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Theorem ghmnsgpreima 18377
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 18304 . . 3 (𝑉 ∈ (NrmSGrp‘𝑇) → 𝑉 ∈ (SubGrp‘𝑇))
2 ghmpreima 18374 . . 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 18354 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
54ad2antrr 724 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑆 ∈ Grp)
6 simprl 769 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑥 ∈ (Base‘𝑆))
7 simprr 771 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑦 ∈ (𝐹𝑉))
8 simpll 765 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
9 eqid 2821 . . . . . . . . . . . 12 (Base‘𝑆) = (Base‘𝑆)
10 eqid 2821 . . . . . . . . . . . 12 (Base‘𝑇) = (Base‘𝑇)
119, 10ghmf 18356 . . . . . . . . . . 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 6510 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝐹 Fn (Base‘𝑆))
14 elpreima 6823 . . . . . . . . 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 234 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) ∈ 𝑉))
1716simpld 497 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑦 ∈ (Base‘𝑆))
18 eqid 2821 . . . . . . 7 (+g𝑆) = (+g𝑆)
199, 18grpcl 18105 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
205, 6, 17, 19syl3anc 1367 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
21 eqid 2821 . . . . . 6 (-g𝑆) = (-g𝑆)
229, 21grpsubcl 18173 . . . . 5 ((𝑆 ∈ Grp ∧ (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (Base‘𝑆))
235, 20, 6, 22syl3anc 1367 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → ((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (Base‘𝑆))
24 eqid 2821 . . . . . . . 8 (-g𝑇) = (-g𝑇)
259, 21, 24ghmsub 18360 . . . . . . 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 1367 . . . . . 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 2821 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
289, 18, 27ghmlin 18357 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
298, 6, 17, 28syl3anc 1367 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3029oveq1d 7165 . . . . . 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 2856 . . . . 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 767 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑉 ∈ (NrmSGrp‘𝑇))
3312, 6ffvelrnd 6847 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹𝑥) ∈ (Base‘𝑇))
3416simprd 498 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹𝑦) ∈ 𝑉)
3510, 27, 24nsgconj 18305 . . . . . 6 ((𝑉 ∈ (NrmSGrp‘𝑇) ∧ (𝐹𝑥) ∈ (Base‘𝑇) ∧ (𝐹𝑦) ∈ 𝑉) → (((𝐹𝑥)(+g𝑇)(𝐹𝑦))(-g𝑇)(𝐹𝑥)) ∈ 𝑉)
3632, 33, 34, 35syl3anc 1367 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (((𝐹𝑥)(+g𝑇)(𝐹𝑦))(-g𝑇)(𝐹𝑥)) ∈ 𝑉)
3731, 36eqeltrd 2913 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹‘((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥)) ∈ 𝑉)
38 elpreima 6823 . . . . 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 711 . . 3 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → ((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (𝐹𝑉))
4140ralrimivva 3191 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (𝐹𝑉)((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (𝐹𝑉))
429, 18, 21isnsg3 18306 . 2 ((𝐹𝑉) ∈ (NrmSGrp‘𝑆) ↔ ((𝐹𝑉) ∈ (SubGrp‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (𝐹𝑉)((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (𝐹𝑉)))
433, 41, 42sylanbrc 585 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (𝐹𝑉) ∈ (NrmSGrp‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  wral 3138  ccnv 5549  cima 5553   Fn wfn 6345  wf 6346  cfv 6350  (class class class)co 7150  Basecbs 16477  +gcplusg 16559  Grpcgrp 18097  -gcsg 18099  SubGrpcsubg 18267  NrmSGrpcnsg 18268   GrpHom cghm 18349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-plusg 16572  df-0g 16709  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-grp 18100  df-minusg 18101  df-sbg 18102  df-subg 18270  df-nsg 18271  df-ghm 18350
This theorem is referenced by:  ghmker  18378
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