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Theorem ringccatidALTV 44251
Description: Lemma for ringccatALTV 44252. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringccatALTV.c 𝐶 = (RingCatALTV‘𝑈)
ringccatidALTV.b 𝐵 = (Base‘𝐶)
Assertion
Ref Expression
ringccatidALTV (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝐵 ↦ ( I ↾ (Base‘𝑥)))))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝑈   𝑥,𝑉

Proof of Theorem ringccatidALTV
Dummy variables 𝑓 𝑔 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringccatidALTV.b . . 3 𝐵 = (Base‘𝐶)
21a1i 11 . 2 (𝑈𝑉𝐵 = (Base‘𝐶))
3 eqidd 2819 . 2 (𝑈𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶))
4 eqidd 2819 . 2 (𝑈𝑉 → (comp‘𝐶) = (comp‘𝐶))
5 ringccatALTV.c . . . 4 𝐶 = (RingCatALTV‘𝑈)
65fvexi 6677 . . 3 𝐶 ∈ V
76a1i 11 . 2 (𝑈𝑉𝐶 ∈ V)
8 biid 262 . 2 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) ↔ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))))
9 simpl 483 . . . . . 6 ((𝑈𝑉𝑥𝐵) → 𝑈𝑉)
105, 1, 9ringcbasALTV 44245 . . . . 5 ((𝑈𝑉𝑥𝐵) → 𝐵 = (𝑈 ∩ Ring))
11 eleq2 2898 . . . . . . . 8 (𝐵 = (𝑈 ∩ Ring) → (𝑥𝐵𝑥 ∈ (𝑈 ∩ Ring)))
12 elin 4166 . . . . . . . . 9 (𝑥 ∈ (𝑈 ∩ Ring) ↔ (𝑥𝑈𝑥 ∈ Ring))
1312simprbi 497 . . . . . . . 8 (𝑥 ∈ (𝑈 ∩ Ring) → 𝑥 ∈ Ring)
1411, 13syl6bi 254 . . . . . . 7 (𝐵 = (𝑈 ∩ Ring) → (𝑥𝐵𝑥 ∈ Ring))
1514com12 32 . . . . . 6 (𝑥𝐵 → (𝐵 = (𝑈 ∩ Ring) → 𝑥 ∈ Ring))
1615adantl 482 . . . . 5 ((𝑈𝑉𝑥𝐵) → (𝐵 = (𝑈 ∩ Ring) → 𝑥 ∈ Ring))
1710, 16mpd 15 . . . 4 ((𝑈𝑉𝑥𝐵) → 𝑥 ∈ Ring)
18 eqid 2818 . . . . 5 (Base‘𝑥) = (Base‘𝑥)
1918idrhm 19412 . . . 4 (𝑥 ∈ Ring → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
2017, 19syl 17 . . 3 ((𝑈𝑉𝑥𝐵) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
21 eqid 2818 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
22 simpr 485 . . . 4 ((𝑈𝑉𝑥𝐵) → 𝑥𝐵)
235, 1, 9, 21, 22, 22ringchomALTV 44247 . . 3 ((𝑈𝑉𝑥𝐵) → (𝑥(Hom ‘𝐶)𝑥) = (𝑥 RingHom 𝑥))
2420, 23eleqtrrd 2913 . 2 ((𝑈𝑉𝑥𝐵) → ( I ↾ (Base‘𝑥)) ∈ (𝑥(Hom ‘𝐶)𝑥))
25 simpl 483 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑈𝑉)
26 eqid 2818 . . . 4 (comp‘𝐶) = (comp‘𝐶)
27 simpl 483 . . . . . 6 ((𝑤𝐵𝑥𝐵) → 𝑤𝐵)
28273ad2ant1 1125 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑤𝐵)
2928adantl 482 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑤𝐵)
30 simpr 485 . . . . . 6 ((𝑤𝐵𝑥𝐵) → 𝑥𝐵)
31303ad2ant1 1125 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥𝐵)
3231adantl 482 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑥𝐵)
33 simp1 1128 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → 𝑈𝑉)
34273ad2ant3 1127 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → 𝑤𝐵)
35303ad2ant3 1127 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → 𝑥𝐵)
365, 1, 33, 21, 34, 35ringchomALTV 44247 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑤(Hom ‘𝐶)𝑥) = (𝑤 RingHom 𝑥))
3736eleq2d 2895 . . . . . . . . . . 11 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ↔ 𝑓 ∈ (𝑤 RingHom 𝑥)))
3837biimpd 230 . . . . . . . . . 10 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓 ∈ (𝑤 RingHom 𝑥)))
39383exp 1111 . . . . . . . . 9 (𝑈𝑉 → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓 ∈ (𝑤 RingHom 𝑥)))))
4039com14 96 . . . . . . . 8 (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉𝑓 ∈ (𝑤 RingHom 𝑥)))))
41403ad2ant1 1125 . . . . . . 7 ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉𝑓 ∈ (𝑤 RingHom 𝑥)))))
4241com13 88 . . . . . 6 ((𝑤𝐵𝑥𝐵) → ((𝑦𝐵𝑧𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈𝑉𝑓 ∈ (𝑤 RingHom 𝑥)))))
43423imp 1103 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑉𝑓 ∈ (𝑤 RingHom 𝑥)))
4443impcom 408 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓 ∈ (𝑤 RingHom 𝑥))
4520expcom 414 . . . . . . 7 (𝑥𝐵 → (𝑈𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥)))
4645adantl 482 . . . . . 6 ((𝑤𝐵𝑥𝐵) → (𝑈𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥)))
47463ad2ant1 1125 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥)))
4847impcom 408 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
495, 1, 25, 26, 29, 32, 32, 44, 48ringccoALTV 44250 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ (Base‘𝑥))(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = (( I ↾ (Base‘𝑥)) ∘ 𝑓))
50 simpl 483 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑤𝐵𝑥𝐵)) → 𝑈𝑉)
51 simprl 767 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑤𝐵𝑥𝐵)) → 𝑤𝐵)
52 simprr 769 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑤𝐵𝑥𝐵)) → 𝑥𝐵)
535, 1, 50, 21, 51, 52elringchomALTV 44248 . . . . . . . . . . 11 ((𝑈𝑉 ∧ (𝑤𝐵𝑥𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓:(Base‘𝑤)⟶(Base‘𝑥)))
5453ex 413 . . . . . . . . . 10 (𝑈𝑉 → ((𝑤𝐵𝑥𝐵) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓:(Base‘𝑤)⟶(Base‘𝑥))))
5554com13 88 . . . . . . . . 9 (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉𝑓:(Base‘𝑤)⟶(Base‘𝑥))))
56 fcoi2 6546 . . . . . . . . 9 (𝑓:(Base‘𝑤)⟶(Base‘𝑥) → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)
5755, 56syl8 76 . . . . . . . 8 (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)))
58573ad2ant1 1125 . . . . . . 7 ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)))
5958com12 32 . . . . . 6 ((𝑤𝐵𝑥𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)))
6059a1d 25 . . . . 5 ((𝑤𝐵𝑥𝐵) → ((𝑦𝐵𝑧𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓))))
61603imp 1103 . . . 4 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓))
6261impcom 408 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)
6349, 62eqtrd 2853 . 2 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ (Base‘𝑥))(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
64 simp3 1130 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑈𝑉)
6530adantr 481 . . . . . . . . . 10 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → 𝑥𝐵)
66653ad2ant2 1126 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑥𝐵)
67 simprl 767 . . . . . . . . . 10 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → 𝑦𝐵)
68673ad2ant2 1126 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑦𝐵)
6946adantr 481 . . . . . . . . . . 11 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑈𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥)))
7069a1i 11 . . . . . . . . . 10 (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑈𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))))
71703imp 1103 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
72 simpl 483 . . . . . . . . . . . . . . 15 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → 𝑈𝑉)
7365adantl 482 . . . . . . . . . . . . . . 15 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → 𝑥𝐵)
7467adantl 482 . . . . . . . . . . . . . . 15 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → 𝑦𝐵)
755, 1, 72, 21, 73, 74ringchomALTV 44247 . . . . . . . . . . . . . 14 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 RingHom 𝑦))
7675eleq2d 2895 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑔 ∈ (𝑥 RingHom 𝑦)))
7776biimpd 230 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RingHom 𝑦)))
7877ex 413 . . . . . . . . . . 11 (𝑈𝑉 → (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RingHom 𝑦))))
7978com13 88 . . . . . . . . . 10 (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑈𝑉𝑔 ∈ (𝑥 RingHom 𝑦))))
80793imp 1103 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑔 ∈ (𝑥 RingHom 𝑦))
815, 1, 64, 26, 66, 66, 68, 71, 80ringccoALTV 44250 . . . . . . . 8 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = (𝑔 ∘ ( I ↾ (Base‘𝑥))))
825, 1, 72, 21, 73, 74elringchomALTV 44248 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦)))
8382ex 413 . . . . . . . . . . 11 (𝑈𝑉 → (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦))))
8483com13 88 . . . . . . . . . 10 (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑈𝑉𝑔:(Base‘𝑥)⟶(Base‘𝑦))))
85843imp 1103 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦))
86 fcoi1 6545 . . . . . . . . 9 (𝑔:(Base‘𝑥)⟶(Base‘𝑦) → (𝑔 ∘ ( I ↾ (Base‘𝑥))) = 𝑔)
8785, 86syl 17 . . . . . . . 8 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → (𝑔 ∘ ( I ↾ (Base‘𝑥))) = 𝑔)
8881, 87eqtrd 2853 . . . . . . 7 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)
89883exp 1111 . . . . . 6 (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑈𝑉 → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)))
90893ad2ant2 1126 . . . . 5 ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑈𝑉 → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)))
9190expdcom 415 . . . 4 ((𝑤𝐵𝑥𝐵) → ((𝑦𝐵𝑧𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈𝑉 → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔))))
92913imp 1103 . . 3 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑉 → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔))
9392impcom 408 . 2 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)
94 simpl 483 . . . . . . . . . . . . . 14 ((𝑦𝐵𝑧𝐵) → 𝑦𝐵)
95943ad2ant2 1126 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → 𝑦𝐵)
965, 1, 33, 21, 35, 95ringchomALTV 44247 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 RingHom 𝑦))
9796eleq2d 2895 . . . . . . . . . . 11 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑔 ∈ (𝑥 RingHom 𝑦)))
9897biimpd 230 . . . . . . . . . 10 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RingHom 𝑦)))
99983exp 1111 . . . . . . . . 9 (𝑈𝑉 → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RingHom 𝑦)))))
10099com14 96 . . . . . . . 8 (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉𝑔 ∈ (𝑥 RingHom 𝑦)))))
1011003ad2ant2 1126 . . . . . . 7 ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉𝑔 ∈ (𝑥 RingHom 𝑦)))))
102101com13 88 . . . . . 6 ((𝑤𝐵𝑥𝐵) → ((𝑦𝐵𝑧𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈𝑉𝑔 ∈ (𝑥 RingHom 𝑦)))))
1031023imp 1103 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑉𝑔 ∈ (𝑥 RingHom 𝑦)))
104103impcom 408 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑥 RingHom 𝑦))
105 rhmco 19418 . . . 4 ((𝑔 ∈ (𝑥 RingHom 𝑦) ∧ 𝑓 ∈ (𝑤 RingHom 𝑥)) → (𝑔𝑓) ∈ (𝑤 RingHom 𝑦))
106104, 44, 105syl2anc 584 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔𝑓) ∈ (𝑤 RingHom 𝑦))
107943ad2ant2 1126 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦𝐵)
108107adantl 482 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑦𝐵)
1095, 1, 25, 26, 29, 32, 108, 44, 104ringccoALTV 44250 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓) = (𝑔𝑓))
1105, 1, 25, 21, 29, 108ringchomALTV 44247 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑤(Hom ‘𝐶)𝑦) = (𝑤 RingHom 𝑦))
111106, 109, 1103eltr4d 2925 . 2 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓) ∈ (𝑤(Hom ‘𝐶)𝑦))
112 coass 6111 . . . 4 ((𝑔) ∘ 𝑓) = ( ∘ (𝑔𝑓))
113 simp2r 1192 . . . . . 6 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧𝐵)
114113adantl 482 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑧𝐵)
115 simp2r 1192 . . . . . . . . . . . . . . 15 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → 𝑧𝐵)
1165, 1, 33, 21, 95, 115ringchomALTV 44247 . . . . . . . . . . . . . 14 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑦(Hom ‘𝐶)𝑧) = (𝑦 RingHom 𝑧))
117116eleq2d 2895 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → ( ∈ (𝑦(Hom ‘𝐶)𝑧) ↔ ∈ (𝑦 RingHom 𝑧)))
118117biimpd 230 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → ( ∈ (𝑦(Hom ‘𝐶)𝑧) → ∈ (𝑦 RingHom 𝑧)))
1191183exp 1111 . . . . . . . . . . 11 (𝑈𝑉 → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → ( ∈ (𝑦(Hom ‘𝐶)𝑧) → ∈ (𝑦 RingHom 𝑧)))))
120119com14 96 . . . . . . . . . 10 ( ∈ (𝑦(Hom ‘𝐶)𝑧) → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉 ∈ (𝑦 RingHom 𝑧)))))
1211203ad2ant3 1127 . . . . . . . . 9 ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉 ∈ (𝑦 RingHom 𝑧)))))
122121com13 88 . . . . . . . 8 ((𝑤𝐵𝑥𝐵) → ((𝑦𝐵𝑧𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈𝑉 ∈ (𝑦 RingHom 𝑧)))))
1231223imp 1103 . . . . . . 7 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑉 ∈ (𝑦 RingHom 𝑧)))
124123impcom 408 . . . . . 6 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ∈ (𝑦 RingHom 𝑧))
125 rhmco 19418 . . . . . 6 (( ∈ (𝑦 RingHom 𝑧) ∧ 𝑔 ∈ (𝑥 RingHom 𝑦)) → (𝑔) ∈ (𝑥 RingHom 𝑧))
126124, 104, 125syl2anc 584 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔) ∈ (𝑥 RingHom 𝑧))
1275, 1, 25, 26, 29, 32, 114, 44, 126ringccoALTV 44250 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = ((𝑔) ∘ 𝑓))
1285, 1, 25, 26, 29, 108, 114, 106, 124ringccoALTV 44250 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔𝑓)) = ( ∘ (𝑔𝑓)))
129112, 127, 1283eqtr4a 2879 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = ((⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔𝑓)))
1305, 1, 25, 26, 32, 108, 114, 104, 124ringccoALTV 44250 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔) = (𝑔))
131130oveq1d 7160 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (((⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = ((𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓))
132109oveq2d 7161 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓)) = ((⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔𝑓)))
133129, 131, 1323eqtr4d 2863 . 2 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (((⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = ((⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓)))
1342, 3, 4, 7, 8, 24, 63, 93, 111, 133iscatd2 16940 1 (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝐵 ↦ ( I ↾ (Base‘𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  Vcvv 3492  cin 3932  cop 4563  cmpt 5137   I cid 5452  cres 5550  ccom 5552  wf 6344  cfv 6348  (class class class)co 7145  Basecbs 16471  Hom chom 16564  compcco 16565  Catccat 16923  Idccid 16924  Ringcrg 19226   RingHom crh 19393  RingCatALTVcringcALTV 44203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12881  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-plusg 16566  df-hom 16577  df-cco 16578  df-0g 16703  df-cat 16927  df-cid 16928  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-mhm 17944  df-grp 18044  df-ghm 18294  df-mgp 19169  df-ur 19181  df-ring 19228  df-rnghom 19396  df-ringcALTV 44205
This theorem is referenced by:  ringccatALTV  44252  ringcidALTV  44253
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