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Theorem qusinv 13306
Description: Value of the group inverse operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
Hypotheses
Ref Expression
qusgrp.h |- H = ( G /.s ( G ~QG S ) )
qusinv.v |- V = ( Base ` G )
qusinv.i |- I = ( invg ` G )
qusinv.n |- N = ( invg ` H )
Assertion
Ref Expression
qusinv |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> ( N ` [ X ] ( G ~QG S ) ) = [ ( I ` X ) ] ( G ~QG S ) )
Proof of Theorem qusinv
StepHypRef Expression
1 nsgsubg 13275 . . . . . 6 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
2 subgrcl 13249 . . . . . 6 |- ( S e. (SubGrp ` G ) -> G e. Grp )
31, 2syl 14 . . . . 5 |- ( S e. (NrmSGrp ` G ) -> G e. Grp )
4 qusinv.v . . . . . 6 |- V = ( Base ` G )
5 qusinv.i . . . . . 6 |- I = ( invg ` G )
64, 5grpinvcl 13120 . . . . 5 |- ( ( G e. Grp /\ X e. V ) -> ( I ` X ) e. V )
73, 6sylan 283 . . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> ( I ` X ) e. V )
8 qusgrp.h . . . . 5 |- H = ( G /.s ( G ~QG S ) )
9 eqid 2193 . . . . 5 |- ( +g ` G ) = ( +g ` G )
10 eqid 2193 . . . . 5 |- ( +g ` H ) = ( +g ` H )
118, 4, 9, 10qusadd 13304 . . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ ( I ` X ) e. V ) -> ( [ X ] ( G ~QG S ) ( +g ` H ) [ ( I ` X ) ] ( G ~QG S ) ) = [ ( X ( +g ` G ) ( I ` X ) ) ] ( G ~QG S ) )
127, 11mpd3an3 1349 . . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> ( [ X ] ( G ~QG S ) ( +g ` H ) [ ( I ` X ) ] ( G ~QG S ) ) = [ ( X ( +g ` G ) ( I ` X ) ) ] ( G ~QG S ) )
13 eqid 2193 . . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
144, 9, 13, 5grprinv 13123 . . . . 5 |- ( ( G e. Grp /\ X e. V ) -> ( X ( +g ` G ) ( I ` X ) ) = ( 0g ` G ) )
153, 14sylan 283 . . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> ( X ( +g ` G ) ( I ` X ) ) = ( 0g ` G ) )
1615eceq1d 6623 . . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> [ ( X ( +g ` G ) ( I ` X ) ) ] ( G ~QG S ) = [ ( 0g ` G ) ] ( G ~QG S ) )
178, 13qus0 13305 . . . 4 |- ( S e. (NrmSGrp ` G ) -> [ ( 0g ` G ) ] ( G ~QG S ) = ( 0g ` H ) )
1817adantr 276 . . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> [ ( 0g ` G ) ] ( G ~QG S ) = ( 0g ` H ) )
1912, 16, 183eqtrd 2230 . 2 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> ( [ X ] ( G ~QG S ) ( +g ` H ) [ ( I ` X ) ] ( G ~QG S ) ) = ( 0g ` H ) )
208qusgrp 13302 . . . 4 |- ( S e. (NrmSGrp ` G ) -> H e. Grp )
2120adantr 276 . . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> H e. Grp )
22 eqid 2193 . . . 4 |- ( Base ` H ) = ( Base ` H )
238, 4, 22quseccl 13303 . . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> [ X ] ( G ~QG S ) e. ( Base ` H ) )
248, 4, 22quseccl 13303 . . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ ( I ` X ) e. V ) -> [ ( I ` X ) ] ( G ~QG S ) e. ( Base ` H ) )
257, 24syldan 282 . . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> [ ( I ` X ) ] ( G ~QG S ) e. ( Base ` H ) )
26 eqid 2193 . . . 4 |- ( 0g ` H ) = ( 0g ` H )
27 qusinv.n . . . 4 |- N = ( invg ` H )
2822, 10, 26, 27grpinvid1 13124 . . 3 |- ( ( H e. Grp /\ [ X ] ( G ~QG S ) e. ( Base ` H ) /\ [ ( I ` X ) ] ( G ~QG S ) e. ( Base ` H ) ) -> ( ( N ` [ X ] ( G ~QG S ) ) = [ ( I ` X ) ] ( G ~QG S ) <-> ( [ X ] ( G ~QG S ) ( +g ` H ) [ ( I ` X ) ] ( G ~QG S ) ) = ( 0g ` H ) ) )
2921, 23, 25, 28syl3anc 1249 . 2 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> ( ( N ` [ X ] ( G ~QG S ) ) = [ ( I ` X ) ] ( G ~QG S ) <-> ( [ X ] ( G ~QG S ) ( +g ` H ) [ ( I ` X ) ] ( G ~QG S ) ) = ( 0g ` H ) ) )
3019, 29mpbird 167 1 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> ( N ` [ X ] ( G ~QG S ) ) = [ ( I ` X ) ] ( G ~QG S ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   ` cfv 5254  (class class class)co 5918   [cec 6585   Basecbs 12618   +g cplusg 12695   0gc0g 12867   /.s cqus 12883   Grpcgrp 13072   invgcminusg 13073  SubGrpcsubg 13237  NrmSGrpcnsg 13238   ~QG cqg 13239
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-tp 3626  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-er 6587  df-ec 6589  df-qs 6593  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-0g 12869  df-iimas 12885  df-qus 12886  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-subg 13240  df-nsg 13241  df-eqg 13242
This theorem is referenced by:  qussub  13307
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