Theorem rhmsubc 20921

Description: According to df-subc 17967, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17995 and subcss2 17998). 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 2762	. . . 4 ⊢ (𝜑 → (Rng ∩ 𝑈) = (Rng ∩ 𝑈))
4	1, 2, 3	rhmsscrnghm 20897	. . 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 2767	. . . . 5 ⊢ (𝜑 → (RngCat‘𝑈) = 𝐶)
10	9	fveq2d 6881	. . . 4 ⊢ (𝜑 → (Homf ‘(RngCat‘𝑈)) = (Homf ‘𝐶))
11		eqid 2761	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
12	7, 11, 1	rngchomfeqhom 20857	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Hom ‘𝐶))
13		eqid 2761	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
14	7, 11, 1, 13	rngchomfval 20854	. . . . 5 ⊢ (𝜑 → (Hom ‘𝐶) = ( RngHom ↾ ((Base‘𝐶) × (Base‘𝐶))))
15	7, 11, 1	rngcbas 20853	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
16		incom 4155	. . . . . . . 8 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
17	15, 16	eqtrdi 2812	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐶) = (Rng ∩ 𝑈))
18	17	sqxpeqd 5683	. . . . . 6 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Rng ∩ 𝑈) × (Rng ∩ 𝑈)))
19	18	reseq2d 5970	. . . . 5 ⊢ (𝜑 → ( RngHom ↾ ((Base‘𝐶) × (Base‘𝐶))) = ( RngHom ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
20	14, 19	eqtrd 2796	. . . 4 ⊢ (𝜑 → (Hom ‘𝐶) = ( RngHom ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
21	10, 12, 20	3eqtrd 2800	. . 3 ⊢ (𝜑 → (Homf ‘(RngCat‘𝑈)) = ( RngHom ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
22	4, 6, 21	3brtr4d 5137	. 2 ⊢ (𝜑 → 𝐻 ⊆cat (Homf ‘(RngCat‘𝑈)))
23	1, 7, 2, 5	rhmsubclem3 20919	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥))
24	1, 7, 2, 5	rhmsubclem4 20920	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
25	24	ralrimivva 3206	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
26	25	ralrimivva 3206	. . . 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 521	. . 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 3155	. 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 2761	. . 3 ⊢ (Homf ‘(RngCat‘𝑈)) = (Homf ‘(RngCat‘𝑈))
30		eqid 2761	. . 3 ⊢ (Id‘(RngCat‘𝑈)) = (Id‘(RngCat‘𝑈))
31		eqid 2761	. . 3 ⊢ (comp‘(RngCat‘𝑈)) = (comp‘(RngCat‘𝑈))
32		eqid 2761	. . . . 5 ⊢ (RngCat‘𝑈) = (RngCat‘𝑈)
33	32	rngccat 20866	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (RngCat‘𝑈) ∈ Cat)
34	1, 33	syl 18	. . 3 ⊢ (𝜑 → (RngCat‘𝑈) ∈ Cat)
35	1, 7, 2, 5	rhmsubclem1 20917	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝑅 × 𝑅))
36	29, 30, 31, 34, 35	issubc2 17991	. 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 726	1 ⊢ (𝜑 → 𝐻 ∈ (Subcat‘(RngCat‘𝑈)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ∩ cin 3898  ⟨cop 4590   class class class wbr 5103   × cxp 5649   ↾ cres 5653  ‘cfv 6531  (class class class)co 7412  Basecbs 17367  Hom chom 17419  compcco 17420  Catccat 17818  Idccid 17819  Hom_f chomf 17820   ⊆_cat cssc 17962  Subcatcsubc 17964  Rngcrng 20354  Ringcrg 20439   RngHom crnghm 20644   RingHom crh 20679  RngCatcrngc 20848

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-er 8701  df-map 8833  df-pm 8834  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-2 12386  df-3 12387  df-4 12388  df-5 12389  df-6 12390  df-7 12391  df-8 12392  df-9 12393  df-n0 12588  df-z 12675  df-dec 12796  df-uz 12947  df-fz 13621  df-struct 17305  df-sets 17322  df-slot 17340  df-ndx 17352  df-base 17368  df-ress 17389  df-plusg 17421  df-hom 17432  df-cco 17433  df-0g 17592  df-cat 17822  df-cid 17823  df-homf 17824  df-ssc 17965  df-resc 17966  df-subc 17967  df-estrc 18277  df-mgm 18796  df-mgmhm 18861  df-sgrp 18888  df-mnd 18904  df-mhm 18958  df-grp 19127  df-minusg 19128  df-ghm 19408  df-cmn 19976  df-abl 19977  df-mgp 20341  df-rng 20355  df-ur 20388  df-ring 20441  df-rnghm 20646  df-rhm 20682  df-rngc 20849

This theorem is used by:  rhmsubccat  20922