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Theorem ghmnsgpreima 13801
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 13737 . . 3 |- ( V e. (NrmSGrp ` T ) -> V e. (SubGrp ` T ) )
2 ghmpreima 13798 . . 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 13777 . . . . . 6 |- ( F e. ( S GrpHom T ) -> S e. Grp )
54ad2antrr 488 . . . . 5 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> S e. Grp )
6 simprl 529 . . . . . 6 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> x e. ( Base ` S ) )
7 simprr 531 . . . . . . . 8 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> y e. ( `' F " V ) )
8 simpll 527 . . . . . . . . . . 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 2229 . . . . . . . . . . . 12 |- ( Base ` S ) = ( Base ` S )
10 eqid 2229 . . . . . . . . . . . 12 |- ( Base ` T ) = ( Base ` T )
119, 10ghmf 13779 . . . . . . . . . . 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 5473 . . . . . . . . 9 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> F Fn ( Base ` S ) )
14 elpreima 5753 . . . . . . . . 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 2229 . . . . . . 7 |- ( +g ` S ) = ( +g ` S )
199, 18grpcl 13536 . . . . . 6 |- ( ( S e. Grp /\ x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( x ( +g ` S ) y ) e. ( Base ` S ) )
205, 6, 17, 19syl3anc 1271 . . . . 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 2229 . . . . . 6 |- ( -g ` S ) = ( -g ` S )
229, 21grpsubcl 13608 . . . . 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 1271 . . . 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 2229 . . . . . . . 8 |- ( -g ` T ) = ( -g ` T )
259, 21, 24ghmsub 13783 . . . . . . 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 1271 . . . . . 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 2229 . . . . . . . . 9 |- ( +g ` T ) = ( +g ` T )
289, 18, 27ghmlin 13780 . . . . . . . 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 1271 . . . . . . 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 6015 . . . . . 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 2262 . . . . 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 528 . . . . . 6 |- ( ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( `' F " V ) ) ) -> V e. (NrmSGrp ` T ) )
3312, 6ffvelcdmd 5770 . . . . . 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 13738 . . . . . 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 1271 . . . . 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 2306 . . . 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 5753 . . . . 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 950 . . 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 2612 . 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 13739 . 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 417 1 |- ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) -> ( `' F " V ) e. (NrmSGrp ` S ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   `'ccnv 4717   "cima 4721   Fn wfn 5312   -->wf 5313   ` cfv 5317  (class class class)co 6000   Basecbs 13027   +g cplusg 13105   Grpcgrp 13528   -gcsg 13530  SubGrpcsubg 13699  NrmSGrpcnsg 13700   GrpHom cghm 13772
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-pre-ltirr 8107  ax-pre-ltadd 8111
This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-pnf 8179  df-mnf 8180  df-ltxr 8182  df-inn 9107  df-2 9165  df-ndx 13030  df-slot 13031  df-base 13033  df-sets 13034  df-iress 13035  df-plusg 13118  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-grp 13531  df-minusg 13532  df-sbg 13533  df-subg 13702  df-nsg 13703  df-ghm 13773
This theorem is referenced by:  ghmker  13802
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