Theorem rhmsubc 20598

Description: According to df-subc 17774, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17802 and subcss2 17805). 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 2730	. . . 4 ⊢ (𝜑 → (Rng ∩ 𝑈) = (Rng ∩ 𝑈))
4	1, 2, 3	rhmsscrnghm 20574	. . 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 2735	. . . . 5 ⊢ (𝜑 → (RngCat‘𝑈) = 𝐶)
10	9	fveq2d 6862	. . . 4 ⊢ (𝜑 → (Homf ‘(RngCat‘𝑈)) = (Homf ‘𝐶))
11		eqid 2729	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
12	7, 11, 1	rngchomfeqhom 20534	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Hom ‘𝐶))
13		eqid 2729	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
14	7, 11, 1, 13	rngchomfval 20531	. . . . 5 ⊢ (𝜑 → (Hom ‘𝐶) = ( RngHom ↾ ((Base‘𝐶) × (Base‘𝐶))))
15	7, 11, 1	rngcbas 20530	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
16		incom 4172	. . . . . . . 8 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
17	15, 16	eqtrdi 2780	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐶) = (Rng ∩ 𝑈))
18	17	sqxpeqd 5670	. . . . . 6 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Rng ∩ 𝑈) × (Rng ∩ 𝑈)))
19	18	reseq2d 5950	. . . . 5 ⊢ (𝜑 → ( RngHom ↾ ((Base‘𝐶) × (Base‘𝐶))) = ( RngHom ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
20	14, 19	eqtrd 2764	. . . 4 ⊢ (𝜑 → (Hom ‘𝐶) = ( RngHom ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
21	10, 12, 20	3eqtrd 2768	. . 3 ⊢ (𝜑 → (Homf ‘(RngCat‘𝑈)) = ( RngHom ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
22	4, 6, 21	3brtr4d 5139	. 2 ⊢ (𝜑 → 𝐻 ⊆cat (Homf ‘(RngCat‘𝑈)))
23	1, 7, 2, 5	rhmsubclem3 20596	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥))
24	1, 7, 2, 5	rhmsubclem4 20597	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
25	24	ralrimivva 3180	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
26	25	ralrimivva 3180	. . . 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 3125	. 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 2729	. . 3 ⊢ (Homf ‘(RngCat‘𝑈)) = (Homf ‘(RngCat‘𝑈))
30		eqid 2729	. . 3 ⊢ (Id‘(RngCat‘𝑈)) = (Id‘(RngCat‘𝑈))
31		eqid 2729	. . 3 ⊢ (comp‘(RngCat‘𝑈)) = (comp‘(RngCat‘𝑈))
32		eqid 2729	. . . . 5 ⊢ (RngCat‘𝑈) = (RngCat‘𝑈)
33	32	rngccat 20543	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (RngCat‘𝑈) ∈ Cat)
34	1, 33	syl 17	. . 3 ⊢ (𝜑 → (RngCat‘𝑈) ∈ Cat)
35	1, 7, 2, 5	rhmsubclem1 20594	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝑅 × 𝑅))
36	29, 30, 31, 34, 35	issubc2 17798	. 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 1540   ∈ wcel 2109  ∀wral 3044   ∩ cin 3913  ⟨cop 4595   class class class wbr 5107   × cxp 5636   ↾ cres 5640  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  Hom chom 17231  compcco 17232  Catccat 17625  Idccid 17626  Hom_f chomf 17627   ⊆_cat cssc 17769  Subcatcsubc 17771  Rngcrng 20061  Ringcrg 20142   RngHom crnghm 20343   RingHom crh 20378  RngCatcrngc 20525

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-pm 8802  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-hom 17244  df-cco 17245  df-0g 17404  df-cat 17629  df-cid 17630  df-homf 17631  df-ssc 17772  df-resc 17773  df-subc 17774  df-estrc 18084  df-mgm 18567  df-mgmhm 18619  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-grp 18868  df-minusg 18869  df-ghm 19145  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-ring 20144  df-rnghm 20345  df-rhm 20381  df-rngc 20526

This theorem is referenced by:  rhmsubccat  20599