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Theorem ghmnsgpreima 17886
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 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (𝐹𝑉) ∈ (NrmSGrp‘𝑆))

Proof of Theorem ghmnsgpreima
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nsgsubg 17827 . . 3 (𝑉 ∈ (NrmSGrp‘𝑇) → 𝑉 ∈ (SubGrp‘𝑇))
2 ghmpreima 17883 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹𝑉) ∈ (SubGrp‘𝑆))
31, 2sylan2 492 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (𝐹𝑉) ∈ (SubGrp‘𝑆))
4 ghmgrp1 17863 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
54ad2antrr 764 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑆 ∈ Grp)
6 simprl 811 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑥 ∈ (Base‘𝑆))
7 simprr 813 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑦 ∈ (𝐹𝑉))
8 simpll 807 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
9 eqid 2760 . . . . . . . . . . . 12 (Base‘𝑆) = (Base‘𝑆)
10 eqid 2760 . . . . . . . . . . . 12 (Base‘𝑇) = (Base‘𝑇)
119, 10ghmf 17865 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
128, 11syl 17 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
13 ffn 6206 . . . . . . . . . 10 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
1412, 13syl 17 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝐹 Fn (Base‘𝑆))
15 elpreima 6500 . . . . . . . . 9 (𝐹 Fn (Base‘𝑆) → (𝑦 ∈ (𝐹𝑉) ↔ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) ∈ 𝑉)))
1614, 15syl 17 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝑦 ∈ (𝐹𝑉) ↔ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) ∈ 𝑉)))
177, 16mpbid 222 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) ∈ 𝑉))
1817simpld 477 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑦 ∈ (Base‘𝑆))
19 eqid 2760 . . . . . . 7 (+g𝑆) = (+g𝑆)
209, 19grpcl 17631 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
215, 6, 18, 20syl3anc 1477 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
22 eqid 2760 . . . . . 6 (-g𝑆) = (-g𝑆)
239, 22grpsubcl 17696 . . . . 5 ((𝑆 ∈ Grp ∧ (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (Base‘𝑆))
245, 21, 6, 23syl3anc 1477 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → ((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (Base‘𝑆))
25 eqid 2760 . . . . . . . 8 (-g𝑇) = (-g𝑇)
269, 22, 25ghmsub 17869 . . . . . . 7 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐹‘((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥)) = ((𝐹‘(𝑥(+g𝑆)𝑦))(-g𝑇)(𝐹𝑥)))
278, 21, 6, 26syl3anc 1477 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹‘((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥)) = ((𝐹‘(𝑥(+g𝑆)𝑦))(-g𝑇)(𝐹𝑥)))
28 eqid 2760 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
299, 19, 28ghmlin 17866 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
308, 6, 18, 29syl3anc 1477 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3130oveq1d 6828 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → ((𝐹‘(𝑥(+g𝑆)𝑦))(-g𝑇)(𝐹𝑥)) = (((𝐹𝑥)(+g𝑇)(𝐹𝑦))(-g𝑇)(𝐹𝑥)))
3227, 31eqtrd 2794 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹‘((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥)) = (((𝐹𝑥)(+g𝑇)(𝐹𝑦))(-g𝑇)(𝐹𝑥)))
33 simplr 809 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑉 ∈ (NrmSGrp‘𝑇))
3412, 6ffvelrnd 6523 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹𝑥) ∈ (Base‘𝑇))
3517simprd 482 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹𝑦) ∈ 𝑉)
3610, 28, 25nsgconj 17828 . . . . . 6 ((𝑉 ∈ (NrmSGrp‘𝑇) ∧ (𝐹𝑥) ∈ (Base‘𝑇) ∧ (𝐹𝑦) ∈ 𝑉) → (((𝐹𝑥)(+g𝑇)(𝐹𝑦))(-g𝑇)(𝐹𝑥)) ∈ 𝑉)
3733, 34, 35, 36syl3anc 1477 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (((𝐹𝑥)(+g𝑇)(𝐹𝑦))(-g𝑇)(𝐹𝑥)) ∈ 𝑉)
3832, 37eqeltrd 2839 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹‘((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥)) ∈ 𝑉)
39 elpreima 6500 . . . . 5 (𝐹 Fn (Base‘𝑆) → (((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (𝐹𝑉) ↔ (((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (Base‘𝑆) ∧ (𝐹‘((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥)) ∈ 𝑉)))
4014, 39syl 17 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (𝐹𝑉) ↔ (((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (Base‘𝑆) ∧ (𝐹‘((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥)) ∈ 𝑉)))
4124, 38, 40mpbir2and 995 . . 3 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → ((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (𝐹𝑉))
4241ralrimivva 3109 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (𝐹𝑉)((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (𝐹𝑉))
439, 19, 22isnsg3 17829 . 2 ((𝐹𝑉) ∈ (NrmSGrp‘𝑆) ↔ ((𝐹𝑉) ∈ (SubGrp‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (𝐹𝑉)((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (𝐹𝑉)))
443, 42, 43sylanbrc 701 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (𝐹𝑉) ∈ (NrmSGrp‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  ccnv 5265  cima 5269   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6813  Basecbs 16059  +gcplusg 16143  Grpcgrp 17623  -gcsg 17625  SubGrpcsubg 17789  NrmSGrpcnsg 17790   GrpHom cghm 17858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-0g 16304  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-grp 17626  df-minusg 17627  df-sbg 17628  df-subg 17792  df-nsg 17793  df-ghm 17859
This theorem is referenced by:  ghmker  17887
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