Theorem qus0 13885

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 13855	. . . . . . 7 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
2		subgrcl 13829	. . . . . . 7 |- ( S e. (SubGrp ` G ) -> G e. Grp )
3	1, 2	syl 14	. . . . . 6 |- ( S e. (NrmSGrp ` G ) -> G e. Grp )
4		eqid 2231	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
5		qus0.p	. . . . . . 7 |- .0. = ( 0g ` G )
6	4, 5	grpidcl 13675	. . . . . 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 2231	. . . . . 6 |- ( +g ` G ) = ( +g ` G )
10		eqid 2231	. . . . . 6 |- ( +g ` H ) = ( +g ` H )
11	8, 4, 9, 10	qusadd 13884	. . . . 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 13677	. . . . . 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 6781	. . . 4 |- ( S e. (NrmSGrp ` G ) -> [ ( .0. ( +g ` G ) .0. ) ] ( G ~QG S ) = [ .0. ] ( G ~QG S ) )
16	12, 15	eqtrd 2264	. . 3 |- ( S e. (NrmSGrp ` G ) -> ( [ .0. ] ( G ~QG S ) ( +g ` H ) [ .0. ] ( G ~QG S ) ) = [ .0. ] ( G ~QG S ) )
17	8	qusgrp 13882	. . . 4 |- ( S e. (NrmSGrp ` G ) -> H e. Grp )
18		eqid 2231	. . . . . 6 |- ( Base ` H ) = ( Base ` H )
19	8, 4, 18	quseccl 13883	. . . . 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 2231	. . . . 5 |- ( 0g ` H ) = ( 0g ` H )
22	18, 10, 21	grpid 13685	. . . 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 2237	1 |- ( S e. (NrmSGrp ` G ) -> [ .0. ] ( G ~QG S ) = ( 0g ` H ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2202   ` cfv 5333  (class class class)co 6028   [cec 6743   Basecbs 13145   +g cplusg 13223   0gc0g 13402   /._s cqus 13446   Grpcgrp 13646  SubGrpcsubg 13817  NrmSGrpcnsg 13818   ~_QG cqg 13819

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-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-lttrn 8189  ax-pre-ltadd 8191

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-er 6745  df-ec 6747  df-qs 6751  df-pnf 8258  df-mnf 8259  df-ltxr 8261  df-inn 9186  df-2 9244  df-3 9245  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-iress 13153  df-plusg 13236  df-mulr 13237  df-0g 13404  df-iimas 13448  df-qus 13449  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-grp 13649  df-minusg 13650  df-subg 13820  df-nsg 13821  df-eqg 13822

This theorem is referenced by:  qusinv  13886