Theorem qusrhm 14788

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 2234	. 2 |- ( 1r ` R ) = ( 1r ` R )
3		eqid 2234	. 2 |- ( 1r ` U ) = ( 1r ` U )
4		eqid 2234	. 2 |- ( .r ` R ) = ( .r ` R )
5		eqid 2234	. 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 14787	. 2 |- ( ( R e. Ring /\ S e. I ) -> U e. Ring )
10		eqid 2234	. . . . . . . 8 |- (LIdeal ` R ) = (LIdeal ` R )
11		eqid 2234	. . . . . . . 8 |- (oppr ` R ) = (oppr ` R )
12		eqid 2234	. . . . . . . 8 |- (LIdeal ` (oppr ` R ) ) = (LIdeal ` (oppr ` R ) )
13	10, 11, 12, 8	2idlelb 14765	. . . . . . 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 14746	. . . . . 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 2234	. . . . . 6 |- ( R ~QG S ) = ( R ~QG S )
18	1, 17	eqger 14025	. . . . 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 13355	. . . . . 6 |- Base Fn _V
21	6	elexd 2829	. . . . . 6 |- ( ( R e. Ring /\ S e. I ) -> R e. _V )
22		funfvex 5692	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
23	22	funfni 5463	. . . . . 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 2321	. . . 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 13625	. . 3 |- ( ( R e. Ring /\ S e. I ) -> ( F ` ( 1r ` R ) ) = [ ( 1r ` R ) ] ( R ~QG S ) )
28	7, 8, 2	qus1 14786	. . . 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 2267	. 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 14784	. . . . 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 14241	. . . . . . . 8 |- ( ( R e. Ring /\ y e. X /\ z e. X ) -> ( y ( .r ` R ) z ) e. X )
35	34	3expb 1231	. . . . . . 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 6215	. . . . 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 13634	. . . 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 1231	. . 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 13625	. . . 4 |- ( ( ( R e. Ring /\ S e. I ) /\ ( y e. X /\ z e. X ) ) -> ( F ` y ) = [ y ] ( R ~QG S ) )
43	40, 41, 26	divsfval 13625	. . . 4 |- ( ( ( R e. Ring /\ S e. I ) /\ ( y e. X /\ z e. X ) ) -> ( F ` z ) = [ z ] ( R ~QG S ) )
44	42, 43	oveq12d 6076	. . 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 13625	. . 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 2278	. 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 14260	. . . . . 6 |- ( R e. Ring -> R e. Abel )
48	47	adantr 276	. . . . 5 |- ( ( R e. Ring /\ S e. I ) -> R e. Abel )
49		ablnsg 14135	. . . . 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 2314	. . 3 |- ( ( R e. Ring /\ S e. I ) -> S e. (NrmSGrp ` R ) )
52	1, 7, 26	qusghm 14083	. . 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 14395	1 |- ( ( R e. Ring /\ S e. I ) -> F e. ( R RingHom U ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   _Vcvv 2815   |-> cmpt 4176   Fn wfn 5352   ` cfv 5357  (class class class)co 6058   Er wer 6777   [cec 6778   Basecbs 13296   .rcmulr 13375   /._s cqus 13597  SubGrpcsubg 13968  NrmSGrpcnsg 13969   ~_QG cqg 13970   GrpHom cghm 14041   Abelcabl 14086   1rcur 14187   Ringcrg 14224  opp_rcoppr 14295   RingHom crh 14380  LIdealclidl 14727  2Idealc2idl 14759

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-tp 3702  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-tpos 6489  df-er 6780  df-ec 6782  df-qs 6786  df-map 6897  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-sca 13390  df-vsca 13391  df-ip 13392  df-0g 13555  df-iimas 13599  df-qus 13600  df-mgm 13653  df-sgrp 13699  df-mnd 13714  df-mhm 13756  df-grp 13800  df-minusg 13801  df-sbg 13802  df-subg 13971  df-nsg 13972  df-eqg 13973  df-ghm 14042  df-cmn 14087  df-abl 14088  df-mgp 14149  df-rng 14161  df-ur 14188  df-srg 14192  df-ring 14226  df-oppr 14296  df-rhm 14382  df-subrg 14450  df-lmod 14549  df-lssm 14613  df-sra 14695  df-rgmod 14696  df-lidl 14729  df-2idl 14760

This theorem is referenced by:  znzrh2  14906