Theorem ringccatidALTV 48416

Description: Lemma for ringccatALTV 48417. (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 2732	. 2 ⊢ (𝑈 ∈ 𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶))
4		eqidd 2732	. 2 ⊢ (𝑈 ∈ 𝑉 → (comp‘𝐶) = (comp‘𝐶))
5		ringccatALTV.c	. . . 4 ⊢ 𝐶 = (RingCatALTV‘𝑈)
6	5	fvexi 6836	. . 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 48410	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝐵 = (𝑈 ∩ Ring))
11		eleq2 2820	. . . . . . . 8 ⊢ (𝐵 = (𝑈 ∩ Ring) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (𝑈 ∩ Ring)))
12		elin 3913	. . . . . . . . 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 2731	. . . . 5 ⊢ (Base‘𝑥) = (Base‘𝑥)
19	18	idrhm 20407	. . . 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 2731	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
22		simpr 484	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
23	5, 1, 9, 21, 22, 22	ringchomALTV 48412	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → (𝑥(Hom ‘𝐶)𝑥) = (𝑥 RingHom 𝑥))
24	20, 23	eleqtrrd 2834	. 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 2731	. . . 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 48412	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑤(Hom ‘𝐶)𝑥) = (𝑤 RingHom 𝑥))
37	36	eleq2d 2817	. . . . . . . . . . 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 48415	. . 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 48413	. . . . . . . . . . 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 6698	. . . . . . . . 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 2766	. 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 48412	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 RingHom 𝑦))
76	75	eleq2d 2817	. . . . . . . . . . . . 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 48415	. . . . . . . 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 48413	. . . . . . . . . . . 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 6697	. . . . . . . . 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 2766	. . . . . . 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 48412	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 RingHom 𝑦))
97	96	eleq2d 2817	. . . . . . . . . . 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 20416	. . . 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 48415	. . 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 48412	. . 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 2846	. 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 6213	. . . 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 48412	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(Hom ‘𝐶)𝑧) = (𝑦 RingHom 𝑧))
117	116	eleq2d 2817	. . . . . . . . . . . . 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 20416	. . . . . 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 48415	. . . 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 48415	. . . 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 2792	. . 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 48415	. . . 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 7361	. . 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 7362	. . 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 2776	. 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 17587	1 ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ ( I ↾ (Base‘𝑥)))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∩ cin 3896  ⟨cop 4579   ↦ cmpt 5170   I cid 5508   ↾ cres 5616   ∘ ccom 5618  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  Hom chom 17172  compcco 17173  Catccat 17570  Idccid 17571  Ringcrg 20151   RingHom crh 20387  RingCatALTVcringcALTV 48397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-z 12469  df-dec 12589  df-uz 12733  df-fz 13408  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-plusg 17174  df-hom 17185  df-cco 17186  df-0g 17345  df-cat 17574  df-cid 17575  df-mgm 18548  df-sgrp 18627  df-mnd 18643  df-mhm 18691  df-grp 18849  df-ghm 19125  df-mgp 20059  df-ur 20100  df-ring 20153  df-rhm 20390  df-ringcALTV 48398

This theorem is referenced by:  ringccatALTV  48417  ringcidALTV  48418