Theorem qus1 14674

Description: The multiplicative identity of the quotient ring. (Contributed by Mario Carneiro, 14-Jun-2015.)

Hypotheses

Ref	Expression
qusring.u	|- U = ( R /.s ( R ~QG S ) )
qusring.i	|- I = (2Ideal ` R )
qus1.o	|- .1. = ( 1r ` R )

Assertion

Ref	Expression
qus1	|- ( ( R e. Ring /\ S e. I ) -> ( U e. Ring /\ [ .1. ] ( R ~QG S ) = ( 1r ` U ) ) )

Proof of Theorem qus1

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

Step	Hyp	Ref	Expression
1		qusring.u	. . 3 |- U = ( R /.s ( R ~QG S ) )
2	1	a1i 9	. 2 |- ( ( R e. Ring /\ S e. I ) -> U = ( R /.s ( R ~QG S ) ) )
3		eqid 2232	. . 3 |- ( Base ` R ) = ( Base ` R )
4	3	a1i 9	. 2 |- ( ( R e. Ring /\ S e. I ) -> ( Base ` R ) = ( Base ` R ) )
5		eqid 2232	. 2 |- ( +g ` R ) = ( +g ` R )
6		eqid 2232	. 2 |- ( .r ` R ) = ( .r ` R )
7		qus1.o	. 2 |- .1. = ( 1r ` R )
8		simpr 110	. . . . . 6 |- ( ( R e. Ring /\ S e. I ) -> S e. I )
9		eqid 2232	. . . . . . . 8 |- (LIdeal ` R ) = (LIdeal ` R )
10		eqid 2232	. . . . . . . 8 |- (oppr ` R ) = (oppr ` R )
11		eqid 2232	. . . . . . . 8 |- (LIdeal ` (oppr ` R ) ) = (LIdeal ` (oppr ` R ) )
12		qusring.i	. . . . . . . 8 |- I = (2Ideal ` R )
13	9, 10, 11, 12	2idlvalg 14651	. . . . . . 7 |- ( R e. Ring -> I = ( (LIdeal ` R ) i^i (LIdeal ` (oppr ` R ) ) ) )
14	13	adantr 276	. . . . . 6 |- ( ( R e. Ring /\ S e. I ) -> I = ( (LIdeal ` R ) i^i (LIdeal ` (oppr ` R ) ) ) )
15	8, 14	eleqtrd 2311	. . . . 5 |- ( ( R e. Ring /\ S e. I ) -> S e. ( (LIdeal ` R ) i^i (LIdeal ` (oppr ` R ) ) ) )
16	15	elin1d 3408	. . . 4 |- ( ( R e. Ring /\ S e. I ) -> S e. (LIdeal ` R ) )
17	9	lidlsubg 14634	. . . 4 |- ( ( R e. Ring /\ S e. (LIdeal ` R ) ) -> S e. (SubGrp ` R ) )
18	16, 17	syldan 282	. . 3 |- ( ( R e. Ring /\ S e. I ) -> S e. (SubGrp ` R ) )
19		eqid 2232	. . . 4 |- ( R ~QG S ) = ( R ~QG S )
20	3, 19	eqger 13941	. . 3 |- ( S e. (SubGrp ` R ) -> ( R ~QG S ) Er ( Base ` R ) )
21	18, 20	syl 14	. 2 |- ( ( R e. Ring /\ S e. I ) -> ( R ~QG S ) Er ( Base ` R ) )
22		ringabl 14176	. . . . . 6 |- ( R e. Ring -> R e. Abel )
23	22	adantr 276	. . . . 5 |- ( ( R e. Ring /\ S e. I ) -> R e. Abel )
24		ablnsg 14051	. . . . 5 |- ( R e. Abel -> (NrmSGrp ` R ) = (SubGrp ` R ) )
25	23, 24	syl 14	. . . 4 |- ( ( R e. Ring /\ S e. I ) -> (NrmSGrp ` R ) = (SubGrp ` R ) )
26	18, 25	eleqtrrd 2312	. . 3 |- ( ( R e. Ring /\ S e. I ) -> S e. (NrmSGrp ` R ) )
27	3, 19, 5	eqgcpbl 13945	. . 3 |- ( S e. (NrmSGrp ` R ) -> ( ( a ( R ~QG S ) c /\ b ( R ~QG S ) d ) -> ( a ( +g ` R ) b ) ( R ~QG S ) ( c ( +g ` R ) d ) ) )
28	26, 27	syl 14	. 2 |- ( ( R e. Ring /\ S e. I ) -> ( ( a ( R ~QG S ) c /\ b ( R ~QG S ) d ) -> ( a ( +g ` R ) b ) ( R ~QG S ) ( c ( +g ` R ) d ) ) )
29	3, 19, 12, 6	2idlcpbl 14672	. 2 |- ( ( 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 ) ) )
30		simpl 109	. 2 |- ( ( R e. Ring /\ S e. I ) -> R e. Ring )
31	2, 4, 5, 6, 7, 21, 28, 29, 30	qusring2 14210	1 |- ( ( R e. Ring /\ S e. I ) -> ( U e. Ring /\ [ .1. ] ( R ~QG S ) = ( 1r ` U ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   i^i cin 3210   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   Er wer 6764   [cec 6765   Basecbs 13212   +g cplusg 13290   .rcmulr 13291   /._s cqus 13513  SubGrpcsubg 13884  NrmSGrpcnsg 13885   ~_QG cqg 13886   Abelcabl 14002   1rcur 14103   Ringcrg 14140  opp_rcoppr 14211  LIdealclidl 14615  2Idealc2idl 14647

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-ltadd 8243

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-tp 3697  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-tpos 6476  df-er 6767  df-ec 6769  df-qs 6773  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220  df-plusg 13303  df-mulr 13304  df-sca 13306  df-vsca 13307  df-ip 13308  df-0g 13471  df-iimas 13515  df-qus 13516  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-minusg 13717  df-sbg 13718  df-subg 13887  df-nsg 13888  df-eqg 13889  df-cmn 14003  df-abl 14004  df-mgp 14065  df-rng 14077  df-ur 14104  df-srg 14108  df-ring 14142  df-oppr 14212  df-subrg 14364  df-lmod 14437  df-lssm 14501  df-sra 14583  df-rgmod 14584  df-lidl 14617  df-2idl 14648

This theorem is referenced by:  qusring  14675  qusrhm  14676