Theorem rhmsubcrngc 20644

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 2741	. . . . . . 7 ⊢ (RngCat‘𝑈) = (RngCat‘𝑈)
4		eqid 2741	. . . . . . 7 ⊢ (Base‘(RngCat‘𝑈)) = (Base‘(RngCat‘𝑈))
5	3, 4, 1	rngcbas 20597	. . . . . 6 ⊢ (𝜑 → (Base‘(RngCat‘𝑈)) = (𝑈 ∩ Rng))
6		incom 4141	. . . . . 6 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
7	5, 6	eqtrdi 2792	. . . . 5 ⊢ (𝜑 → (Base‘(RngCat‘𝑈)) = (Rng ∩ 𝑈))
8	1, 2, 7	rhmsscrnghm 20641	. . . 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 6835	. . . . . 6 ⊢ (𝜑 → (Base‘𝐶) = (Base‘(RngCat‘𝑈)))
12	11	sqxpeqd 5653	. . . . 5 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈))))
13	12	reseq2d 5938	. . . 4 ⊢ (𝜑 → ( RngHom ↾ ((Base‘𝐶) × (Base‘𝐶))) = ( RngHom ↾ ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈)))))
14	8, 13	breqtrrd 5103	. . 3 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ⊆cat ( RngHom ↾ ((Base‘𝐶) × (Base‘𝐶))))
15		rhmsubcrngc.h	. . 3 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
16		eqid 2741	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
17	9, 16, 1	rngchomfeqhom 20601	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Hom ‘𝐶))
18		eqid 2741	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
19	9, 16, 1, 18	rngchomfval 20598	. . . 4 ⊢ (𝜑 → (Hom ‘𝐶) = ( RngHom ↾ ((Base‘𝐶) × (Base‘𝐶))))
20	17, 19	eqtrd 2776	. . 3 ⊢ (𝜑 → (Homf ‘𝐶) = ( RngHom ↾ ((Base‘𝐶) × (Base‘𝐶))))
21	14, 15, 20	3brtr4d 5107	. 2 ⊢ (𝜑 → 𝐻 ⊆cat (Homf ‘𝐶))
22	9, 1, 2, 15	rhmsubcrngclem1 20642	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))
23	9, 1, 2, 15	rhmsubcrngclem2 20643	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))
24	22, 23	jca 517	. . 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 3133	. 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 2741	. . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
27		eqid 2741	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
28		eqid 2741	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
29	9	rngccat 20610	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
30	1, 29	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
31		incom 4141	. . . . 5 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring)
32	2, 31	eqtrdi 2792	. . . 4 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))
33	32, 15	rhmresfn 20624	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))
34	26, 27, 28, 30, 33	issubc2 17798	. 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 720	1 ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055   ∩ cin 3884  ⟨cop 4564   class class class wbr 5075   × cxp 5619   ↾ cres 5623  ‘cfv 6489  (class class class)co 7360  Basecbs 17174  Hom chom 17226  compcco 17227  Catccat 17625  Idccid 17626  Hom_f chomf 17627   ⊆_cat cssc 17769  Subcatcsubc 17771  Rngcrng 20128  Ringcrg 20209   RngHom crnghm 20409   RingHom crh 20444  RngCatcrngc 20592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-map 8769  df-pm 8770  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-dec 12640  df-uz 12784  df-fz 13457  df-struct 17112  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-plusg 17228  df-hom 17239  df-cco 17240  df-0g 17399  df-cat 17629  df-cid 17630  df-homf 17631  df-ssc 17772  df-resc 17773  df-subc 17774  df-estrc 18084  df-mgm 18603  df-mgmhm 18655  df-sgrp 18682  df-mnd 18698  df-mhm 18746  df-grp 18907  df-minusg 18908  df-ghm 19183  df-cmn 19752  df-abl 19753  df-mgp 20117  df-rng 20129  df-ur 20158  df-ring 20211  df-rnghm 20411  df-rhm 20447  df-rngc 20593  df-ringc 20622

This theorem is referenced by: (None)