Theorem qusrhm 14160

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 2196	. 2 |- ( 1r ` R ) = ( 1r ` R )
3		eqid 2196	. 2 |- ( 1r ` U ) = ( 1r ` U )
4		eqid 2196	. 2 |- ( .r ` R ) = ( .r ` R )
5		eqid 2196	. 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 14159	. 2 |- ( ( R e. Ring /\ S e. I ) -> U e. Ring )
10		eqid 2196	. . . . . . . 8 |- (LIdeal ` R ) = (LIdeal ` R )
11		eqid 2196	. . . . . . . 8 |- (oppr ` R ) = (oppr ` R )
12		eqid 2196	. . . . . . . 8 |- (LIdeal ` (oppr ` R ) ) = (LIdeal ` (oppr ` R ) )
13	10, 11, 12, 8	2idlelb 14137	. . . . . . 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 14118	. . . . . 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 2196	. . . . . 6 |- ( R ~QG S ) = ( R ~QG S )
18	1, 17	eqger 13430	. . . . 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 12761	. . . . . 6 |- Base Fn _V
21	6	elexd 2776	. . . . . 6 |- ( ( R e. Ring /\ S e. I ) -> R e. _V )
22		funfvex 5578	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
23	22	funfni 5361	. . . . . 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 2283	. . . 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 13030	. . 3 |- ( ( R e. Ring /\ S e. I ) -> ( F ` ( 1r ` R ) ) = [ ( 1r ` R ) ] ( R ~QG S ) )
28	7, 8, 2	qus1 14158	. . . 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 2229	. 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 14156	. . . . 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 13645	. . . . . . . 8 |- ( ( R e. Ring /\ y e. X /\ z e. X ) -> ( y ( .r ` R ) z ) e. X )
35	34	3expb 1206	. . . . . . 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 6080	. . . . 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 13039	. . . 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 1206	. . 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 13030	. . . 4 |- ( ( ( R e. Ring /\ S e. I ) /\ ( y e. X /\ z e. X ) ) -> ( F ` y ) = [ y ] ( R ~QG S ) )
43	40, 41, 26	divsfval 13030	. . . 4 |- ( ( ( R e. Ring /\ S e. I ) /\ ( y e. X /\ z e. X ) ) -> ( F ` z ) = [ z ] ( R ~QG S ) )
44	42, 43	oveq12d 5943	. . 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 13030	. . 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 2240	. 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 13664	. . . . . 6 |- ( R e. Ring -> R e. Abel )
48	47	adantr 276	. . . . 5 |- ( ( R e. Ring /\ S e. I ) -> R e. Abel )
49		ablnsg 13540	. . . . 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 2276	. . 3 |- ( ( R e. Ring /\ S e. I ) -> S e. (NrmSGrp ` R ) )
52	1, 7, 26	qusghm 13488	. . 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 13797	1 |- ( ( R e. Ring /\ S e. I ) -> F e. ( R RingHom U ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   _Vcvv 2763   |-> cmpt 4095   Fn wfn 5254   ` cfv 5259  (class class class)co 5925   Er wer 6598   [cec 6599   Basecbs 12703   .rcmulr 12781   /._s cqus 13002  SubGrpcsubg 13373  NrmSGrpcnsg 13374   ~_QG cqg 13375   GrpHom cghm 13446   Abelcabl 13491   1rcur 13591   Ringcrg 13628  opp_rcoppr 13699   RingHom crh 13782  LIdealclidl 14099  2Idealc2idl 14131

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-tp 3631  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-tpos 6312  df-er 6601  df-ec 6603  df-qs 6607  df-map 6718  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-mulr 12794  df-sca 12796  df-vsca 12797  df-ip 12798  df-0g 12960  df-iimas 13004  df-qus 13005  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-mhm 13161  df-grp 13205  df-minusg 13206  df-sbg 13207  df-subg 13376  df-nsg 13377  df-eqg 13378  df-ghm 13447  df-cmn 13492  df-abl 13493  df-mgp 13553  df-rng 13565  df-ur 13592  df-srg 13596  df-ring 13630  df-oppr 13700  df-rhm 13784  df-subrg 13851  df-lmod 13921  df-lssm 13985  df-sra 14067  df-rgmod 14068  df-lidl 14101  df-2idl 14132

This theorem is referenced by:  znzrh2  14278