Theorem qusrhm 14513

Description: If S is a two-sided ideal in R, then the "natural map" from elements to their cosets is a ring homomorphism from R to R / S. (Contributed by Mario Carneiro, 15-Jun-2015.)

Hypotheses

Ref	Expression
qusring.u	|- U = ( R /.s ( R ~QG S ) )
qusring.i	|- I = (2Ideal ` R )
qusrhm.x	|- X = ( Base ` R )
qusrhm.f	|- F = ( x e. X |-> [ x ] ( R ~QG S ) )

Assertion

Ref	Expression
qusrhm	|- ( ( R e. Ring /\ S e. I ) -> F e. ( R RingHom U ) )

Distinct variable groups:   x, I   x, R   x, S   x, U   x, X

Allowed substitution hint:   F( x)

Proof of Theorem qusrhm

Dummy variables y z a b c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qusrhm.x	. 2 |- X = ( Base ` R )
2		eqid 2229	. 2 |- ( 1r ` R ) = ( 1r ` R )
3		eqid 2229	. 2 |- ( 1r ` U ) = ( 1r ` U )
4		eqid 2229	. 2 |- ( .r ` R ) = ( .r ` R )
5		eqid 2229	. 2 |- ( .r ` U ) = ( .r ` U )
6		simpl 109	. 2 |- ( ( R e. Ring /\ S e. I ) -> R e. Ring )
7		qusring.u	. . 3 |- U = ( R /.s ( R ~QG S ) )
8		qusring.i	. . 3 |- I = (2Ideal ` R )
9	7, 8	qusring 14512	. 2 |- ( ( R e. Ring /\ S e. I ) -> U e. Ring )
10		eqid 2229	. . . . . . . 8 |- (LIdeal ` R ) = (LIdeal ` R )
11		eqid 2229	. . . . . . . 8 |- (oppr ` R ) = (oppr ` R )
12		eqid 2229	. . . . . . . 8 |- (LIdeal ` (oppr ` R ) ) = (LIdeal ` (oppr ` R ) )
13	10, 11, 12, 8	2idlelb 14490	. . . . . . 7 |- ( S e. I <-> ( S e. (LIdeal ` R ) /\ S e. (LIdeal ` (oppr ` R ) ) ) )
14	13	simplbi 274	. . . . . 6 |- ( S e. I -> S e. (LIdeal ` R ) )
15	10	lidlsubg 14471	. . . . . 6 |- ( ( R e. Ring /\ S e. (LIdeal ` R ) ) -> S e. (SubGrp ` R ) )
16	14, 15	sylan2 286	. . . . 5 |- ( ( R e. Ring /\ S e. I ) -> S e. (SubGrp ` R ) )
17		eqid 2229	. . . . . 6 |- ( R ~QG S ) = ( R ~QG S )
18	1, 17	eqger 13782	. . . . 5 |- ( S e. (SubGrp ` R ) -> ( R ~QG S ) Er X )
19	16, 18	syl 14	. . . 4 |- ( ( R e. Ring /\ S e. I ) -> ( R ~QG S ) Er X )
20		basfn 13112	. . . . . 6 |- Base Fn _V
21	6	elexd 2813	. . . . . 6 |- ( ( R e. Ring /\ S e. I ) -> R e. _V )
22		funfvex 5649	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
23	22	funfni 5426	. . . . . 6 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
24	20, 21, 23	sylancr 414	. . . . 5 |- ( ( R e. Ring /\ S e. I ) -> ( Base ` R ) e. _V )
25	1, 24	eqeltrid 2316	. . . 4 |- ( ( R e. Ring /\ S e. I ) -> X e. _V )
26		qusrhm.f	. . . 4 |- F = ( x e. X |-> [ x ] ( R ~QG S ) )
27	19, 25, 26	divsfval 13382	. . 3 |- ( ( R e. Ring /\ S e. I ) -> ( F ` ( 1r ` R ) ) = [ ( 1r ` R ) ] ( R ~QG S ) )
28	7, 8, 2	qus1 14511	. . . 4 |- ( ( R e. Ring /\ S e. I ) -> ( U e. Ring /\ [ ( 1r ` R ) ] ( R ~QG S ) = ( 1r ` U ) ) )
29	28	simprd 114	. . 3 |- ( ( R e. Ring /\ S e. I ) -> [ ( 1r ` R ) ] ( R ~QG S ) = ( 1r ` U ) )
30	27, 29	eqtrd 2262	. 2 |- ( ( R e. Ring /\ S e. I ) -> ( F ` ( 1r ` R ) ) = ( 1r ` U ) )
31	7	a1i 9	. . . . 5 |- ( ( R e. Ring /\ S e. I ) -> U = ( R /.s ( R ~QG S ) ) )
32	1	a1i 9	. . . . 5 |- ( ( R e. Ring /\ S e. I ) -> X = ( Base ` R ) )
33	1, 17, 8, 4	2idlcpbl 14509	. . . . 5 |- ( ( R e. Ring /\ S e. I ) -> ( ( a ( R ~QG S ) c /\ b ( R ~QG S ) d ) -> ( a ( .r ` R ) b ) ( R ~QG S ) ( c ( .r ` R ) d ) ) )
34	1, 4	ringcl 13997	. . . . . . . 8 |- ( ( R e. Ring /\ y e. X /\ z e. X ) -> ( y ( .r ` R ) z ) e. X )
35	34	3expb 1228	. . . . . . 7 |- ( ( R e. Ring /\ ( y e. X /\ z e. X ) ) -> ( y ( .r ` R ) z ) e. X )
36	35	adantlr 477	. . . . . 6 |- ( ( ( R e. Ring /\ S e. I ) /\ ( y e. X /\ z e. X ) ) -> ( y ( .r ` R ) z ) e. X )
37	36	caovclg 6167	. . . . 5 |- ( ( ( R e. Ring /\ S e. I ) /\ ( c e. X /\ d e. X ) ) -> ( c ( .r ` R ) d ) e. X )
38	31, 32, 19, 6, 33, 37, 4, 5	qusmulval 13391	. . . 4 |- ( ( ( R e. Ring /\ S e. I ) /\ y e. X /\ z e. X ) -> ( [ y ] ( R ~QG S ) ( .r ` U ) [ z ] ( R ~QG S ) ) = [ ( y ( .r ` R ) z ) ] ( R ~QG S ) )
39	38	3expb 1228	. . 3 |- ( ( ( R e. Ring /\ S e. I ) /\ ( y e. X /\ z e. X ) ) -> ( [ y ] ( R ~QG S ) ( .r ` U ) [ z ] ( R ~QG S ) ) = [ ( y ( .r ` R ) z ) ] ( R ~QG S ) )
40	19	adantr 276	. . . . 5 |- ( ( ( R e. Ring /\ S e. I ) /\ ( y e. X /\ z e. X ) ) -> ( R ~QG S ) Er X )
41	25	adantr 276	. . . . 5 |- ( ( ( R e. Ring /\ S e. I ) /\ ( y e. X /\ z e. X ) ) -> X e. _V )
42	40, 41, 26	divsfval 13382	. . . 4 |- ( ( ( R e. Ring /\ S e. I ) /\ ( y e. X /\ z e. X ) ) -> ( F ` y ) = [ y ] ( R ~QG S ) )
43	40, 41, 26	divsfval 13382	. . . 4 |- ( ( ( R e. Ring /\ S e. I ) /\ ( y e. X /\ z e. X ) ) -> ( F ` z ) = [ z ] ( R ~QG S ) )
44	42, 43	oveq12d 6028	. . 3 |- ( ( ( R e. Ring /\ S e. I ) /\ ( y e. X /\ z e. X ) ) -> ( ( F ` y ) ( .r ` U ) ( F ` z ) ) = ( [ y ] ( R ~QG S ) ( .r ` U ) [ z ] ( R ~QG S ) ) )
45	40, 41, 26	divsfval 13382	. . 3 |- ( ( ( R e. Ring /\ S e. I ) /\ ( y e. X /\ z e. X ) ) -> ( F ` ( y ( .r ` R ) z ) ) = [ ( y ( .r ` R ) z ) ] ( R ~QG S ) )
46	39, 44, 45	3eqtr4rd 2273	. 2 |- ( ( ( R e. Ring /\ S e. I ) /\ ( y e. X /\ z e. X ) ) -> ( F ` ( y ( .r ` R ) z ) ) = ( ( F ` y ) ( .r ` U ) ( F ` z ) ) )
47		ringabl 14016	. . . . . 6 |- ( R e. Ring -> R e. Abel )
48	47	adantr 276	. . . . 5 |- ( ( R e. Ring /\ S e. I ) -> R e. Abel )
49		ablnsg 13892	. . . . 5 |- ( R e. Abel -> (NrmSGrp ` R ) = (SubGrp ` R ) )
50	48, 49	syl 14	. . . 4 |- ( ( R e. Ring /\ S e. I ) -> (NrmSGrp ` R ) = (SubGrp ` R ) )
51	16, 50	eleqtrrd 2309	. . 3 |- ( ( R e. Ring /\ S e. I ) -> S e. (NrmSGrp ` R ) )
52	1, 7, 26	qusghm 13840	. . 3 |- ( S e. (NrmSGrp ` R ) -> F e. ( R GrpHom U ) )
53	51, 52	syl 14	. 2 |- ( ( R e. Ring /\ S e. I ) -> F e. ( R GrpHom U ) )
54	1, 2, 3, 4, 5, 6, 9, 30, 46, 53	isrhm2d 14150	1 |- ( ( R e. Ring /\ S e. I ) -> F e. ( R RingHom U ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2799   |-> cmpt 4145   Fn wfn 5316   ` cfv 5321  (class class class)co 6010   Er wer 6690   [cec 6691   Basecbs 13053   .rcmulr 13132   /._s cqus 13354  SubGrpcsubg 13725  NrmSGrpcnsg 13726   ~_QG cqg 13727   GrpHom cghm 13798   Abelcabl 13843   1rcur 13943   Ringcrg 13980  opp_rcoppr 14051   RingHom crh 14135  LIdealclidl 14452  2Idealc2idl 14484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-pre-ltirr 8127  ax-pre-lttrn 8129  ax-pre-ltadd 8131

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-tpos 6402  df-er 6693  df-ec 6695  df-qs 6699  df-map 6810  df-pnf 8199  df-mnf 8200  df-ltxr 8202  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-5 9188  df-6 9189  df-7 9190  df-8 9191  df-ndx 13056  df-slot 13057  df-base 13059  df-sets 13060  df-iress 13061  df-plusg 13144  df-mulr 13145  df-sca 13147  df-vsca 13148  df-ip 13149  df-0g 13312  df-iimas 13356  df-qus 13357  df-mgm 13410  df-sgrp 13456  df-mnd 13471  df-mhm 13513  df-grp 13557  df-minusg 13558  df-sbg 13559  df-subg 13728  df-nsg 13729  df-eqg 13730  df-ghm 13799  df-cmn 13844  df-abl 13845  df-mgp 13905  df-rng 13917  df-ur 13944  df-srg 13948  df-ring 13982  df-oppr 14052  df-rhm 14137  df-subrg 14204  df-lmod 14274  df-lssm 14338  df-sra 14420  df-rgmod 14421  df-lidl 14454  df-2idl 14485

This theorem is referenced by:  znzrh2  14631