Theorem qus0 13365

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 13335	. . . . . . 7 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
2		subgrcl 13309	. . . . . . 7 |- ( S e. (SubGrp ` G ) -> G e. Grp )
3	1, 2	syl 14	. . . . . 6 |- ( S e. (NrmSGrp ` G ) -> G e. Grp )
4		eqid 2196	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
5		qus0.p	. . . . . . 7 |- .0. = ( 0g ` G )
6	4, 5	grpidcl 13161	. . . . . 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 2196	. . . . . 6 |- ( +g ` G ) = ( +g ` G )
10		eqid 2196	. . . . . 6 |- ( +g ` H ) = ( +g ` H )
11	8, 4, 9, 10	qusadd 13364	. . . . 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 1350	. . . 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 13163	. . . . . 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 6628	. . . 4 |- ( S e. (NrmSGrp ` G ) -> [ ( .0. ( +g ` G ) .0. ) ] ( G ~QG S ) = [ .0. ] ( G ~QG S ) )
16	12, 15	eqtrd 2229	. . 3 |- ( S e. (NrmSGrp ` G ) -> ( [ .0. ] ( G ~QG S ) ( +g ` H ) [ .0. ] ( G ~QG S ) ) = [ .0. ] ( G ~QG S ) )
17	8	qusgrp 13362	. . . 4 |- ( S e. (NrmSGrp ` G ) -> H e. Grp )
18		eqid 2196	. . . . . 6 |- ( Base ` H ) = ( Base ` H )
19	8, 4, 18	quseccl 13363	. . . . 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 2196	. . . . 5 |- ( 0g ` H ) = ( 0g ` H )
22	18, 10, 21	grpid 13171	. . . 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 2202	1 |- ( S e. (NrmSGrp ` G ) -> [ .0. ] ( G ~QG S ) = ( 0g ` H ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2167   ` cfv 5258  (class class class)co 5922   [cec 6590   Basecbs 12678   +g cplusg 12755   0gc0g 12927   /._s cqus 12943   Grpcgrp 13132  SubGrpcsubg 13297  NrmSGrpcnsg 13298   ~_QG cqg 13299

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-tp 3630  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-er 6592  df-ec 6594  df-qs 6598  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-0g 12929  df-iimas 12945  df-qus 12946  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-subg 13300  df-nsg 13301  df-eqg 13302

This theorem is referenced by:  qusinv  13366