Theorem qus0 13827

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 13797	. . . . . . 7 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
2		subgrcl 13771	. . . . . . 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 13617	. . . . . 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 13826	. . . . 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 13619	. . . . . 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 6738	. . . 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 13824	. . . 4 |- ( S e. (NrmSGrp ` G ) -> H e. Grp )
18		eqid 2231	. . . . . 6 |- ( Base ` H ) = ( Base ` H )
19	8, 4, 18	quseccl 13825	. . . . 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 13627	. . . 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 6018   [cec 6700   Basecbs 13087   +g cplusg 13165   0gc0g 13344   /._s cqus 13388   Grpcgrp 13588  SubGrpcsubg 13759  NrmSGrpcnsg 13760   ~_QG cqg 13761

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 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-lttrn 8146  ax-pre-ltadd 8148

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 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-er 6702  df-ec 6704  df-qs 6708  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13090  df-slot 13091  df-base 13093  df-sets 13094  df-iress 13095  df-plusg 13178  df-mulr 13179  df-0g 13346  df-iimas 13390  df-qus 13391  df-mgm 13444  df-sgrp 13490  df-mnd 13505  df-grp 13591  df-minusg 13592  df-subg 13762  df-nsg 13763  df-eqg 13764

This theorem is referenced by:  qusinv  13828