Theorem qus0 13821

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 13791	. . . . . . 7 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
2		subgrcl 13765	. . . . . . 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 13611	. . . . . 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 13820	. . . . 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 1375	. . . 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 13613	. . . . . 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 6737	. . . 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 13818	. . . 4 |- ( S e. (NrmSGrp ` G ) -> H e. Grp )
18		eqid 2231	. . . . . 6 |- ( Base ` H ) = ( Base ` H )
19	8, 4, 18	quseccl 13819	. . . . 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 13621	. . . 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 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   [cec 6699   Basecbs 13081   +g cplusg 13159   0gc0g 13338   /._s cqus 13382   Grpcgrp 13582  SubGrpcsubg 13753  NrmSGrpcnsg 13754   ~_QG cqg 13755

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-tp 3677  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-er 6701  df-ec 6703  df-qs 6707  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089  df-plusg 13172  df-mulr 13173  df-0g 13340  df-iimas 13384  df-qus 13385  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-subg 13756  df-nsg 13757  df-eqg 13758

This theorem is referenced by:  qusinv  13822