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Theorem ringccatidALTV 44155
Description: Lemma for ringccatALTV 44156. (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 2827 . 2 (𝑈𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶))
4 eqidd 2827 . 2 (𝑈𝑉 → (comp‘𝐶) = (comp‘𝐶))
5 ringccatALTV.c . . . 4 𝐶 = (RingCatALTV‘𝑈)
65fvexi 6681 . . 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 44149 . . . . 5 ((𝑈𝑉𝑥𝐵) → 𝐵 = (𝑈 ∩ Ring))
11 eleq2 2906 . . . . . . . 8 (𝐵 = (𝑈 ∩ Ring) → (𝑥𝐵𝑥 ∈ (𝑈 ∩ Ring)))
12 elin 4173 . . . . . . . . 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 2826 . . . . 5 (Base‘𝑥) = (Base‘𝑥)
1918idrhm 19403 . . . 4 (𝑥 ∈ Ring → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
2017, 19syl 17 . . 3 ((𝑈𝑉𝑥𝐵) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
21 eqid 2826 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
22 simpr 485 . . . 4 ((𝑈𝑉𝑥𝐵) → 𝑥𝐵)
235, 1, 9, 21, 22, 22ringchomALTV 44151 . . 3 ((𝑈𝑉𝑥𝐵) → (𝑥(Hom ‘𝐶)𝑥) = (𝑥 RingHom 𝑥))
2420, 23eleqtrrd 2921 . 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 2826 . . . 4 (comp‘𝐶) = (comp‘𝐶)
27 simpl 483 . . . . . 6 ((𝑤𝐵𝑥𝐵) → 𝑤𝐵)
28273ad2ant1 1127 . . . . 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 1127 . . . . 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 1130 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → 𝑈𝑉)
34273ad2ant3 1129 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → 𝑤𝐵)
35303ad2ant3 1129 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → 𝑥𝐵)
365, 1, 33, 21, 34, 35ringchomALTV 44151 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑤(Hom ‘𝐶)𝑥) = (𝑤 RingHom 𝑥))
3736eleq2d 2903 . . . . . . . . . . 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 1113 . . . . . . . . 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 1127 . . . . . . 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 1105 . . . . 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 1127 . . . . 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 44154 . . 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 44152 . . . . . . . . . . 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 6550 . . . . . . . . 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 1127 . . . . . . 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 1105 . . . 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 2861 . 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 1132 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑈𝑉)
6530adantr 481 . . . . . . . . . 10 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → 𝑥𝐵)
66653ad2ant2 1128 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑥𝐵)
67 simprl 767 . . . . . . . . . 10 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → 𝑦𝐵)
68673ad2ant2 1128 . . . . . . . . 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 1105 . . . . . . . . 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 44151 . . . . . . . . . . . . . 14 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 RingHom 𝑦))
7675eleq2d 2903 . . . . . . . . . . . . 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 1105 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑔 ∈ (𝑥 RingHom 𝑦))
815, 1, 64, 26, 66, 66, 68, 71, 80ringccoALTV 44154 . . . . . . . 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 44152 . . . . . . . . . . . 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 1105 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦))
86 fcoi1 6549 . . . . . . . . 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 2861 . . . . . . 7 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)
89883exp 1113 . . . . . 6 (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑈𝑉 → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)))
90893ad2ant2 1128 . . . . 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 1105 . . 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 1128 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → 𝑦𝐵)
965, 1, 33, 21, 35, 95ringchomALTV 44151 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 RingHom 𝑦))
9796eleq2d 2903 . . . . . . . . . . 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 1113 . . . . . . . . 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 1128 . . . . . . 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 1105 . . . . 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 19409 . . . 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 1128 . . . . 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 44154 . . 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 44151 . . 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 2933 . 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 6116 . . . 4 ((𝑔) ∘ 𝑓) = ( ∘ (𝑔𝑓))
113 simp2r 1194 . . . . . 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 1194 . . . . . . . . . . . . . . 15 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → 𝑧𝐵)
1165, 1, 33, 21, 95, 115ringchomALTV 44151 . . . . . . . . . . . . . 14 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑦(Hom ‘𝐶)𝑧) = (𝑦 RingHom 𝑧))
117116eleq2d 2903 . . . . . . . . . . . . 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 1113 . . . . . . . . . . 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 1129 . . . . . . . . 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 1105 . . . . . . 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 19409 . . . . . 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 44154 . . . 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 44154 . . . 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 2887 . . 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 44154 . . . 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 7163 . . 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 7164 . . 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 2871 . 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 16942 1 (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝐵 ↦ ( I ↾ (Base‘𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107  Vcvv 3500  cin 3939  cop 4570  cmpt 5143   I cid 5458  cres 5556  ccom 5558  wf 6348  cfv 6352  (class class class)co 7148  Basecbs 16473  Hom chom 16566  compcco 16567  Catccat 16925  Idccid 16926  Ringcrg 19217   RingHom crh 19384  RingCatALTVcringcALTV 44107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-3 11690  df-4 11691  df-5 11692  df-6 11693  df-7 11694  df-8 11695  df-9 11696  df-n0 11887  df-z 11971  df-dec 12088  df-uz 12233  df-fz 12883  df-struct 16475  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-plusg 16568  df-hom 16579  df-cco 16580  df-0g 16705  df-cat 16929  df-cid 16930  df-mgm 17842  df-sgrp 17890  df-mnd 17901  df-mhm 17944  df-grp 18036  df-ghm 18286  df-mgp 19160  df-ur 19172  df-ring 19219  df-rnghom 19387  df-ringcALTV 44109
This theorem is referenced by:  ringccatALTV  44156  ringcidALTV  44157
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