Theorem qusrhm 14848

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 2238	. 2 |- ( 1r ` R ) = ( 1r ` R )
3		eqid 2238	. 2 |- ( 1r ` U ) = ( 1r ` U )
4		eqid 2238	. 2 |- ( .r ` R ) = ( .r ` R )
5		eqid 2238	. 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 14847	. 2 |- ( ( R e. Ring /\ S e. I ) -> U e. Ring )
10		eqid 2238	. . . . . . . 8 |- (LIdeal ` R ) = (LIdeal ` R )
11		eqid 2238	. . . . . . . 8 |- (oppr ` R ) = (oppr ` R )
12		eqid 2238	. . . . . . . 8 |- (LIdeal ` (oppr ` R ) ) = (LIdeal ` (oppr ` R ) )
13	10, 11, 12, 8	2idlelb 14825	. . . . . . 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 14806	. . . . . 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 2238	. . . . . 6 |- ( R ~QG S ) = ( R ~QG S )
18	1, 17	eqger 14010	. . . . 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 13394	. . . . . 6 |- Base Fn _V
21	6	elexd 2835	. . . . . 6 |- ( ( R e. Ring /\ S e. I ) -> R e. _V )
22		funfvex 5710	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
23	22	funfni 5481	. . . . . 6 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
24	20, 21, 23	sylancr 418	. . . . 5 |- ( ( R e. Ring /\ S e. I ) -> ( Base ` R ) e. _V )
25	1, 24	eqeltrid 2325	. . . 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 13632	. . 3 |- ( ( R e. Ring /\ S e. I ) -> ( F ` ( 1r ` R ) ) = [ ( 1r ` R ) ] ( R ~QG S ) )
28	7, 8, 2	qus1 14846	. . . 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 2271	. 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 14844	. . . . 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 14300	. . . . . . . 8 |- ( ( R e. Ring /\ y e. X /\ z e. X ) -> ( y ( .r ` R ) z ) e. X )
35	34	3expb 1235	. . . . . . 7 |- ( ( R e. Ring /\ ( y e. X /\ z e. X ) ) -> ( y ( .r ` R ) z ) e. X )
36	35	adantlr 481	. . . . . 6 |- ( ( ( R e. Ring /\ S e. I ) /\ ( y e. X /\ z e. X ) ) -> ( y ( .r ` R ) z ) e. X )
37	36	caovclg 6236	. . . . 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 13641	. . . 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 1235	. . 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 13632	. . . 4 |- ( ( ( R e. Ring /\ S e. I ) /\ ( y e. X /\ z e. X ) ) -> ( F ` y ) = [ y ] ( R ~QG S ) )
43	40, 41, 26	divsfval 13632	. . . 4 |- ( ( ( R e. Ring /\ S e. I ) /\ ( y e. X /\ z e. X ) ) -> ( F ` z ) = [ z ] ( R ~QG S ) )
44	42, 43	oveq12d 6097	. . 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 13632	. . 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 2282	. 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 14320	. . . . . 6 |- ( R e. Ring -> R e. Abel )
48	47	adantr 276	. . . . 5 |- ( ( R e. Ring /\ S e. I ) -> R e. Abel )
49		ablnsg 14121	. . . . 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 2318	. . 3 |- ( ( R e. Ring /\ S e. I ) -> S e. (NrmSGrp ` R ) )
52	1, 7, 26	qusghm 14068	. . 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 14455	1 |- ( ( R e. Ring /\ S e. I ) -> F e. ( R RingHom U ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   |-> cmpt 4190   Fn wfn 5370   ` cfv 5375  (class class class)co 6079   Er wer 6798   [cec 6799   Basecbs 13335   .rcmulr 13415   /._s cqus 13606  SubGrpcsubg 13953  NrmSGrpcnsg 13954   ~_QG cqg 13955   GrpHom cghm 14026   Abelcabl 14071   1rcur 14245   Ringcrg 14283  opp_rcoppr 14355   RingHom crh 14440  LIdealclidl 14787  2Idealc2idl 14819

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-pre-ltirr 8285  ax-pre-lttrn 8287  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-tp 3716  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-tpos 6510  df-er 6801  df-ec 6803  df-qs 6807  df-map 6918  df-pnf 8356  df-mnf 8357  df-ltxr 8359  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-iress 13343  df-plusg 13427  df-mulr 13428  df-sca 13430  df-vsca 13431  df-ip 13432  df-0g 13595  df-iimas 13607  df-qus 13608  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-mhm 13749  df-grp 13791  df-minusg 13792  df-sbg 13793  df-subg 13956  df-nsg 13957  df-eqg 13958  df-ghm 14027  df-cmn 14072  df-abl 14073  df-mgp 14201  df-rng 14215  df-ur 14246  df-srg 14251  df-ring 14285  df-oppr 14356  df-rhm 14442  df-subrg 14510  df-lmod 14608  df-lssm 14673  df-sra 14755  df-rgmod 14756  df-lidl 14789  df-2idl 14820

This theorem is referenced by:  znzrh2  14964