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

Theorem xpcpropd 18097
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 2736 . . 3 (𝐴 ×c 𝐶) = (𝐴 ×c 𝐶)
2 eqid 2736 . . 3 (Base‘𝐴) = (Base‘𝐴)
3 eqid 2736 . . 3 (Base‘𝐶) = (Base‘𝐶)
4 eqid 2736 . . 3 (Hom ‘𝐴) = (Hom ‘𝐴)
5 eqid 2736 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2736 . . 3 (comp‘𝐴) = (comp‘𝐴)
7 eqid 2736 . . 3 (comp‘𝐶) = (comp‘𝐶)
8 xpcpropd.a . . 3 (𝜑𝐴𝑉)
9 xpcpropd.c . . 3 (𝜑𝐶𝑉)
10 eqidd 2737 . . 3 (𝜑 → ((Base‘𝐴) × (Base‘𝐶)) = ((Base‘𝐴) × (Base‘𝐶)))
111, 2, 3xpcbas 18066 . . . . 5 ((Base‘𝐴) × (Base‘𝐶)) = (Base‘(𝐴 ×c 𝐶))
12 eqid 2736 . . . . 5 (Hom ‘(𝐴 ×c 𝐶)) = (Hom ‘(𝐴 ×c 𝐶))
131, 11, 4, 5, 12xpchomfval 18067 . . . 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 2737 . . 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 18065 . 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 2736 . . 3 (𝐵 ×c 𝐷) = (𝐵 ×c 𝐷)
18 eqid 2736 . . 3 (Base‘𝐵) = (Base‘𝐵)
19 eqid 2736 . . 3 (Base‘𝐷) = (Base‘𝐷)
20 eqid 2736 . . 3 (Hom ‘𝐵) = (Hom ‘𝐵)
21 eqid 2736 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
22 eqid 2736 . . 3 (comp‘𝐵) = (comp‘𝐵)
23 eqid 2736 . . 3 (comp‘𝐷) = (comp‘𝐷)
24 xpcpropd.b . . 3 (𝜑𝐵𝑉)
25 xpcpropd.d . . 3 (𝜑𝐷𝑉)
26 xpcpropd.1 . . . . 5 (𝜑 → (Homf𝐴) = (Homf𝐵))
2726homfeqbas 17576 . . . 4 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
28 xpcpropd.3 . . . . 5 (𝜑 → (Homf𝐶) = (Homf𝐷))
2928homfeqbas 17576 . . . 4 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
3027, 29xpeq12d 5664 . . 3 (𝜑 → ((Base‘𝐴) × (Base‘𝐶)) = ((Base‘𝐵) × (Base‘𝐷)))
31263ad2ant1 1133 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (Homf𝐴) = (Homf𝐵))
32 xp1st 7953 . . . . . . . 8 (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1st𝑢) ∈ (Base‘𝐴))
33323ad2ant2 1134 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (1st𝑢) ∈ (Base‘𝐴))
34 xp1st 7953 . . . . . . . 8 (𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (1st𝑣) ∈ (Base‘𝐴))
35343ad2ant3 1135 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (1st𝑣) ∈ (Base‘𝐴))
362, 4, 20, 31, 33, 35homfeqval 17577 . . . . . 6 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → ((1st𝑢)(Hom ‘𝐴)(1st𝑣)) = ((1st𝑢)(Hom ‘𝐵)(1st𝑣)))
37283ad2ant1 1133 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (Homf𝐶) = (Homf𝐷))
38 xp2nd 7954 . . . . . . . 8 (𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2nd𝑢) ∈ (Base‘𝐶))
39383ad2ant2 1134 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (2nd𝑢) ∈ (Base‘𝐶))
40 xp2nd 7954 . . . . . . . 8 (𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶)) → (2nd𝑣) ∈ (Base‘𝐶))
41403ad2ant3 1135 . . . . . . 7 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → (2nd𝑣) ∈ (Base‘𝐶))
423, 5, 21, 37, 39, 41homfeqval 17577 . . . . . 6 ((𝜑𝑢 ∈ ((Base‘𝐴) × (Base‘𝐶)) ∧ 𝑣 ∈ ((Base‘𝐴) × (Base‘𝐶))) → ((2nd𝑢)(Hom ‘𝐶)(2nd𝑣)) = ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣)))
4336, 42xpeq12d 5664 . . . . 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 7434 . . . 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 2788 . . 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 730 . . . . . . . . 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 730 . . . . . . . . 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 782 . . . . . . . . . . 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 7953 . . . . . . . . . . 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 7953 . . . . . . . . . 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 7954 . . . . . . . . . . 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 7953 . . . . . . . . . 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 774 . . . . . . . . . 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 7953 . . . . . . . . . 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 485 . . . . . . . . . . 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 7960 . . . . . . . . . . . . . . 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 6846 . . . . . . . . . . . . 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 7360 . . . . . . . . . . . . 13 ((1st𝑥)(Hom ‘(𝐴 ×c 𝐶))(2nd𝑥)) = ((Hom ‘(𝐴 ×c 𝐶))‘⟨(1st𝑥), (2nd𝑥)⟩)
6664, 65eqtr4di 2794 . . . . . . . . . . . 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 18068 . . . . . . . . . . . 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 2776 . . . . . . . . . . 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 2840 . . . . . . . . . 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 7953 . . . . . . . . . 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 767 . . . . . . . . . . 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 18068 . . . . . . . . . . 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 2840 . . . . . . . . . 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 7953 . . . . . . . . . 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 17588 . . . . . . . 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 730 . . . . . . . . 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 730 . . . . . . . . 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 7954 . . . . . . . . . 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 7954 . . . . . . . . . 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 7954 . . . . . . . . . 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 7954 . . . . . . . . . 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 7954 . . . . . . . . . 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 17588 . . . . . . . 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 4838 . . . . . . 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 1110 . . . . . 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 7434 . . . . 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 1110 . . . 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 7434 . . 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 18065 . 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 2779 1 (𝜑 → (𝐴 ×c 𝐶) = (𝐵 ×c 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  {ctp 4590  cop 4592   × cxp 5631  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  ndxcnx 17065  Basecbs 17083  Hom chom 17144  compcco 17145  Homf chomf 17546  compfccomf 17547   ×c cxpc 18056
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-slot 17054  df-ndx 17066  df-base 17084  df-hom 17157  df-cco 17158  df-homf 17550  df-comf 17551  df-xpc 18060
This theorem is referenced by:  curfpropd  18122  oppchofcl  18149
  Copyright terms: Public domain W3C validator