝜑 → ((Base‘𝐴) × (Base‘𝐶)) = ((Base‘𝐵) × (Base‘𝐷)))
31263ad2ant1 1102 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (Homf𝐴) = (Homf𝐵))
32 xp1st 7242 . . . . . . . 8 (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1st𝑢) ∈ (Base‘𝐴))
33323ad2ant2 1103 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (1st𝑢) ∈ (Base‘𝐴))
34 xp1st 7242 . . . . . . . 8 (𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1st𝑣) ∈ (Base‘𝐴))
35343ad2ant3 1104 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (1st𝑣) ∈ (Base‘𝐴))
362, 4, 20, 31, 33, 35homfeqval 16404 . . . . . 6 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → ((1st𝑢)(Hom ‘𝐴)(1st𝑣)) = ((1st𝑢)(Hom ‘𝐵)(1st𝑣)))
37283ad2ant1 1102 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (Homf𝐶) = (Homf𝐷))
38 xp2nd 7243 . . . . . . . 8 (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2nd𝑢) ∈ (Base‘𝐶))
39383ad2ant2 1103 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (2nd𝑢) ∈ (Base‘𝐶))
40 xp2nd 7243 . . . . . . . 8 (𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2nd𝑣) ∈ (Base‘𝐶))
41403ad2ant3 1104 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (2nd𝑣) ∈ (Base‘𝐶))
423, 5, 21, 37, 39, 41homfeqval 16404 . . . . . 6 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → ((2nd𝑢)(Hom ‘𝐶)(2nd𝑣)) = ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣)))
4336, 42xpeq12d 5174 . . . . 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 6761 . . . 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 2697 . . 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 769 . . . . . . . . 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 769 . . . . . . . . 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 824 . . . . . . . . . . 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 7242 . . . . . . . . . . 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 7242 . . . . . . . . . 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 7243 . . . . . . . . . . 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 7242 . . . . . . . . . 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 815 . . . . . . . . . 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 7242 . . . . . . . . . 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 476 . . . . . . . . . . 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 7249 . . . . . . . . . . . . . . 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 6233 . . . . . . . . . . . . 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 6693 . . . . . . . . . . . . 13 ((1st𝑥)(Hom ‘(𝐴 ×c 𝐶))(2nd𝑥)) = ((Hom ‘(𝐴 ×c 𝐶))‘⟨(1st𝑥), (2nd𝑥)⟩)
6664, 65syl6eqr 2703 . . . . . . . . . . . 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 16867 . . . . . . . . . . . 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 2685 . . . . . . . . . . 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 2732 . . . . . . . . . 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 7242 . . . . . . . . . 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 807 . . . . . . . . . . 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 16867 . . . . . . . . . . 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 2732 . . . . . . . . . 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 7242 . . . . . . . . . 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 16415 . . . . . . . 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 769 . . . . . . . . 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 769 . . . . . . . . 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 7243 . . . . . . . . . 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 7243 . . . . . . . . . 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 7243 . . . . . . . . . 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 7243 . . . . . . . . . 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 7243 . . . . . . . . . 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 16415 . . . . . . . 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 4441 . . . . . . 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 1278 . . . . . 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 6761 . . . . 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 1278 . . . 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 6761 . . 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 16864 . 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 2688 1 (𝜑 → (𝐴 ×c 𝐶) = (𝐵 ×c 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  {ctp 4214  ⟨cop 4216   × cxp 5141  ‘cfv 5926  (class class class)co 6690   ↦ cmpt2 6692  1st c1st 7208  2nd c2nd 7209  ndxcnx 15901  Basecbs 15904  Hom chom 15999  compcco 16000  Homf chomf 16374  compfccomf 16375   ×c cxpc 16855 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-hom 16013  df-cco 16014  df-homf 16378  df-comf 16379  df-xpc 16859 This theorem is referenced by:  curfpropd  16920  oppchofcl  16947
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