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Theorem ghmnsgpreima 14072
Description: The inverse image of a normal subgroup under a homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
Assertion
Ref Expression
ghmnsgpreima |- ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) -> ( `' F " V ) e. (NrmSGrp ` S ) )
Proof of Theorem ghmnsgpreima
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nsgsubg 14008 . . 3 |- ( V e. (NrmSGrp ` T ) -> V e. (SubGrp ` T ) )
2 ghmpreima 14069 . . 3 |- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( `' F " V ) e. (SubGrp ` S ) )
31, 2sylan2 286 . 2 |- ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) -> ( `' F " V ) e. (SubGrp ` S ) )
4 ghmgrp1 14048 . . . . . 6 |- ( F e. ( S GrpHom T ) -> S e. Grp )
54ad2antrr 492 . . . . 5 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> S e. Grp )
6 simprl 535 . . . . . 6 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> x e. ( Base ` S ) )
7 simprr 537 . . . . . . . 8 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> y e. ( `' F " V ) )
8 simpll 531 . . . . . . . . . . 11 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> F e. ( S GrpHom T ) )
9 eqid 2238 . . . . . . . . . . . 12 |- ( Base ` S ) = ( Base ` S )
10 eqid 2238 . . . . . . . . . . . 12 |- ( Base ` T ) = ( Base ` T )
119, 10ghmf 14050 . . . . . . . . . . 11 |- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
128, 11syl 14 . . . . . . . . . 10 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> F : ( Base ` S ) --> ( Base ` T ) )
1312ffnd 5534 . . . . . . . . 9 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> F Fn ( Base ` S ) )
14 elpreima 5828 . . . . . . . . 9 |- ( F Fn ( Base ` S ) -> ( y e. ( `' F " V ) <-> ( y e. ( Base ` S ) /\ ( F ` y ) e. V ) ) )
1513, 14syl 14 . . . . . . . 8 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> ( y e. ( `' F " V ) <-> ( y e. ( Base ` S ) /\ ( F ` y ) e. V ) ) )
167, 15mpbid 147 . . . . . . 7 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> ( y e. ( Base ` S ) /\ ( F ` y ) e. V ) )
1716simpld 112 . . . . . 6 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> y e. ( Base ` S ) )
18 eqid 2238 . . . . . . 7 |- ( +g ` S ) = ( +g ` S )
199, 18grpcl 13813 . . . . . 6 |- ( ( S e. Grp /\ x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( x ( +g ` S ) y ) e. ( Base ` S ) )
205, 6, 17, 19syl3anc 1278 . . . . 5 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> ( x ( +g ` S ) y ) e. ( Base ` S ) )
21 eqid 2238 . . . . . 6 |- ( -g ` S ) = ( -g ` S )
229, 21grpsubcl 13885 . . . . 5 |- ( ( S e. Grp /\ ( x ( +g ` S ) y ) e. ( Base ` S ) /\ x e. ( Base ` S ) ) -> ( ( x ( +g ` S ) y ) ( -g ` S ) x ) e. ( Base ` S ) )
235, 20, 6, 22syl3anc 1278 . . . 4 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> ( ( x ( +g ` S ) y ) ( -g ` S ) x ) e. ( Base ` S ) )
24 eqid 2238 . . . . . . . 8 |- ( -g ` T ) = ( -g ` T )
259, 21, 24ghmsub 14054 . . . . . . 7 |- ( ( F e. ( S GrpHom T ) /\ ( x ( +g ` S ) y ) e. ( Base ` S ) /\ x e. ( Base ` S ) ) -> ( F ` ( ( x ( +g ` S ) y ) ( -g ` S ) x ) ) = ( ( F ` ( x ( +g ` S ) y ) ) ( -g ` T ) ( F ` x ) ) )
268, 20, 6, 25syl3anc 1278 . . . . . 6 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> ( F ` ( ( x ( +g ` S ) y ) ( -g ` S ) x ) ) = ( ( F ` ( x ( +g ` S ) y ) ) ( -g ` T ) ( F ` x ) ) )
27 eqid 2238 . . . . . . . . 9 |- ( +g ` T ) = ( +g ` T )
289, 18, 27ghmlin 14051 . . . . . . . 8 |- ( ( F e. ( S GrpHom T ) /\ x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
298, 6, 17, 28syl3anc 1278 . . . . . . 7 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
3029oveq1d 6100 . . . . . 6 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> ( ( F ` ( x ( +g ` S ) y ) ) ( -g ` T ) ( F ` x ) ) = ( ( ( F ` x ) ( +g ` T ) ( F ` y ) ) ( -g ` T ) ( F ` x ) ) )
3126, 30eqtrd 2271 . . . . 5 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> ( F ` ( ( x ( +g ` S ) y ) ( -g ` S ) x ) ) = ( ( ( F ` x ) ( +g ` T ) ( F ` y ) ) ( -g ` T ) ( F ` x ) ) )
32 simplr 533 . . . . . 6 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> V e. (NrmSGrp ` T ) )
3312, 6ffvelcdmd 5844 . . . . . 6 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> ( F ` x ) e. ( Base ` T ) )
3416simprd 114 . . . . . 6 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> ( F ` y ) e. V )
3510, 27, 24nsgconj 14009 . . . . . 6 |- ( ( V e. (NrmSGrp ` T ) /\ ( F ` x ) e. ( Base ` T ) /\ ( F ` y ) e. V ) -> ( ( ( F ` x ) ( +g ` T ) ( F ` y ) ) ( -g ` T ) ( F ` x ) ) e. V )
3632, 33, 34, 35syl3anc 1278 . . . . 5 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> ( ( ( F ` x ) ( +g ` T ) ( F ` y ) ) ( -g ` T ) ( F ` x ) ) e. V )
3731, 36eqeltrd 2315 . . . 4 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> ( F ` ( ( x ( +g ` S ) y ) ( -g ` S ) x ) ) e. V )
38 elpreima 5828 . . . . 5 |- ( F Fn ( Base ` S ) -> ( ( ( x ( +g ` S ) y ) ( -g ` S ) x ) e. ( `' F " V ) <-> ( ( ( x ( +g ` S ) y ) ( -g ` S ) x ) e. ( Base ` S ) /\ ( F ` ( ( x ( +g ` S ) y ) ( -g ` S ) x ) ) e. V ) ) )
3913, 38syl 14 . . . 4 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> ( ( ( x ( +g ` S ) y ) ( -g ` S ) x ) e. ( `' F " V ) <-> ( ( ( x ( +g ` S ) y ) ( -g ` S ) x ) e. ( Base ` S ) /\ ( F ` ( ( x ( +g ` S ) y ) ( -g ` S ) x ) ) e. V ) ) )
4023, 37, 39mpbir2and 957 . . 3 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> ( ( x ( +g ` S ) y ) ( -g ` S ) x ) e. ( `' F " V ) )
4140ralrimivva 2632 . 2 |- ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) -> A. x e. ( Base ` S ) A. y e. ( `' F " V ) ( ( x ( +g ` S ) y ) ( -g ` S ) x ) e. ( `' F " V ) )
429, 18, 21isnsg3 14010 . 2 |- ( ( `' F " V ) e. (NrmSGrp ` S ) <-> ( ( `' F " V ) e. (SubGrp ` S ) /\ A. x e. ( Base ` S ) A. y e. ( `' F " V ) ( ( x ( +g ` S ) y ) ( -g ` S ) x ) e. ( `' F " V ) ) )
433, 41, 42sylanbrc 421 1 |- ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) -> ( `' F " V ) e. (NrmSGrp ` S ) )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   `'ccnv 4773   "cima 4777   Fn wfn 5372   -->wf 5373   ` cfv 5377  (class class class)co 6085   Basecbs 13352   +g cplusg 13431   Grpcgrp 13805   -gcsg 13807  SubGrpcsubg 13970  NrmSGrpcnsg 13971   GrpHom cghm 14043
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295
This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809  df-sbg 13810  df-subg 13973  df-nsg 13974  df-ghm 14044
This theorem is used by:  ghmker  14073
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