Theorem qus1 14622

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 2231	. . 3 |- ( Base ` R ) = ( Base ` R )
4	3	a1i 9	. 2 |- ( ( R e. Ring /\ S e. I ) -> ( Base ` R ) = ( Base ` R ) )
5		eqid 2231	. 2 |- ( +g ` R ) = ( +g ` R )
6		eqid 2231	. 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 2231	. . . . . . . 8 |- (LIdeal ` R ) = (LIdeal ` R )
10		eqid 2231	. . . . . . . 8 |- (oppr ` R ) = (oppr ` R )
11		eqid 2231	. . . . . . . 8 |- (LIdeal ` (oppr ` R ) ) = (LIdeal ` (oppr ` R ) )
12		qusring.i	. . . . . . . 8 |- I = (2Ideal ` R )
13	9, 10, 11, 12	2idlvalg 14599	. . . . . . 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 2310	. . . . 5 |- ( ( R e. Ring /\ S e. I ) -> S e. ( (LIdeal ` R ) i^i (LIdeal ` (oppr ` R ) ) ) )
16	15	elin1d 3398	. . . 4 |- ( ( R e. Ring /\ S e. I ) -> S e. (LIdeal ` R ) )
17	9	lidlsubg 14582	. . . 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 2231	. . . 4 |- ( R ~QG S ) = ( R ~QG S )
20	3, 19	eqger 13891	. . 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 14126	. . . . . 6 |- ( R e. Ring -> R e. Abel )
23	22	adantr 276	. . . . 5 |- ( ( R e. Ring /\ S e. I ) -> R e. Abel )
24		ablnsg 14001	. . . . 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 2311	. . 3 |- ( ( R e. Ring /\ S e. I ) -> S e. (NrmSGrp ` R ) )
27	3, 19, 5	eqgcpbl 13895	. . 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 14620	. 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 14160	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 2202   i^i cin 3200   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   Er wer 6742   [cec 6743   Basecbs 13162   +g cplusg 13240   .rcmulr 13241   /._s cqus 13463  SubGrpcsubg 13834  NrmSGrpcnsg 13835   ~_QG cqg 13836   Abelcabl 13952   1rcur 14053   Ringcrg 14090  opp_rcoppr 14161  LIdealclidl 14563  2Idealc2idl 14595

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-lttrn 8206  ax-pre-ltadd 8208

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-tpos 6454  df-er 6745  df-ec 6747  df-qs 6751  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-5 9264  df-6 9265  df-7 9266  df-8 9267  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170  df-plusg 13253  df-mulr 13254  df-sca 13256  df-vsca 13257  df-ip 13258  df-0g 13421  df-iimas 13465  df-qus 13466  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-sbg 13668  df-subg 13837  df-nsg 13838  df-eqg 13839  df-cmn 13953  df-abl 13954  df-mgp 14015  df-rng 14027  df-ur 14054  df-srg 14058  df-ring 14092  df-oppr 14162  df-subrg 14314  df-lmod 14385  df-lssm 14449  df-sra 14531  df-rgmod 14532  df-lidl 14565  df-2idl 14596

This theorem is referenced by:  qusring  14623  qusrhm  14624