Theorem rhmsubc 20602

Description: According to df-subc 17716, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17744 and subcss2 17747). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.)

Hypotheses

Ref	Expression
rngcrescrhm.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
rngcrescrhm.c	⊢ 𝐶 = (RngCat‘𝑈)
rngcrescrhm.r	⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))
rngcrescrhm.h	⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))

Assertion

Ref	Expression
rhmsubc	⊢ (𝜑 → 𝐻 ∈ (Subcat‘(RngCat‘𝑈)))

Proof of Theorem rhmsubc

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

Step	Hyp	Ref	Expression
1		rngcrescrhm.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
2		rngcrescrhm.r	. . . 4 ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))
3		eqidd 2732	. . . 4 ⊢ (𝜑 → (Rng ∩ 𝑈) = (Rng ∩ 𝑈))
4	1, 2, 3	rhmsscrnghm 20578	. . 3 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHom ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
5		rngcrescrhm.h	. . . 4 ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))
6	5	a1i 11	. . 3 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)))
7		rngcrescrhm.c	. . . . . . 7 ⊢ 𝐶 = (RngCat‘𝑈)
8	7	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐶 = (RngCat‘𝑈))
9	8	eqcomd 2737	. . . . 5 ⊢ (𝜑 → (RngCat‘𝑈) = 𝐶)
10	9	fveq2d 6826	. . . 4 ⊢ (𝜑 → (Homf ‘(RngCat‘𝑈)) = (Homf ‘𝐶))
11		eqid 2731	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
12	7, 11, 1	rngchomfeqhom 20538	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Hom ‘𝐶))
13		eqid 2731	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
14	7, 11, 1, 13	rngchomfval 20535	. . . . 5 ⊢ (𝜑 → (Hom ‘𝐶) = ( RngHom ↾ ((Base‘𝐶) × (Base‘𝐶))))
15	7, 11, 1	rngcbas 20534	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
16		incom 4159	. . . . . . . 8 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
17	15, 16	eqtrdi 2782	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐶) = (Rng ∩ 𝑈))
18	17	sqxpeqd 5648	. . . . . 6 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Rng ∩ 𝑈) × (Rng ∩ 𝑈)))
19	18	reseq2d 5928	. . . . 5 ⊢ (𝜑 → ( RngHom ↾ ((Base‘𝐶) × (Base‘𝐶))) = ( RngHom ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
20	14, 19	eqtrd 2766	. . . 4 ⊢ (𝜑 → (Hom ‘𝐶) = ( RngHom ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
21	10, 12, 20	3eqtrd 2770	. . 3 ⊢ (𝜑 → (Homf ‘(RngCat‘𝑈)) = ( RngHom ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
22	4, 6, 21	3brtr4d 5123	. 2 ⊢ (𝜑 → 𝐻 ⊆cat (Homf ‘(RngCat‘𝑈)))
23	1, 7, 2, 5	rhmsubclem3 20600	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥))
24	1, 7, 2, 5	rhmsubclem4 20601	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
25	24	ralrimivva 3175	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
26	25	ralrimivva 3175	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ∀𝑦 ∈ 𝑅 ∀𝑧 ∈ 𝑅 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
27	23, 26	jca 511	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → (((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝑅 ∀𝑧 ∈ 𝑅 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧)))
28	27	ralrimiva 3124	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑅 (((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝑅 ∀𝑧 ∈ 𝑅 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧)))
29		eqid 2731	. . 3 ⊢ (Homf ‘(RngCat‘𝑈)) = (Homf ‘(RngCat‘𝑈))
30		eqid 2731	. . 3 ⊢ (Id‘(RngCat‘𝑈)) = (Id‘(RngCat‘𝑈))
31		eqid 2731	. . 3 ⊢ (comp‘(RngCat‘𝑈)) = (comp‘(RngCat‘𝑈))
32		eqid 2731	. . . . 5 ⊢ (RngCat‘𝑈) = (RngCat‘𝑈)
33	32	rngccat 20547	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (RngCat‘𝑈) ∈ Cat)
34	1, 33	syl 17	. . 3 ⊢ (𝜑 → (RngCat‘𝑈) ∈ Cat)
35	1, 7, 2, 5	rhmsubclem1 20598	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝑅 × 𝑅))
36	29, 30, 31, 34, 35	issubc2 17740	. 2 ⊢ (𝜑 → (𝐻 ∈ (Subcat‘(RngCat‘𝑈)) ↔ (𝐻 ⊆cat (Homf ‘(RngCat‘𝑈)) ∧ ∀𝑥 ∈ 𝑅 (((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝑅 ∀𝑧 ∈ 𝑅 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧)))))
37	22, 28, 36	mpbir2and 713	1 ⊢ (𝜑 → 𝐻 ∈ (Subcat‘(RngCat‘𝑈)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ∩ cin 3901  ⟨cop 4582   class class class wbr 5091   × cxp 5614   ↾ cres 5618  ‘cfv 6481  (class class class)co 7346  Basecbs 17117  Hom chom 17169  compcco 17170  Catccat 17567  Idccid 17568  Hom_f chomf 17569   ⊆_cat cssc 17711  Subcatcsubc 17713  Rngcrng 20068  Ringcrg 20149   RngHom crnghm 20350   RingHom crh 20385  RngCatcrngc 20529

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 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080

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 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  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-pm 8753  df-ixp 8822  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-nn 12123  df-2 12185  df-3 12186  df-4 12187  df-5 12188  df-6 12189  df-7 12190  df-8 12191  df-9 12192  df-n0 12379  df-z 12466  df-dec 12586  df-uz 12730  df-fz 13405  df-struct 17055  df-sets 17072  df-slot 17090  df-ndx 17102  df-base 17118  df-ress 17139  df-plusg 17171  df-hom 17182  df-cco 17183  df-0g 17342  df-cat 17571  df-cid 17572  df-homf 17573  df-ssc 17714  df-resc 17715  df-subc 17716  df-estrc 18026  df-mgm 18545  df-mgmhm 18597  df-sgrp 18624  df-mnd 18640  df-mhm 18688  df-grp 18846  df-minusg 18847  df-ghm 19123  df-cmn 19692  df-abl 19693  df-mgp 20057  df-rng 20069  df-ur 20098  df-ring 20151  df-rnghm 20352  df-rhm 20388  df-rngc 20530

This theorem is referenced by:  rhmsubccat  20603