Theorem ringccatidALTV 48149

Description: Lemma for ringccatALTV 48150. (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	⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ ( I ↾ (Base‘𝑥)))))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝑈   𝑥,𝑉

Proof of Theorem ringccatidALTV

Dummy variables 𝑓 𝑔 ℎ 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ringccatidALTV.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2	1	a1i 11	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐵 = (Base‘𝐶))
3		eqidd 2735	. 2 ⊢ (𝑈 ∈ 𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶))
4		eqidd 2735	. 2 ⊢ (𝑈 ∈ 𝑉 → (comp‘𝐶) = (comp‘𝐶))
5		ringccatALTV.c	. . . 4 ⊢ 𝐶 = (RingCatALTV‘𝑈)
6	5	fvexi 6920	. . 3 ⊢ 𝐶 ∈ V
7	6	a1i 11	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ V)
8		biid 261	. 2 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) ↔ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))))
9		simpl 482	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑈 ∈ 𝑉)
10	5, 1, 9	ringcbasALTV 48143	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝐵 = (𝑈 ∩ Ring))
11		eleq2 2827	. . . . . . . 8 ⊢ (𝐵 = (𝑈 ∩ Ring) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (𝑈 ∩ Ring)))
12		elin 3978	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑈 ∩ Ring) ↔ (𝑥 ∈ 𝑈 ∧ 𝑥 ∈ Ring))
13	12	simprbi 496	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑈 ∩ Ring) → 𝑥 ∈ Ring)
14	11, 13	biimtrdi 253	. . . . . . 7 ⊢ (𝐵 = (𝑈 ∩ Ring) → (𝑥 ∈ 𝐵 → 𝑥 ∈ Ring))
15	14	com12 32	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → (𝐵 = (𝑈 ∩ Ring) → 𝑥 ∈ Ring))
16	15	adantl 481	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → (𝐵 = (𝑈 ∩ Ring) → 𝑥 ∈ Ring))
17	10, 16	mpd 15	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ Ring)
18		eqid 2734	. . . . 5 ⊢ (Base‘𝑥) = (Base‘𝑥)
19	18	idrhm 20506	. . . 4 ⊢ (𝑥 ∈ Ring → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
20	17, 19	syl 17	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
21		eqid 2734	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
22		simpr 484	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
23	5, 1, 9, 21, 22, 22	ringchomALTV 48145	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → (𝑥(Hom ‘𝐶)𝑥) = (𝑥 RingHom 𝑥))
24	20, 23	eleqtrrd 2841	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → ( I ↾ (Base‘𝑥)) ∈ (𝑥(Hom ‘𝐶)𝑥))
25		simpl 482	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑈 ∈ 𝑉)
26		eqid 2734	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
27		simpl 482	. . . . . 6 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑤 ∈ 𝐵)
28	27	3ad2ant1 1132	. . . . 5 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑤 ∈ 𝐵)
29	28	adantl 481	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑤 ∈ 𝐵)
30		simpr 484	. . . . . 6 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
31	30	3ad2ant1 1132	. . . . 5 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥 ∈ 𝐵)
32	31	adantl 481	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑥 ∈ 𝐵)
33		simp1 1135	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑈 ∈ 𝑉)
34	27	3ad2ant3 1134	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
35	30	3ad2ant3 1134	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
36	5, 1, 33, 21, 34, 35	ringchomALTV 48145	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑤(Hom ‘𝐶)𝑥) = (𝑤 RingHom 𝑥))
37	36	eleq2d 2824	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ↔ 𝑓 ∈ (𝑤 RingHom 𝑥)))
38	37	biimpd 229	. . . . . . . . . 10 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓 ∈ (𝑤 RingHom 𝑥)))
39	38	3exp 1118	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑉 → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓 ∈ (𝑤 RingHom 𝑥)))))
40	39	com14 96	. . . . . . . 8 ⊢ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → 𝑓 ∈ (𝑤 RingHom 𝑥)))))
41	40	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → 𝑓 ∈ (𝑤 RingHom 𝑥)))))
42	41	com13 88	. . . . . 6 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈 ∈ 𝑉 → 𝑓 ∈ (𝑤 RingHom 𝑥)))))
43	42	3imp 1110	. . . . 5 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈 ∈ 𝑉 → 𝑓 ∈ (𝑤 RingHom 𝑥)))
44	43	impcom 407	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓 ∈ (𝑤 RingHom 𝑥))
45	20	expcom 413	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (𝑈 ∈ 𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥)))
46	45	adantl 481	. . . . . 6 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥)))
47	46	3ad2ant1 1132	. . . . 5 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈 ∈ 𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥)))
48	47	impcom 407	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
49	5, 1, 25, 26, 29, 32, 32, 44, 48	ringccoALTV 48148	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ (Base‘𝑥))(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = (( I ↾ (Base‘𝑥)) ∘ 𝑓))
50		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑈 ∈ 𝑉)
51		simprl 771	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
52		simprr 773	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
53	5, 1, 50, 21, 51, 52	elringchomALTV 48146	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓:(Base‘𝑤)⟶(Base‘𝑥)))
54	53	ex 412	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝑉 → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓:(Base‘𝑤)⟶(Base‘𝑥))))
55	54	com13 88	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → 𝑓:(Base‘𝑤)⟶(Base‘𝑥))))
56		fcoi2 6783	. . . . . . . . 9 ⊢ (𝑓:(Base‘𝑤)⟶(Base‘𝑥) → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)
57	55, 56	syl8 76	. . . . . . . 8 ⊢ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)))
58	57	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)))
59	58	com12 32	. . . . . 6 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈 ∈ 𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)))
60	59	a1d 25	. . . . 5 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈 ∈ 𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓))))
61	60	3imp 1110	. . . 4 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈 ∈ 𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓))
62	61	impcom 407	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)
63	49, 62	eqtrd 2774	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ (Base‘𝑥))(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
64		simp3 1137	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → 𝑈 ∈ 𝑉)
65	30	adantr 480	. . . . . . . . . 10 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
66	65	3ad2ant2 1133	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → 𝑥 ∈ 𝐵)
67		simprl 771	. . . . . . . . . 10 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
68	67	3ad2ant2 1133	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → 𝑦 ∈ 𝐵)
69	46	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑈 ∈ 𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥)))
70	69	a1i 11	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑈 ∈ 𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))))
71	70	3imp 1110	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
72		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → 𝑈 ∈ 𝑉)
73	65	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → 𝑥 ∈ 𝐵)
74	67	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → 𝑦 ∈ 𝐵)
75	5, 1, 72, 21, 73, 74	ringchomALTV 48145	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 RingHom 𝑦))
76	75	eleq2d 2824	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑔 ∈ (𝑥 RingHom 𝑦)))
77	76	biimpd 229	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RingHom 𝑦)))
78	77	ex 412	. . . . . . . . . . 11 ⊢ (𝑈 ∈ 𝑉 → (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RingHom 𝑦))))
79	78	com13 88	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑈 ∈ 𝑉 → 𝑔 ∈ (𝑥 RingHom 𝑦))))
80	79	3imp 1110	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → 𝑔 ∈ (𝑥 RingHom 𝑦))
81	5, 1, 64, 26, 66, 66, 68, 71, 80	ringccoALTV 48148	. . . . . . . 8 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = (𝑔 ∘ ( I ↾ (Base‘𝑥))))
82	5, 1, 72, 21, 73, 74	elringchomALTV 48146	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦)))
83	82	ex 412	. . . . . . . . . . 11 ⊢ (𝑈 ∈ 𝑉 → (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦))))
84	83	com13 88	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑈 ∈ 𝑉 → 𝑔:(Base‘𝑥)⟶(Base‘𝑦))))
85	84	3imp 1110	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦))
86		fcoi1 6782	. . . . . . . . 9 ⊢ (𝑔:(Base‘𝑥)⟶(Base‘𝑦) → (𝑔 ∘ ( I ↾ (Base‘𝑥))) = 𝑔)
87	85, 86	syl 17	. . . . . . . 8 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → (𝑔 ∘ ( I ↾ (Base‘𝑥))) = 𝑔)
88	81, 87	eqtrd 2774	. . . . . . 7 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)
89	88	3exp 1118	. . . . . 6 ⊢ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑈 ∈ 𝑉 → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)))
90	89	3ad2ant2 1133	. . . . 5 ⊢ ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑈 ∈ 𝑉 → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)))
91	90	expdcom 414	. . . 4 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈 ∈ 𝑉 → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔))))
92	91	3imp 1110	. . 3 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈 ∈ 𝑉 → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔))
93	92	impcom 407	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)
94		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → 𝑦 ∈ 𝐵)
95	94	3ad2ant2 1133	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
96	5, 1, 33, 21, 35, 95	ringchomALTV 48145	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 RingHom 𝑦))
97	96	eleq2d 2824	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑔 ∈ (𝑥 RingHom 𝑦)))
98	97	biimpd 229	. . . . . . . . . 10 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RingHom 𝑦)))
99	98	3exp 1118	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑉 → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RingHom 𝑦)))))
100	99	com14 96	. . . . . . . 8 ⊢ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → 𝑔 ∈ (𝑥 RingHom 𝑦)))))
101	100	3ad2ant2 1133	. . . . . . 7 ⊢ ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → 𝑔 ∈ (𝑥 RingHom 𝑦)))))
102	101	com13 88	. . . . . 6 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈 ∈ 𝑉 → 𝑔 ∈ (𝑥 RingHom 𝑦)))))
103	102	3imp 1110	. . . . 5 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈 ∈ 𝑉 → 𝑔 ∈ (𝑥 RingHom 𝑦)))
104	103	impcom 407	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑥 RingHom 𝑦))
105		rhmco 20517	. . . 4 ⊢ ((𝑔 ∈ (𝑥 RingHom 𝑦) ∧ 𝑓 ∈ (𝑤 RingHom 𝑥)) → (𝑔 ∘ 𝑓) ∈ (𝑤 RingHom 𝑦))
106	104, 44, 105	syl2anc 584	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔 ∘ 𝑓) ∈ (𝑤 RingHom 𝑦))
107	94	3ad2ant2 1133	. . . . 5 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦 ∈ 𝐵)
108	107	adantl 481	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑦 ∈ 𝐵)
109	5, 1, 25, 26, 29, 32, 108, 44, 104	ringccoALTV 48148	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓) = (𝑔 ∘ 𝑓))
110	5, 1, 25, 21, 29, 108	ringchomALTV 48145	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑤(Hom ‘𝐶)𝑦) = (𝑤 RingHom 𝑦))
111	106, 109, 110	3eltr4d 2853	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓) ∈ (𝑤(Hom ‘𝐶)𝑦))
112		coass 6286	. . . 4 ⊢ ((ℎ ∘ 𝑔) ∘ 𝑓) = (ℎ ∘ (𝑔 ∘ 𝑓))
113		simp2r 1199	. . . . . 6 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧 ∈ 𝐵)
114	113	adantl 481	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑧 ∈ 𝐵)
115		simp2r 1199	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑧 ∈ 𝐵)
116	5, 1, 33, 21, 95, 115	ringchomALTV 48145	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(Hom ‘𝐶)𝑧) = (𝑦 RingHom 𝑧))
117	116	eleq2d 2824	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (ℎ ∈ (𝑦(Hom ‘𝐶)𝑧) ↔ ℎ ∈ (𝑦 RingHom 𝑧)))
118	117	biimpd 229	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (ℎ ∈ (𝑦(Hom ‘𝐶)𝑧) → ℎ ∈ (𝑦 RingHom 𝑧)))
119	118	3exp 1118	. . . . . . . . . . 11 ⊢ (𝑈 ∈ 𝑉 → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (ℎ ∈ (𝑦(Hom ‘𝐶)𝑧) → ℎ ∈ (𝑦 RingHom 𝑧)))))
120	119	com14 96	. . . . . . . . . 10 ⊢ (ℎ ∈ (𝑦(Hom ‘𝐶)𝑧) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → ℎ ∈ (𝑦 RingHom 𝑧)))))
121	120	3ad2ant3 1134	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → ℎ ∈ (𝑦 RingHom 𝑧)))))
122	121	com13 88	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈 ∈ 𝑉 → ℎ ∈ (𝑦 RingHom 𝑧)))))
123	122	3imp 1110	. . . . . . 7 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈 ∈ 𝑉 → ℎ ∈ (𝑦 RingHom 𝑧)))
124	123	impcom 407	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ℎ ∈ (𝑦 RingHom 𝑧))
125		rhmco 20517	. . . . . 6 ⊢ ((ℎ ∈ (𝑦 RingHom 𝑧) ∧ 𝑔 ∈ (𝑥 RingHom 𝑦)) → (ℎ ∘ 𝑔) ∈ (𝑥 RingHom 𝑧))
126	124, 104, 125	syl2anc 584	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ ∘ 𝑔) ∈ (𝑥 RingHom 𝑧))
127	5, 1, 25, 26, 29, 32, 114, 44, 126	ringccoALTV 48148	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((ℎ ∘ 𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = ((ℎ ∘ 𝑔) ∘ 𝑓))
128	5, 1, 25, 26, 29, 108, 114, 106, 124	ringccoALTV 48148	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ(⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔 ∘ 𝑓)) = (ℎ ∘ (𝑔 ∘ 𝑓)))
129	112, 127, 128	3eqtr4a 2800	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((ℎ ∘ 𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = (ℎ(⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔 ∘ 𝑓)))
130	5, 1, 25, 26, 32, 108, 114, 104, 124	ringccoALTV 48148	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔) = (ℎ ∘ 𝑔))
131	130	oveq1d 7445	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = ((ℎ ∘ 𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓))
132	109	oveq2d 7446	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ(⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓)) = (ℎ(⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔 ∘ 𝑓)))
133	129, 131, 132	3eqtr4d 2784	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = (ℎ(⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓)))
134	2, 3, 4, 7, 8, 24, 63, 93, 111, 133	iscatd2 17725	1 ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ ( I ↾ (Base‘𝑥)))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105  Vcvv 3477   ∩ cin 3961  ⟨cop 4636   ↦ cmpt 5230   I cid 5581   ↾ cres 5690   ∘ ccom 5692  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430  Basecbs 17244  Hom chom 17308  compcco 17309  Catccat 17708  Idccid 17709  Ringcrg 20250   RingHom crh 20485  RingCatALTVcringcALTV 48130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-plusg 17310  df-hom 17321  df-cco 17322  df-0g 17487  df-cat 17712  df-cid 17713  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-grp 18966  df-ghm 19243  df-mgp 20152  df-ur 20199  df-ring 20252  df-rhm 20488  df-ringcALTV 48131

This theorem is referenced by:  ringccatALTV  48150  ringcidALTV  48151