Theorem rhmsubc 20775

Description: According to df-subc 17870, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17898 and subcss2 17901). 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 2764	. . . 4 ⊢ (𝜑 → (Rng ∩ 𝑈) = (Rng ∩ 𝑈))
4	1, 2, 3	rhmsscrnghm 20751	. . 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 2769	. . . . 5 ⊢ (𝜑 → (RngCat‘𝑈) = 𝐶)
10	9	fveq2d 6887	. . . 4 ⊢ (𝜑 → (Homf ‘(RngCat‘𝑈)) = (Homf ‘𝐶))
11		eqid 2763	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
12	7, 11, 1	rngchomfeqhom 20711	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Hom ‘𝐶))
13		eqid 2763	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
14	7, 11, 1, 13	rngchomfval 20708	. . . . 5 ⊢ (𝜑 → (Hom ‘𝐶) = ( RngHom ↾ ((Base‘𝐶) × (Base‘𝐶))))
15	7, 11, 1	rngcbas 20707	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
16		incom 4163	. . . . . . . 8 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
17	15, 16	eqtrdi 2814	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐶) = (Rng ∩ 𝑈))
18	17	sqxpeqd 5695	. . . . . 6 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Rng ∩ 𝑈) × (Rng ∩ 𝑈)))
19	18	reseq2d 5980	. . . . 5 ⊢ (𝜑 → ( RngHom ↾ ((Base‘𝐶) × (Base‘𝐶))) = ( RngHom ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
20	14, 19	eqtrd 2798	. . . 4 ⊢ (𝜑 → (Hom ‘𝐶) = ( RngHom ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
21	10, 12, 20	3eqtrd 2802	. . 3 ⊢ (𝜑 → (Homf ‘(RngCat‘𝑈)) = ( RngHom ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈))))
22	4, 6, 21	3brtr4d 5144	. 2 ⊢ (𝜑 → 𝐻 ⊆cat (Homf ‘(RngCat‘𝑈)))
23	1, 7, 2, 5	rhmsubclem3 20773	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥))
24	1, 7, 2, 5	rhmsubclem4 20774	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
25	24	ralrimivva 3208	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
26	25	ralrimivva 3208	. . . 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 520	. . 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 3157	. 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 2763	. . 3 ⊢ (Homf ‘(RngCat‘𝑈)) = (Homf ‘(RngCat‘𝑈))
30		eqid 2763	. . 3 ⊢ (Id‘(RngCat‘𝑈)) = (Id‘(RngCat‘𝑈))
31		eqid 2763	. . 3 ⊢ (comp‘(RngCat‘𝑈)) = (comp‘(RngCat‘𝑈))
32		eqid 2763	. . . . 5 ⊢ (RngCat‘𝑈) = (RngCat‘𝑈)
33	32	rngccat 20720	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (RngCat‘𝑈) ∈ Cat)
34	1, 33	syl 18	. . 3 ⊢ (𝜑 → (RngCat‘𝑈) ∈ Cat)
35	1, 7, 2, 5	rhmsubclem1 20771	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝑅 × 𝑅))
36	29, 30, 31, 34, 35	issubc2 17894	. 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 725	1 ⊢ (𝜑 → 𝐻 ∈ (Subcat‘(RngCat‘𝑈)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ∩ cin 3905  ⟨cop 4596   class class class wbr 5110   × cxp 5661   ↾ cres 5665  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  Hom chom 17322  compcco 17323  Catccat 17721  Idccid 17722  Hom_f chomf 17723   ⊆_cat cssc 17865  Subcatcsubc 17867  Rngcrng 20231  Ringcrg 20316   RngHom crnghm 20517   RingHom crh 20552  RngCatcrngc 20702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-er 8695  df-map 8827  df-pm 8828  df-ixp 8897  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-4 12306  df-5 12307  df-6 12308  df-7 12309  df-8 12310  df-9 12311  df-n0 12506  df-z 12593  df-dec 12713  df-uz 12864  df-fz 13537  df-struct 17208  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-plusg 17324  df-hom 17335  df-cco 17336  df-0g 17495  df-cat 17725  df-cid 17726  df-homf 17727  df-ssc 17868  df-resc 17869  df-subc 17870  df-estrc 18180  df-mgm 18699  df-mgmhm 18751  df-sgrp 18778  df-mnd 18794  df-mhm 18842  df-grp 19004  df-minusg 19005  df-ghm 19285  df-cmn 19853  df-abl 19854  df-mgp 20218  df-rng 20232  df-ur 20265  df-ring 20318  df-rnghm 20519  df-rhm 20555  df-rngc 20703

This theorem is referenced by:  rhmsubccat  20776