ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  qusinv Unicode version
Theorem qusinv 13309
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 13278 . . . . . 6 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
2 subgrcl 13252 . . . . . 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 13123 . . . . 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 13307 . . . 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 13126 . . . . 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 6625 . . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> [ ( X ( +g ` G ) ( I ` X ) ) ] ( G ~QG S ) = [ ( 0g ` G ) ] ( G ~QG S ) )
178, 13qus0 13308 . . . 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 13305 . . . 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 13306 . . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> [ X ] ( G ~QG S ) e. ( Base ` H ) )
248, 4, 22quseccl 13306 . . . 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 13127 . . 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 5255  (class class class)co 5919   [cec 6587   Basecbs 12621   +g cplusg 12698   0gc0g 12870   /.s cqus 12886   Grpcgrp 13075   invgcminusg 13076  SubGrpcsubg 13240  NrmSGrpcnsg 13241   ~QG cqg 13242
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 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-lttrn 7988  ax-pre-ltadd 7990
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 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-tp 3627  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-er 6589  df-ec 6591  df-qs 6595  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629  df-plusg 12711  df-mulr 12712  df-0g 12872  df-iimas 12888  df-qus 12889  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-minusg 13079  df-subg 13243  df-nsg 13244  df-eqg 13245
This theorem is referenced by:  qussub  13310
  Copyright terms: Public domain W3C validator