Theorem rhmsubcrngc 20618

Description: The unital ring homomorphisms between unital rings (in a universe) are a subcategory of the category of non-unital rings. (Contributed by AV, 12-Mar-2020.)

Hypotheses

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

Assertion

Ref	Expression
rhmsubcrngc	⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶))

Proof of Theorem rhmsubcrngc

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

Step	Hyp	Ref	Expression
1		rhmsubcrngc.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
2		rhmsubcrngc.b	. . . . 5 ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈))
3		eqid 2737	. . . . . . 7 ⊢ (RngCat‘𝑈) = (RngCat‘𝑈)
4		eqid 2737	. . . . . . 7 ⊢ (Base‘(RngCat‘𝑈)) = (Base‘(RngCat‘𝑈))
5	3, 4, 1	rngcbas 20571	. . . . . 6 ⊢ (𝜑 → (Base‘(RngCat‘𝑈)) = (𝑈 ∩ Rng))
6		incom 4163	. . . . . 6 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
7	5, 6	eqtrdi 2788	. . . . 5 ⊢ (𝜑 → (Base‘(RngCat‘𝑈)) = (Rng ∩ 𝑈))
8	1, 2, 7	rhmsscrnghm 20615	. . . 4 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ⊆cat ( RngHom ↾ ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈)))))
9		rhmsubcrngc.c	. . . . . . . 8 ⊢ 𝐶 = (RngCat‘𝑈)
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐶 = (RngCat‘𝑈))
11	10	fveq2d 6848	. . . . . 6 ⊢ (𝜑 → (Base‘𝐶) = (Base‘(RngCat‘𝑈)))
12	11	sqxpeqd 5666	. . . . 5 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈))))
13	12	reseq2d 5948	. . . 4 ⊢ (𝜑 → ( RngHom ↾ ((Base‘𝐶) × (Base‘𝐶))) = ( RngHom ↾ ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈)))))
14	8, 13	breqtrrd 5128	. . 3 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ⊆cat ( RngHom ↾ ((Base‘𝐶) × (Base‘𝐶))))
15		rhmsubcrngc.h	. . 3 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
16		eqid 2737	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
17	9, 16, 1	rngchomfeqhom 20575	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Hom ‘𝐶))
18		eqid 2737	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
19	9, 16, 1, 18	rngchomfval 20572	. . . 4 ⊢ (𝜑 → (Hom ‘𝐶) = ( RngHom ↾ ((Base‘𝐶) × (Base‘𝐶))))
20	17, 19	eqtrd 2772	. . 3 ⊢ (𝜑 → (Homf ‘𝐶) = ( RngHom ↾ ((Base‘𝐶) × (Base‘𝐶))))
21	14, 15, 20	3brtr4d 5132	. 2 ⊢ (𝜑 → 𝐻 ⊆cat (Homf ‘𝐶))
22	9, 1, 2, 15	rhmsubcrngclem1 20616	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))
23	9, 1, 2, 15	rhmsubcrngclem2 20617	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))
24	22, 23	jca 511	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)))
25	24	ralrimiva 3130	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)))
26		eqid 2737	. . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
27		eqid 2737	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
28		eqid 2737	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
29	9	rngccat 20584	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
30	1, 29	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
31		incom 4163	. . . . 5 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring)
32	2, 31	eqtrdi 2788	. . . 4 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))
33	32, 15	rhmresfn 20598	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))
34	26, 27, 28, 30, 33	issubc2 17774	. 2 ⊢ (𝜑 → (𝐻 ∈ (Subcat‘𝐶) ↔ (𝐻 ⊆cat (Homf ‘𝐶) ∧ ∀𝑥 ∈ 𝐵 (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)))))
35	21, 25, 34	mpbir2and 714	1 ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∩ cin 3902  ⟨cop 4588   class class class wbr 5100   × cxp 5632   ↾ cres 5636  ‘cfv 6502  (class class class)co 7370  Basecbs 17150  Hom chom 17202  compcco 17203  Catccat 17601  Idccid 17602  Hom_f chomf 17603   ⊆_cat cssc 17745  Subcatcsubc 17747  Rngcrng 20104  Ringcrg 20185   RngHom crnghm 20387   RingHom crh 20422  RngCatcrngc 20566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-er 8647  df-map 8779  df-pm 8780  df-ixp 8850  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-2 12222  df-3 12223  df-4 12224  df-5 12225  df-6 12226  df-7 12227  df-8 12228  df-9 12229  df-n0 12416  df-z 12503  df-dec 12622  df-uz 12766  df-fz 13438  df-struct 17088  df-sets 17105  df-slot 17123  df-ndx 17135  df-base 17151  df-ress 17172  df-plusg 17204  df-hom 17215  df-cco 17216  df-0g 17375  df-cat 17605  df-cid 17606  df-homf 17607  df-ssc 17748  df-resc 17749  df-subc 17750  df-estrc 18060  df-mgm 18579  df-mgmhm 18631  df-sgrp 18658  df-mnd 18674  df-mhm 18722  df-grp 18883  df-minusg 18884  df-ghm 19159  df-cmn 19728  df-abl 19729  df-mgp 20093  df-rng 20105  df-ur 20134  df-ring 20187  df-rnghm 20389  df-rhm 20425  df-rngc 20567  df-ringc 20596

This theorem is referenced by: (None)