Theorem qus0 13988

Description: Value of the group identity operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)

Hypotheses

Ref	Expression
qusgrp.h	|- H = ( G /.s ( G ~QG S ) )
qus0.p	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
qus0	|- ( S e. (NrmSGrp ` G ) -> [ .0. ] ( G ~QG S ) = ( 0g ` H ) )

Proof of Theorem qus0

Step	Hyp	Ref	Expression
1		nsgsubg 13958	. . . . . . 7 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
2		subgrcl 13932	. . . . . . 7 |- ( S e. (SubGrp ` G ) -> G e. Grp )
3	1, 2	syl 14	. . . . . 6 |- ( S e. (NrmSGrp ` G ) -> G e. Grp )
4		eqid 2234	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
5		qus0.p	. . . . . . 7 |- .0. = ( 0g ` G )
6	4, 5	grpidcl 13784	. . . . . 6 |- ( G e. Grp -> .0. e. ( Base ` G ) )
7	3, 6	syl 14	. . . . 5 |- ( S e. (NrmSGrp ` G ) -> .0. e. ( Base ` G ) )
8		qusgrp.h	. . . . . 6 |- H = ( G /.s ( G ~QG S ) )
9		eqid 2234	. . . . . 6 |- ( +g ` G ) = ( +g ` G )
10		eqid 2234	. . . . . 6 |- ( +g ` H ) = ( +g ` H )
11	8, 4, 9, 10	qusadd 13987	. . . . 5 |- ( ( S e. (NrmSGrp ` G ) /\ .0. e. ( Base ` G ) /\ .0. e. ( Base ` G ) ) -> ( [ .0. ] ( G ~QG S ) ( +g ` H ) [ .0. ] ( G ~QG S ) ) = [ ( .0. ( +g ` G ) .0. ) ] ( G ~QG S ) )
12	7, 7, 11	mpd3an23 1376	. . . 4 |- ( S e. (NrmSGrp ` G ) -> ( [ .0. ] ( G ~QG S ) ( +g ` H ) [ .0. ] ( G ~QG S ) ) = [ ( .0. ( +g ` G ) .0. ) ] ( G ~QG S ) )
13	4, 9, 5	grplid 13786	. . . . . 6 |- ( ( G e. Grp /\ .0. e. ( Base ` G ) ) -> ( .0. ( +g ` G ) .0. ) = .0. )
14	3, 7, 13	syl2anc 411	. . . . 5 |- ( S e. (NrmSGrp ` G ) -> ( .0. ( +g ` G ) .0. ) = .0. )
15	14	eceq1d 6816	. . . 4 |- ( S e. (NrmSGrp ` G ) -> [ ( .0. ( +g ` G ) .0. ) ] ( G ~QG S ) = [ .0. ] ( G ~QG S ) )
16	12, 15	eqtrd 2267	. . 3 |- ( S e. (NrmSGrp ` G ) -> ( [ .0. ] ( G ~QG S ) ( +g ` H ) [ .0. ] ( G ~QG S ) ) = [ .0. ] ( G ~QG S ) )
17	8	qusgrp 13985	. . . 4 |- ( S e. (NrmSGrp ` G ) -> H e. Grp )
18		eqid 2234	. . . . . 6 |- ( Base ` H ) = ( Base ` H )
19	8, 4, 18	quseccl 13986	. . . . 5 |- ( ( S e. (NrmSGrp ` G ) /\ .0. e. ( Base ` G ) ) -> [ .0. ] ( G ~QG S ) e. ( Base ` H ) )
20	7, 19	mpdan 421	. . . 4 |- ( S e. (NrmSGrp ` G ) -> [ .0. ] ( G ~QG S ) e. ( Base ` H ) )
21		eqid 2234	. . . . 5 |- ( 0g ` H ) = ( 0g ` H )
22	18, 10, 21	grpid 13794	. . . 4 |- ( ( H e. Grp /\ [ .0. ] ( G ~QG S ) e. ( Base ` H ) ) -> ( ( [ .0. ] ( G ~QG S ) ( +g ` H ) [ .0. ] ( G ~QG S ) ) = [ .0. ] ( G ~QG S ) <-> ( 0g ` H ) = [ .0. ] ( G ~QG S ) ) )
23	17, 20, 22	syl2anc 411	. . 3 |- ( S e. (NrmSGrp ` G ) -> ( ( [ .0. ] ( G ~QG S ) ( +g ` H ) [ .0. ] ( G ~QG S ) ) = [ .0. ] ( G ~QG S ) <-> ( 0g ` H ) = [ .0. ] ( G ~QG S ) ) )
24	16, 23	mpbid 147	. 2 |- ( S e. (NrmSGrp ` G ) -> ( 0g ` H ) = [ .0. ] ( G ~QG S ) )
25	24	eqcomd 2240	1 |- ( S e. (NrmSGrp ` G ) -> [ .0. ] ( G ~QG S ) = ( 0g ` H ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2205   ` cfv 5357  (class class class)co 6058   [cec 6778   Basecbs 13296   +g cplusg 13374   0gc0g 13553   /._s cqus 13566   Grpcgrp 13755  SubGrpcsubg 13920  NrmSGrpcnsg 13921   ~_QG cqg 13922

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-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-er 6780  df-ec 6782  df-qs 6786  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-0g 13555  df-iimas 13567  df-qus 13568  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-subg 13923  df-nsg 13924  df-eqg 13925

This theorem is referenced by:  qusinv  13989