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Theorem xpcpropd 17537
 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 2758 . . 3 (𝐴 ×c 𝐶) = (𝐴 ×c 𝐶)
2 eqid 2758 . . 3 (Base‘𝐴) = (Base‘𝐴)
3 eqid 2758 . . 3 (Base‘𝐶) = (Base‘𝐶)
4 eqid 2758 . . 3 (Hom ‘𝐴) = (Hom ‘𝐴)
5 eqid 2758 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2758 . . 3 (comp‘𝐴) = (comp‘𝐴)
7 eqid 2758 . . 3 (comp‘𝐶) = (comp‘𝐶)
8 xpcpropd.a . . 3 (𝜑𝐴𝑉)
9 xpcpropd.c . . 3 (𝜑𝐶𝑉)
10 eqidd 2759 . . 3 (𝜑 → ((Base‘𝐴) × (Base‘𝐶)) = ((Base‘𝐴) × (Base‘𝐶)))
111, 2, 3xpcbas 17507 . . . . 5 ((Base‘𝐴) × (Base‘𝐶)) = (Base‘(𝐴 ×c 𝐶))
12 eqid 2758 . . . . 5 (Hom ‘(𝐴 ×c 𝐶)) = (Hom ‘(𝐴 ×c 𝐶))
131, 11, 4, 5, 12xpchomfval 17508 . . . 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 2759 . . 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 17506 . 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 2758 . . 3 (𝐵 ×c 𝐷) = (𝐵 ×c 𝐷)
18 eqid 2758 . . 3 (Base‘𝐵) = (Base‘𝐵)
19 eqid 2758 . . 3 (Base‘𝐷) = (Base‘𝐷)
20 eqid 2758 . . 3 (Hom ‘𝐵) = (Hom ‘𝐵)
21 eqid 2758 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
22 eqid 2758 . . 3 (comp‘𝐵) = (comp‘𝐵)
23 eqid 2758 . . 3 (comp‘𝐷) = (comp‘𝐷)
24 xpcpropd.b . . 3 (𝜑𝐵𝑉)
25 xpcpropd.d . . 3 (𝜑𝐷𝑉)
26 xpcpropd.1 . . . . 5 (𝜑 → (Homf𝐴) = (Homf𝐵))
2726homfeqbas 17037 . . . 4 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
28 xpcpropd.3 . . . . 5 (𝜑 → (Homf𝐶) = (Homf𝐷))
2928homfeqbas 17037 . . . 4 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
3027, 29xpeq12d 5559 . . 3 (𝜑 → ((Base‘𝐴) × (Base‘𝐶)) = ((Base‘𝐵) × (Base‘𝐷)))
31263ad2ant1 1130 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (Homf𝐴) = (Homf𝐵))
32 xp1st 7731 . . . . . . . 8 (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1st𝑢) ∈ (Base‘𝐴))
33323ad2ant2 1131 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (1st𝑢) ∈ (Base‘𝐴))
34 xp1st 7731 . . . . . . . 8 (𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1st𝑣) ∈ (Base‘𝐴))
35343ad2ant3 1132 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (1st𝑣) ∈ (Base‘𝐴))
362, 4, 20, 31, 33, 35homfeqval 17038 . . . . . 6 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → ((1st𝑢)(Hom ‘𝐴)(1st𝑣)) = ((1st𝑢)(Hom ‘𝐵)(1st𝑣)))
37283ad2ant1 1130 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (Homf𝐶) = (Homf𝐷))
38 xp2nd 7732 . . . . . . . 8 (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2nd𝑢) ∈ (Base‘𝐶))
39383ad2ant2 1131 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (2nd𝑢) ∈ (Base‘𝐶))
40 xp2nd 7732 . . . . . . . 8 (𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2nd𝑣) ∈ (Base‘𝐶))
41403ad2ant3 1132 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (2nd𝑣) ∈ (Base‘𝐶))
423, 5, 21, 37, 39, 41homfeqval 17038 . . . . . 6 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → ((2nd𝑢)(Hom ‘𝐶)(2nd𝑣)) = ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣)))
4336, 42xpeq12d 5559 . . . . 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 7231 . . . 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 2805 . . 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 731 . . . . . . . . 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 731 . . . . . . . . 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 783 . . . . . . . . . . 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 7731 . . . . . . . . . . 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 7731 . . . . . . . . . 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 7732 . . . . . . . . . . 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 7731 . . . . . . . . . 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 775 . . . . . . . . . 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 7731 . . . . . . . . . 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 488 . . . . . . . . . . 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 7738 . . . . . . . . . . . . . . 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 6667 . . . . . . . . . . . . 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 7159 . . . . . . . . . . . . 13 ((1st𝑥)(Hom ‘(𝐴 ×c 𝐶))(2nd𝑥)) = ((Hom ‘(𝐴 ×c 𝐶))‘⟨(1st𝑥), (2nd𝑥)⟩)
6664, 65eqtr4di 2811 . . . . . . . . . . . 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 17509 . . . . . . . . . . . 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 2793 . . . . . . . . . . 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 2854 . . . . . . . . . 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 7731 . . . . . . . . . 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 768 . . . . . . . . . . 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 17509 . . . . . . . . . . 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 2854 . . . . . . . . . 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 7731 . . . . . . . . . 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 17049 . . . . . . . 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 731 . . . . . . . . 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 731 . . . . . . . . 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 7732 . . . . . . . . . 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 7732 . . . . . . . . . 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 7732 . . . . . . . . . 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 7732 . . . . . . . . . 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 7732 . . . . . . . . . 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 17049 . . . . . . . 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 4774 . . . . . . 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 1107 . . . . . 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 7231 . . . . 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 1107 . . . 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 7231 . . 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 17506 . 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 2796 1 (𝜑 → (𝐴 ×c 𝐶) = (𝐵 ×c 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {ctp 4529  ⟨cop 4531   × cxp 5526  ‘cfv 6340  (class class class)co 7156   ∈ cmpo 7158  1st c1st 7697  2nd c2nd 7698  ndxcnx 16551  Basecbs 16554  Hom chom 16647  compcco 16648  Homf chomf 17008  compfccomf 17009   ×c cxpc 17497 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-1o 8118  df-er 8305  df-en 8541  df-dom 8542  df-sdom 8543  df-fin 8544  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-nn 11688  df-2 11750  df-3 11751  df-4 11752  df-5 11753  df-6 11754  df-7 11755  df-8 11756  df-9 11757  df-n0 11948  df-z 12034  df-dec 12151  df-uz 12296  df-fz 12953  df-struct 16556  df-ndx 16557  df-slot 16558  df-base 16560  df-hom 16660  df-cco 16661  df-homf 17012  df-comf 17013  df-xpc 17501 This theorem is referenced by:  curfpropd  17562  oppchofcl  17589
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