Theorem ringccatidALTV 48298

Description: Lemma for ringccatALTV 48299. (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 2731	. 2 ⊢ (𝑈 ∈ 𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶))
4		eqidd 2731	. 2 ⊢ (𝑈 ∈ 𝑉 → (comp‘𝐶) = (comp‘𝐶))
5		ringccatALTV.c	. . . 4 ⊢ 𝐶 = (RingCatALTV‘𝑈)
6	5	fvexi 6875	. . 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 48292	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝐵 = (𝑈 ∩ Ring))
11		eleq2 2818	. . . . . . . 8 ⊢ (𝐵 = (𝑈 ∩ Ring) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (𝑈 ∩ Ring)))
12		elin 3933	. . . . . . . . 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 2730	. . . . 5 ⊢ (Base‘𝑥) = (Base‘𝑥)
19	18	idrhm 20406	. . . 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 2730	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
22		simpr 484	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
23	5, 1, 9, 21, 22, 22	ringchomALTV 48294	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → (𝑥(Hom ‘𝐶)𝑥) = (𝑥 RingHom 𝑥))
24	20, 23	eleqtrrd 2832	. 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 2730	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
27		simpl 482	. . . . . 6 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑤 ∈ 𝐵)
28	27	3ad2ant1 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))) → 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 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))) → 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 1136	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑈 ∈ 𝑉)
34	27	3ad2ant3 1135	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
35	30	3ad2ant3 1135	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
36	5, 1, 33, 21, 34, 35	ringchomALTV 48294	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑤(Hom ‘𝐶)𝑥) = (𝑤 RingHom 𝑥))
37	36	eleq2d 2815	. . . . . . . . . . 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 1119	. . . . . . . . 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 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 → 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 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))) → (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 48297	. . 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 770	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
52		simprr 772	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
53	5, 1, 50, 21, 51, 52	elringchomALTV 48295	. . . . . . . . . . 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 6738	. . . . . . . . 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 1133	. . . . . . 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 2765	. 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 1138	. . . . . . . . 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 1134	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → 𝑥 ∈ 𝐵)
67		simprl 770	. . . . . . . . . 10 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
68	67	3ad2ant2 1134	. . . . . . . . 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 48294	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 RingHom 𝑦))
76	75	eleq2d 2815	. . . . . . . . . . . . 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 48297	. . . . . . . 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 48295	. . . . . . . . . . . 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 6737	. . . . . . . . 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 2765	. . . . . . 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 1119	. . . . . 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 1134	. . . . 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 1134	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
96	5, 1, 33, 21, 35, 95	ringchomALTV 48294	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 RingHom 𝑦))
97	96	eleq2d 2815	. . . . . . . . . . 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 1119	. . . . . . . . 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 1134	. . . . . . 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 20417	. . . 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 1134	. . . . 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 48297	. . 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 48294	. . 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 2844	. 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 6241	. . . 4 ⊢ ((ℎ ∘ 𝑔) ∘ 𝑓) = (ℎ ∘ (𝑔 ∘ 𝑓))
113		simp2r 1201	. . . . . 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 1201	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑧 ∈ 𝐵)
116	5, 1, 33, 21, 95, 115	ringchomALTV 48294	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(Hom ‘𝐶)𝑧) = (𝑦 RingHom 𝑧))
117	116	eleq2d 2815	. . . . . . . . . . . . 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 1119	. . . . . . . . . . 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 1135	. . . . . . . . 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 20417	. . . . . 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 48297	. . . 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 48297	. . . 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 2791	. . 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 48297	. . . 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 7405	. . 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 7406	. . 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 2775	. 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 17649	1 ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ ( I ↾ (Base‘𝑥)))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ∩ cin 3916  ⟨cop 4598   ↦ cmpt 5191   I cid 5535   ↾ cres 5643   ∘ ccom 5645  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  Hom chom 17238  compcco 17239  Catccat 17632  Idccid 17633  Ringcrg 20149   RingHom crh 20385  RingCatALTVcringcALTV 48279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-plusg 17240  df-hom 17251  df-cco 17252  df-0g 17411  df-cat 17636  df-cid 17637  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-mhm 18717  df-grp 18875  df-ghm 19152  df-mgp 20057  df-ur 20098  df-ring 20151  df-rhm 20388  df-ringcALTV 48280

This theorem is referenced by:  ringccatALTV  48299  ringcidALTV  48300