Theorem rhmsubcALTV 42108

Description: According to df-subc 16472, the subcategories (Subcat ` C ) of a category C are subsets of the homomorphisms of C ( see subcssc 16500 and subcss2 16503). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)

Hypotheses

Ref	Expression
rngcrescrhmALTV.u	|- ( ph -> U e. V )
rngcrescrhmALTV.c	|- C = (RngCatALTV ` U )
rngcrescrhmALTV.r	|- ( ph -> R = ( Ring i^i U ) )
rngcrescrhmALTV.h	|- H = ( RingHom |` ( R X. R ) )

Assertion

Ref	Expression
rhmsubcALTV	|- ( ph -> H e. (Subcat ` (RngCatALTV ` U ) ) )

Proof of Theorem rhmsubcALTV

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

Step	Hyp	Ref	Expression
1		rngcrescrhmALTV.u	. . . 4 |- ( ph -> U e. V )
2		rngcrescrhmALTV.r	. . . 4 |- ( ph -> R = ( Ring i^i U ) )
3		eqidd 2623	. . . 4 |- ( ph -> (Rng i^i U ) = (Rng i^i U ) )
4	1, 2, 3	rhmsscrnghm 42026	. . 3 |- ( ph -> ( RingHom |` ( R X. R ) ) C_cat ( RngHomo |` ( (Rng i^i U ) X. (Rng i^i U ) ) ) )
5		rngcrescrhmALTV.h	. . . 4 |- H = ( RingHom |` ( R X. R ) )
6	5	a1i 11	. . 3 |- ( ph -> H = ( RingHom |` ( R X. R ) ) )
7		eqid 2622	. . . 4 |- (RngCatALTV ` U ) = (RngCatALTV ` U )
8		eqid 2622	. . . 4 |- (Rng i^i U ) = (Rng i^i U )
9		eqid 2622	. . . 4 |- ( Hom f ` (RngCatALTV ` U ) ) = ( Hom f ` (RngCatALTV ` U ) )
10	7, 8, 1, 9	rngchomrnghmresALTV 41996	. . 3 |- ( ph -> ( Hom f ` (RngCatALTV ` U ) ) = ( RngHomo |` ( (Rng i^i U ) X. (Rng i^i U ) ) ) )
11	4, 6, 10	3brtr4d 4685	. 2 |- ( ph -> H C_cat ( Hom f ` (RngCatALTV ` U ) ) )
12		rngcrescrhmALTV.c	. . . . 5 |- C = (RngCatALTV ` U )
13	1, 12, 2, 5	rhmsubcALTVlem3 42106	. . . 4 |- ( ( ph /\ x e. R ) -> ( ( Id ` (RngCatALTV ` U ) ) ` x ) e. ( x H x ) )
14	1, 12, 2, 5	rhmsubcALTVlem4 42107	. . . . . 6 |- ( ( ( ( ph /\ x e. R ) /\ ( y e. R /\ z e. R ) ) /\ ( f e. ( x H y ) /\ g e. ( y H z ) ) ) -> ( g ( <. x , y >. (comp ` (RngCatALTV ` U ) ) z ) f ) e. ( x H z ) )
15	14	ralrimivva 2971	. . . . 5 |- ( ( ( ph /\ x e. R ) /\ ( y e. R /\ z e. R ) ) -> A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. (comp ` (RngCatALTV ` U ) ) z ) f ) e. ( x H z ) )
16	15	ralrimivva 2971	. . . 4 |- ( ( ph /\ x e. R ) -> A. y e. R A. z e. R A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. (comp ` (RngCatALTV ` U ) ) z ) f ) e. ( x H z ) )
17	13, 16	jca 554	. . 3 |- ( ( ph /\ x e. R ) -> ( ( ( Id ` (RngCatALTV ` U ) ) ` x ) e. ( x H x ) /\ A. y e. R A. z e. R A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. (comp ` (RngCatALTV ` U ) ) z ) f ) e. ( x H z ) ) )
18	17	ralrimiva 2966	. 2 |- ( ph -> A. x e. R ( ( ( Id ` (RngCatALTV ` U ) ) ` x ) e. ( x H x ) /\ A. y e. R A. z e. R A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. (comp ` (RngCatALTV ` U ) ) z ) f ) e. ( x H z ) ) )
19		eqid 2622	. . 3 |- ( Id ` (RngCatALTV ` U ) ) = ( Id ` (RngCatALTV ` U ) )
20		eqid 2622	. . 3 |- (comp ` (RngCatALTV ` U ) ) = (comp ` (RngCatALTV ` U ) )
21	7	rngccatALTV 41990	. . . 4 |- ( U e. V -> (RngCatALTV ` U ) e. Cat )
22	1, 21	syl 17	. . 3 |- ( ph -> (RngCatALTV ` U ) e. Cat )
23	1, 12, 2, 5	rhmsubcALTVlem1 42104	. . 3 |- ( ph -> H Fn ( R X. R ) )
24	9, 19, 20, 22, 23	issubc2 16496	. 2 |- ( ph -> ( H e. (Subcat ` (RngCatALTV ` U ) ) <-> ( H C_cat ( Hom f ` (RngCatALTV ` U ) ) /\ A. x e. R ( ( ( Id ` (RngCatALTV ` U ) ) ` x ) e. ( x H x ) /\ A. y e. R A. z e. R A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. (comp ` (RngCatALTV ` U ) ) z ) f ) e. ( x H z ) ) ) ) )
25	11, 18, 24	mpbir2and 957	1 |- ( ph -> H e. (Subcat ` (RngCatALTV ` U ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   <.cop 4183   class class class wbr 4653   X. cxp 5112   |` cres 5116   ` cfv 5888  (class class class)co 6650  compcco 15953   Catccat 16325   Idccid 16326   Hom _f chomf 16327   C__cat cssc 16467  Subcatcsubc 16469   Ringcrg 18547   RingHom crh 18712  Rngcrng 41874   RngHomo crngh 41885  RngCatALTVcrngcALTV 41958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-hom 15966  df-cco 15967  df-0g 16102  df-cat 16329  df-cid 16330  df-homf 16331  df-ssc 16470  df-subc 16472  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-ghm 17658  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-rnghom 18715  df-mgmhm 41779  df-rng0 41875  df-rnghomo 41887  df-rngcALTV 41960

This theorem is referenced by:  rhmsubcALTVcat  42109