Theorem rhmsubc 45648

Description: According to df-subc 17524, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17555 and subcss2 17558). 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 2739	. . . 4 ⊢ (𝜑 → (Rng ∩ 𝑈) = (Rng ∩ 𝑈))
4	1, 2, 3	rhmsscrnghm 45584	. . 3 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHomo ↾ ((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 2744	. . . . 5 ⊢ (𝜑 → (RngCat‘𝑈) = 𝐶)
10	9	fveq2d 6778	. . . 4 ⊢ (𝜑 → (Homf ‘(RngCat‘𝑈)) = (Homf ‘𝐶))
11		eqid 2738	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
12	7, 11, 1	rngchomfeqhom 45527	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Hom ‘𝐶))
13		eqid 2738	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
14	7, 11, 1, 13	rngchomfval 45524	. . . . 5 ⊢ (𝜑 → (Hom ‘𝐶) = ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶))))
15	7, 11, 1	rngcbas 45523	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
16		incom 4135	. . . . . . . 8 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
17	15, 16	eqtrdi 2794	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐶) = (Rng ∩ 𝑈))
18	17	sqxpeqd 5621	. . . . . 6 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Rng ∩ 𝑈) × (Rng ∩ 𝑈)))
19	18	reseq2d 5891	. . . . 5 ⊢ (𝜑 → ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶))) = ( RngHomo ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
20	14, 19	eqtrd 2778	. . . 4 ⊢ (𝜑 → (Hom ‘𝐶) = ( RngHomo ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
21	10, 12, 20	3eqtrd 2782	. . 3 ⊢ (𝜑 → (Homf ‘(RngCat‘𝑈)) = ( RngHomo ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
22	4, 6, 21	3brtr4d 5106	. 2 ⊢ (𝜑 → 𝐻 ⊆cat (Homf ‘(RngCat‘𝑈)))
23	1, 7, 2, 5	rhmsubclem3 45646	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥))
24	1, 7, 2, 5	rhmsubclem4 45647	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
25	24	ralrimivva 3123	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
26	25	ralrimivva 3123	. . . 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 512	. . 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 3103	. 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 2738	. . 3 ⊢ (Homf ‘(RngCat‘𝑈)) = (Homf ‘(RngCat‘𝑈))
30		eqid 2738	. . 3 ⊢ (Id‘(RngCat‘𝑈)) = (Id‘(RngCat‘𝑈))
31		eqid 2738	. . 3 ⊢ (comp‘(RngCat‘𝑈)) = (comp‘(RngCat‘𝑈))
32		eqid 2738	. . . . 5 ⊢ (RngCat‘𝑈) = (RngCat‘𝑈)
33	32	rngccat 45536	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (RngCat‘𝑈) ∈ Cat)
34	1, 33	syl 17	. . 3 ⊢ (𝜑 → (RngCat‘𝑈) ∈ Cat)
35	1, 7, 2, 5	rhmsubclem1 45644	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝑅 × 𝑅))
36	29, 30, 31, 34, 35	issubc2 17551	. 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 710	1 ⊢ (𝜑 → 𝐻 ∈ (Subcat‘(RngCat‘𝑈)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ∩ cin 3886  ⟨cop 4567   class class class wbr 5074   × cxp 5587   ↾ cres 5591  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  Hom chom 16973  compcco 16974  Catccat 17373  Idccid 17374  Hom_f chomf 17375   ⊆_cat cssc 17519  Subcatcsubc 17521  Ringcrg 19783   RingHom crh 19956  Rngcrng 45432   RngHomo crngh 45443  RngCatcrngc 45515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-fz 13240  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-hom 16986  df-cco 16987  df-0g 17152  df-cat 17377  df-cid 17378  df-homf 17379  df-ssc 17522  df-resc 17523  df-subc 17524  df-estrc 17839  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-grp 18580  df-minusg 18581  df-ghm 18832  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-ring 19785  df-rnghom 19959  df-mgmhm 45333  df-rng0 45433  df-rnghomo 45445  df-rngc 45517

This theorem is referenced by:  rhmsubccat  45649