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Theorem qusinv 13970
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 13939 . . . . . 6 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
2 subgrcl 13913 . . . . . 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 13778 . . . . 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 2234 . . . . 5 |- ( +g ` G ) = ( +g ` G )
10 eqid 2234 . . . . 5 |- ( +g ` H ) = ( +g ` H )
118, 4, 9, 10qusadd 13968 . . . 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 1375 . . 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 2234 . . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
144, 9, 13, 5grprinv 13781 . . . . 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 6805 . . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> [ ( X ( +g ` G ) ( I ` X ) ) ] ( G ~QG S ) = [ ( 0g ` G ) ] ( G ~QG S ) )
178, 13qus0 13969 . . . 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 2271 . 2 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> ( [ X ] ( G ~QG S ) ( +g ` H ) [ ( I ` X ) ] ( G ~QG S ) ) = ( 0g ` H ) )
208qusgrp 13966 . . . 4 |- ( S e. (NrmSGrp ` G ) -> H e. Grp )
2120adantr 276 . . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> H e. Grp )
22 eqid 2234 . . . 4 |- ( Base ` H ) = ( Base ` H )
238, 4, 22quseccl 13967 . . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> [ X ] ( G ~QG S ) e. ( Base ` H ) )
248, 4, 22quseccl 13967 . . . 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 2234 . . . 4 |- ( 0g ` H ) = ( 0g ` H )
27 qusinv.n . . . 4 |- N = ( invg ` H )
2822, 10, 26, 27grpinvid1 13782 . . 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 1274 . 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 1398   e. wcel 2205   ` cfv 5354  (class class class)co 6052   [cec 6767   Basecbs 13229   +g cplusg 13307   0gc0g 13486   /.s cqus 13530   Grpcgrp 13730   invgcminusg 13731  SubGrpcsubg 13901  NrmSGrpcnsg 13902   ~QG cqg 13903
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 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-pre-ltirr 8241  ax-pre-lttrn 8243  ax-pre-ltadd 8245
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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-tp 3699  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-er 6769  df-ec 6771  df-qs 6775  df-pnf 8312  df-mnf 8313  df-ltxr 8315  df-inn 9240  df-2 9298  df-3 9299  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-iress 13237  df-plusg 13320  df-mulr 13321  df-0g 13488  df-iimas 13532  df-qus 13533  df-mgm 13586  df-sgrp 13632  df-mnd 13647  df-grp 13733  df-minusg 13734  df-subg 13904  df-nsg 13905  df-eqg 13906
This theorem is referenced by:  qussub  13971
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