Theorem srhmsubcALTV 48267

Description: According to df-subc 17830, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17858 and subcss2 17861). Therefore, the set of special ring homomorphisms (i.e., ring homomorphisms from a special ring to another ring of that kind) is a subcategory of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)

Hypotheses

Ref	Expression
srhmsubcALTV.s	⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring
srhmsubcALTV.c	⊢ 𝐶 = (𝑈 ∩ 𝑆)
srhmsubcALTV.j	⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))

Assertion

Ref	Expression
srhmsubcALTV	⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)))

Distinct variable groups:   𝑆,𝑟   𝐶,𝑟,𝑠   𝑈,𝑟,𝑠   𝑉,𝑟,𝑠

Allowed substitution hints:   𝑆(𝑠)   𝐽(𝑠,𝑟)

Proof of Theorem srhmsubcALTV

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

Step	Hyp	Ref	Expression
1		srhmsubcALTV.c	. . . 4 ⊢ 𝐶 = (𝑈 ∩ 𝑆)
2		eleq1w 2818	. . . . . . 7 ⊢ (𝑟 = 𝑥 → (𝑟 ∈ Ring ↔ 𝑥 ∈ Ring))
3		srhmsubcALTV.s	. . . . . . 7 ⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring
4	2, 3	vtoclri 3574	. . . . . 6 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ Ring)
5	4	ssriv 3967	. . . . 5 ⊢ 𝑆 ⊆ Ring
6		sslin 4223	. . . . 5 ⊢ (𝑆 ⊆ Ring → (𝑈 ∩ 𝑆) ⊆ (𝑈 ∩ Ring))
7	5, 6	mp1i 13	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ 𝑆) ⊆ (𝑈 ∩ Ring))
8	1, 7	eqsstrid 4002	. . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ⊆ (𝑈 ∩ Ring))
9		ssid 3986	. . . . . 6 ⊢ (𝑥 RingHom 𝑦) ⊆ (𝑥 RingHom 𝑦)
10		eqid 2736	. . . . . . 7 ⊢ (RingCatALTV‘𝑈) = (RingCatALTV‘𝑈)
11		eqid 2736	. . . . . . 7 ⊢ (Base‘(RingCatALTV‘𝑈)) = (Base‘(RingCatALTV‘𝑈))
12		simpl 482	. . . . . . 7 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑈 ∈ 𝑉)
13		eqid 2736	. . . . . . 7 ⊢ (Hom ‘(RingCatALTV‘𝑈)) = (Hom ‘(RingCatALTV‘𝑈))
14	3, 1	srhmsubcALTVlem1 48265	. . . . . . . 8 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ (Base‘(RingCatALTV‘𝑈)))
15	14	adantrr 717	. . . . . . 7 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ (Base‘(RingCatALTV‘𝑈)))
16	3, 1	srhmsubcALTVlem1 48265	. . . . . . . 8 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ (Base‘(RingCatALTV‘𝑈)))
17	16	adantrl 716	. . . . . . 7 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ (Base‘(RingCatALTV‘𝑈)))
18	10, 11, 12, 13, 15, 17	ringchomALTV 48244	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑦) = (𝑥 RingHom 𝑦))
19	9, 18	sseqtrrid 4007	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 RingHom 𝑦) ⊆ (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑦))
20		srhmsubcALTV.j	. . . . . . 7 ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))
21	20	a1i 11	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)))
22		oveq12 7419	. . . . . . 7 ⊢ ((𝑟 = 𝑥 ∧ 𝑠 = 𝑦) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
23	22	adantl 481	. . . . . 6 ⊢ (((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑦)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
24		simprl 770	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ 𝐶)
25		simprr 772	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ 𝐶)
26		ovexd 7445	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 RingHom 𝑦) ∈ V)
27	21, 23, 24, 25, 26	ovmpod 7564	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥𝐽𝑦) = (𝑥 RingHom 𝑦))
28		eqid 2736	. . . . . 6 ⊢ (Homf ‘(RingCatALTV‘𝑈)) = (Homf ‘(RingCatALTV‘𝑈))
29	28, 11, 13, 15, 17	homfval 17709	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(Homf ‘(RingCatALTV‘𝑈))𝑦) = (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑦))
30	19, 27, 29	3sstr4d 4019	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCatALTV‘𝑈))𝑦))
31	30	ralrimivva 3188	. . 3 ⊢ (𝑈 ∈ 𝑉 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCatALTV‘𝑈))𝑦))
32		ovex 7443	. . . . . 6 ⊢ (𝑟 RingHom 𝑠) ∈ V
33	20, 32	fnmpoi 8074	. . . . 5 ⊢ 𝐽 Fn (𝐶 × 𝐶)
34	33	a1i 11	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐽 Fn (𝐶 × 𝐶))
35	28, 11	homffn 17710	. . . . 5 ⊢ (Homf ‘(RingCatALTV‘𝑈)) Fn ((Base‘(RingCatALTV‘𝑈)) × (Base‘(RingCatALTV‘𝑈)))
36		id 22	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ 𝑉)
37	10, 11, 36	ringcbasALTV 48242	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → (Base‘(RingCatALTV‘𝑈)) = (𝑈 ∩ Ring))
38	37	eqcomd 2742	. . . . . . 7 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) = (Base‘(RingCatALTV‘𝑈)))
39	38	sqxpeqd 5691	. . . . . 6 ⊢ (𝑈 ∈ 𝑉 → ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) = ((Base‘(RingCatALTV‘𝑈)) × (Base‘(RingCatALTV‘𝑈))))
40	39	fneq2d 6637	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → ((Homf ‘(RingCatALTV‘𝑈)) Fn ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) ↔ (Homf ‘(RingCatALTV‘𝑈)) Fn ((Base‘(RingCatALTV‘𝑈)) × (Base‘(RingCatALTV‘𝑈)))))
41	35, 40	mpbiri 258	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (Homf ‘(RingCatALTV‘𝑈)) Fn ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
42		inex1g 5294	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V)
43	34, 41, 42	isssc 17838	. . 3 ⊢ (𝑈 ∈ 𝑉 → (𝐽 ⊆cat (Homf ‘(RingCatALTV‘𝑈)) ↔ (𝐶 ⊆ (𝑈 ∩ Ring) ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝐽𝑦) ⊆ (𝑥(Homf ‘(RingCatALTV‘𝑈))𝑦))))
44	8, 31, 43	mpbir2and 713	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐽 ⊆cat (Homf ‘(RingCatALTV‘𝑈)))
45	1	elin2 4183	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐶 ↔ (𝑥 ∈ 𝑈 ∧ 𝑥 ∈ 𝑆))
46	4	adantl 481	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑈 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ Ring)
47	45, 46	sylbi 217	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ Ring)
48	47	adantl 481	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ Ring)
49		eqid 2736	. . . . . . 7 ⊢ (Base‘𝑥) = (Base‘𝑥)
50	49	idrhm 20455	. . . . . 6 ⊢ (𝑥 ∈ Ring → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
51	48, 50	syl 17	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥))
52		eqid 2736	. . . . . 6 ⊢ (Id‘(RingCatALTV‘𝑈)) = (Id‘(RingCatALTV‘𝑈))
53		simpl 482	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝑈 ∈ 𝑉)
54	10, 11, 52, 53, 14, 49	ringcidALTV 48250	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → ((Id‘(RingCatALTV‘𝑈))‘𝑥) = ( I ↾ (Base‘𝑥)))
55	20	a1i 11	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)))
56		oveq12 7419	. . . . . . 7 ⊢ ((𝑟 = 𝑥 ∧ 𝑠 = 𝑥) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑥))
57	56	adantl 481	. . . . . 6 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑥)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑥))
58		simpr 484	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
59		ovexd 7445	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → (𝑥 RingHom 𝑥) ∈ V)
60	55, 57, 58, 58, 59	ovmpod 7564	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → (𝑥𝐽𝑥) = (𝑥 RingHom 𝑥))
61	51, 54, 60	3eltr4d 2850	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → ((Id‘(RingCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥))
62		eqid 2736	. . . . . . . . 9 ⊢ (comp‘(RingCatALTV‘𝑈)) = (comp‘(RingCatALTV‘𝑈))
63	10	ringccatALTV 48249	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝑉 → (RingCatALTV‘𝑈) ∈ Cat)
64	63	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (RingCatALTV‘𝑈) ∈ Cat)
65	14	adantr 480	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑥 ∈ (Base‘(RingCatALTV‘𝑈)))
66	65	adantr 480	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑥 ∈ (Base‘(RingCatALTV‘𝑈)))
67	16	ad2ant2r 747	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑦 ∈ (Base‘(RingCatALTV‘𝑈)))
68	67	adantr 480	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑦 ∈ (Base‘(RingCatALTV‘𝑈)))
69	3, 1	srhmsubcALTVlem1 48265	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ (Base‘(RingCatALTV‘𝑈)))
70	69	ad2ant2rl 749	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑧 ∈ (Base‘(RingCatALTV‘𝑈)))
71	70	adantr 480	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑧 ∈ (Base‘(RingCatALTV‘𝑈)))
72	53	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑈 ∈ 𝑉)
73		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶) → 𝑦 ∈ 𝐶)
74	58, 73	anim12i 613	. . . . . . . . . . . . . . 15 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶))
75	72, 74	jca 511	. . . . . . . . . . . . . 14 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)))
76	3, 1, 20	srhmsubcALTVlem2 48266	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥𝐽𝑦) = (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑦))
77	75, 76	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥𝐽𝑦) = (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑦))
78	77	eleq2d 2821	. . . . . . . . . . . 12 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑓 ∈ (𝑥𝐽𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑦)))
79	78	biimpcd 249	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑥𝐽𝑦) → (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑓 ∈ (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑦)))
80	79	adantr 480	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑓 ∈ (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑦)))
81	80	impcom 407	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑓 ∈ (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑦))
82	3, 1, 20	srhmsubcALTVlem2 48266	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑦𝐽𝑧) = (𝑦(Hom ‘(RingCatALTV‘𝑈))𝑧))
83	82	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑦𝐽𝑧) = (𝑦(Hom ‘(RingCatALTV‘𝑈))𝑧))
84	83	eleq2d 2821	. . . . . . . . . . . 12 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑔 ∈ (𝑦𝐽𝑧) ↔ 𝑔 ∈ (𝑦(Hom ‘(RingCatALTV‘𝑈))𝑧)))
85	84	biimpd 229	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑔 ∈ (𝑦𝐽𝑧) → 𝑔 ∈ (𝑦(Hom ‘(RingCatALTV‘𝑈))𝑧)))
86	85	adantld 490	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → ((𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑔 ∈ (𝑦(Hom ‘(RingCatALTV‘𝑈))𝑧)))
87	86	imp 406	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → 𝑔 ∈ (𝑦(Hom ‘(RingCatALTV‘𝑈))𝑧))
88	11, 13, 62, 64, 66, 68, 71, 81, 87	catcocl 17702	. . . . . . . 8 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑧))
89	10, 11, 72, 13, 65, 70	ringchomALTV 48244	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑧) = (𝑥 RingHom 𝑧))
90	89	eqcomd 2742	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥 RingHom 𝑧) = (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑧))
91	90	adantr 480	. . . . . . . 8 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑥 RingHom 𝑧) = (𝑥(Hom ‘(RingCatALTV‘𝑈))𝑧))
92	88, 91	eleqtrrd 2838	. . . . . . 7 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥 RingHom 𝑧))
93	20	a1i 11	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)))
94		oveq12 7419	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑥 ∧ 𝑠 = 𝑧) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑧))
95	94	adantl 481	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑧)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑧))
96	58	adantr 480	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑥 ∈ 𝐶)
97		simprr 772	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑧 ∈ 𝐶)
98		ovexd 7445	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥 RingHom 𝑧) ∈ V)
99	93, 95, 96, 97, 98	ovmpod 7564	. . . . . . . 8 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑥𝐽𝑧) = (𝑥 RingHom 𝑧))
100	99	adantr 480	. . . . . . 7 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑥𝐽𝑧) = (𝑥 RingHom 𝑧))
101	92, 100	eleqtrrd 2838	. . . . . 6 ⊢ ((((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧))
102	101	ralrimivva 3188	. . . . 5 ⊢ (((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧))
103	102	ralrimivva 3188	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧))
104	61, 103	jca 511	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐶) → (((Id‘(RingCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
105	104	ralrimiva 3133	. 2 ⊢ (𝑈 ∈ 𝑉 → ∀𝑥 ∈ 𝐶 (((Id‘(RingCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
106	28, 52, 62, 63, 34	issubc2 17854	. 2 ⊢ (𝑈 ∈ 𝑉 → (𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)) ↔ (𝐽 ⊆cat (Homf ‘(RingCatALTV‘𝑈)) ∧ ∀𝑥 ∈ 𝐶 (((Id‘(RingCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RingCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
107	44, 105, 106	mpbir2and 713	1 ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  Vcvv 3464   ∩ cin 3930   ⊆ wss 3931  ⟨cop 4612   class class class wbr 5124   I cid 5552   × cxp 5657   ↾ cres 5661   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  Basecbs 17233  Hom chom 17287  compcco 17288  Catccat 17681  Idccid 17682  Hom_f chomf 17683   ⊆_cat cssc 17825  Subcatcsubc 17827  Ringcrg 20198   RingHom crh 20434  RingCatALTVcringcALTV 48229

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-map 8847  df-pm 8848  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12858  df-fz 13530  df-struct 17171  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-plusg 17289  df-hom 17300  df-cco 17301  df-0g 17460  df-cat 17685  df-cid 17686  df-homf 17687  df-ssc 17828  df-subc 17830  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-mhm 18766  df-grp 18924  df-ghm 19201  df-mgp 20106  df-ur 20147  df-ring 20200  df-rhm 20437  df-ringcALTV 48230

This theorem is referenced by:  sringcatALTV  48268  crhmsubcALTV  48269  drhmsubcALTV  48271  fldhmsubcALTV  48275