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Theorem ghmnsgpreima 19117
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 19038 . . 3 (𝑉 ∈ (NrmSGrp‘𝑇) → 𝑉 ∈ (SubGrp‘𝑇))
2 ghmpreima 19114 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹𝑉) ∈ (SubGrp‘𝑆))
31, 2sylan2 594 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (𝐹𝑉) ∈ (SubGrp‘𝑆))
4 ghmgrp1 19094 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
54ad2antrr 725 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑆 ∈ Grp)
6 simprl 770 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑥 ∈ (Base‘𝑆))
7 simprr 772 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑦 ∈ (𝐹𝑉))
8 simpll 766 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
9 eqid 2733 . . . . . . . . . . . 12 (Base‘𝑆) = (Base‘𝑆)
10 eqid 2733 . . . . . . . . . . . 12 (Base‘𝑇) = (Base‘𝑇)
119, 10ghmf 19096 . . . . . . . . . . 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))
1312ffnd 6719 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝐹 Fn (Base‘𝑆))
14 elpreima 7060 . . . . . . . . 9 (𝐹 Fn (Base‘𝑆) → (𝑦 ∈ (𝐹𝑉) ↔ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) ∈ 𝑉)))
1513, 14syl 17 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝑦 ∈ (𝐹𝑉) ↔ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) ∈ 𝑉)))
167, 15mpbid 231 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) ∈ 𝑉))
1716simpld 496 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑦 ∈ (Base‘𝑆))
18 eqid 2733 . . . . . . 7 (+g𝑆) = (+g𝑆)
199, 18grpcl 18827 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
205, 6, 17, 19syl3anc 1372 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
21 eqid 2733 . . . . . 6 (-g𝑆) = (-g𝑆)
229, 21grpsubcl 18903 . . . . 5 ((𝑆 ∈ Grp ∧ (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (Base‘𝑆))
235, 20, 6, 22syl3anc 1372 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → ((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (Base‘𝑆))
24 eqid 2733 . . . . . . . 8 (-g𝑇) = (-g𝑇)
259, 21, 24ghmsub 19100 . . . . . . 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)))
268, 20, 6, 25syl3anc 1372 . . . . . 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)))
27 eqid 2733 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
289, 18, 27ghmlin 19097 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
298, 6, 17, 28syl3anc 1372 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3029oveq1d 7424 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → ((𝐹‘(𝑥(+g𝑆)𝑦))(-g𝑇)(𝐹𝑥)) = (((𝐹𝑥)(+g𝑇)(𝐹𝑦))(-g𝑇)(𝐹𝑥)))
3126, 30eqtrd 2773 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹‘((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥)) = (((𝐹𝑥)(+g𝑇)(𝐹𝑦))(-g𝑇)(𝐹𝑥)))
32 simplr 768 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑉 ∈ (NrmSGrp‘𝑇))
3312, 6ffvelcdmd 7088 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹𝑥) ∈ (Base‘𝑇))
3416simprd 497 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹𝑦) ∈ 𝑉)
3510, 27, 24nsgconj 19039 . . . . . 6 ((𝑉 ∈ (NrmSGrp‘𝑇) ∧ (𝐹𝑥) ∈ (Base‘𝑇) ∧ (𝐹𝑦) ∈ 𝑉) → (((𝐹𝑥)(+g𝑇)(𝐹𝑦))(-g𝑇)(𝐹𝑥)) ∈ 𝑉)
3632, 33, 34, 35syl3anc 1372 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (((𝐹𝑥)(+g𝑇)(𝐹𝑦))(-g𝑇)(𝐹𝑥)) ∈ 𝑉)
3731, 36eqeltrd 2834 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹‘((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥)) ∈ 𝑉)
38 elpreima 7060 . . . . 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)))
3913, 38syl 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)))
4023, 37, 39mpbir2and 712 . . 3 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → ((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (𝐹𝑉))
4140ralrimivva 3201 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (𝐹𝑉)((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (𝐹𝑉))
429, 18, 21isnsg3 19040 . 2 ((𝐹𝑉) ∈ (NrmSGrp‘𝑆) ↔ ((𝐹𝑉) ∈ (SubGrp‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (𝐹𝑉)((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (𝐹𝑉)))
433, 41, 42sylanbrc 584 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (𝐹𝑉) ∈ (NrmSGrp‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  ccnv 5676  cima 5680   Fn wfn 6539  wf 6540  cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  Grpcgrp 18819  -gcsg 18821  SubGrpcsubg 19000  NrmSGrpcnsg 19001   GrpHom cghm 19089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-sbg 18824  df-subg 19003  df-nsg 19004  df-ghm 19090
This theorem is referenced by:  ghmker  19118
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