Theorem qusrhm 14334

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 2206	. 2 |- ( 1r ` R ) = ( 1r ` R )
3		eqid 2206	. 2 |- ( 1r ` U ) = ( 1r ` U )
4		eqid 2206	. 2 |- ( .r ` R ) = ( .r ` R )
5		eqid 2206	. 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 14333	. 2 |- ( ( R e. Ring /\ S e. I ) -> U e. Ring )
10		eqid 2206	. . . . . . . 8 |- (LIdeal ` R ) = (LIdeal ` R )
11		eqid 2206	. . . . . . . 8 |- (oppr ` R ) = (oppr ` R )
12		eqid 2206	. . . . . . . 8 |- (LIdeal ` (oppr ` R ) ) = (LIdeal ` (oppr ` R ) )
13	10, 11, 12, 8	2idlelb 14311	. . . . . . 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 14292	. . . . . 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 2206	. . . . . 6 |- ( R ~QG S ) = ( R ~QG S )
18	1, 17	eqger 13604	. . . . 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 12934	. . . . . 6 |- Base Fn _V
21	6	elexd 2786	. . . . . 6 |- ( ( R e. Ring /\ S e. I ) -> R e. _V )
22		funfvex 5600	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
23	22	funfni 5381	. . . . . 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 2293	. . . 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 13204	. . 3 |- ( ( R e. Ring /\ S e. I ) -> ( F ` ( 1r ` R ) ) = [ ( 1r ` R ) ] ( R ~QG S ) )
28	7, 8, 2	qus1 14332	. . . 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 2239	. 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 14330	. . . . 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 13819	. . . . . . . 8 |- ( ( R e. Ring /\ y e. X /\ z e. X ) -> ( y ( .r ` R ) z ) e. X )
35	34	3expb 1207	. . . . . . 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 6106	. . . . 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 13213	. . . 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 1207	. . 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 13204	. . . 4 |- ( ( ( R e. Ring /\ S e. I ) /\ ( y e. X /\ z e. X ) ) -> ( F ` y ) = [ y ] ( R ~QG S ) )
43	40, 41, 26	divsfval 13204	. . . 4 |- ( ( ( R e. Ring /\ S e. I ) /\ ( y e. X /\ z e. X ) ) -> ( F ` z ) = [ z ] ( R ~QG S ) )
44	42, 43	oveq12d 5969	. . 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 13204	. . 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 2250	. 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 13838	. . . . . 6 |- ( R e. Ring -> R e. Abel )
48	47	adantr 276	. . . . 5 |- ( ( R e. Ring /\ S e. I ) -> R e. Abel )
49		ablnsg 13714	. . . . 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 2286	. . 3 |- ( ( R e. Ring /\ S e. I ) -> S e. (NrmSGrp ` R ) )
52	1, 7, 26	qusghm 13662	. . 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 13971	1 |- ( ( R e. Ring /\ S e. I ) -> F e. ( R RingHom U ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   _Vcvv 2773   |-> cmpt 4109   Fn wfn 5271   ` cfv 5276  (class class class)co 5951   Er wer 6624   [cec 6625   Basecbs 12876   .rcmulr 12954   /._s cqus 13176  SubGrpcsubg 13547  NrmSGrpcnsg 13548   ~_QG cqg 13549   GrpHom cghm 13620   Abelcabl 13665   1rcur 13765   Ringcrg 13802  opp_rcoppr 13873   RingHom crh 13956  LIdealclidl 14273  2Idealc2idl 14305

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-addcom 8032  ax-addass 8034  ax-i2m1 8037  ax-0lt1 8038  ax-0id 8040  ax-rnegex 8041  ax-pre-ltirr 8044  ax-pre-lttrn 8046  ax-pre-ltadd 8048

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-pw 3619  df-sn 3640  df-pr 3641  df-tp 3642  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-tpos 6338  df-er 6627  df-ec 6629  df-qs 6633  df-map 6744  df-pnf 8116  df-mnf 8117  df-ltxr 8119  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-5 9105  df-6 9106  df-7 9107  df-8 9108  df-ndx 12879  df-slot 12880  df-base 12882  df-sets 12883  df-iress 12884  df-plusg 12966  df-mulr 12967  df-sca 12969  df-vsca 12970  df-ip 12971  df-0g 13134  df-iimas 13178  df-qus 13179  df-mgm 13232  df-sgrp 13278  df-mnd 13293  df-mhm 13335  df-grp 13379  df-minusg 13380  df-sbg 13381  df-subg 13550  df-nsg 13551  df-eqg 13552  df-ghm 13621  df-cmn 13666  df-abl 13667  df-mgp 13727  df-rng 13739  df-ur 13766  df-srg 13770  df-ring 13804  df-oppr 13874  df-rhm 13958  df-subrg 14025  df-lmod 14095  df-lssm 14159  df-sra 14241  df-rgmod 14242  df-lidl 14275  df-2idl 14306

This theorem is referenced by:  znzrh2  14452