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Theorem ghmnsgpreima 17508
 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 17449 . . 3 (𝑉 ∈ (NrmSGrp‘𝑇) → 𝑉 ∈ (SubGrp‘𝑇))
2 ghmpreima 17505 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹𝑉) ∈ (SubGrp‘𝑆))
31, 2sylan2 490 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (𝐹𝑉) ∈ (SubGrp‘𝑆))
4 ghmgrp1 17485 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
54ad2antrr 758 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑆 ∈ Grp)
6 simprl 790 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑥 ∈ (Base‘𝑆))
7 simprr 792 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑦 ∈ (𝐹𝑉))
8 simpll 786 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
9 eqid 2610 . . . . . . . . . . . 12 (Base‘𝑆) = (Base‘𝑆)
10 eqid 2610 . . . . . . . . . . . 12 (Base‘𝑇) = (Base‘𝑇)
119, 10ghmf 17487 . . . . . . . . . . 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 5958 . . . . . . . . . 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 6245 . . . . . . . . 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 221 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) ∈ 𝑉))
1817simpld 474 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑦 ∈ (Base‘𝑆))
19 eqid 2610 . . . . . . 7 (+g𝑆) = (+g𝑆)
209, 19grpcl 17253 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
215, 6, 18, 20syl3anc 1318 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
22 eqid 2610 . . . . . 6 (-g𝑆) = (-g𝑆)
239, 22grpsubcl 17318 . . . . 5 ((𝑆 ∈ Grp ∧ (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (Base‘𝑆))
245, 21, 6, 23syl3anc 1318 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → ((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (Base‘𝑆))
25 eqid 2610 . . . . . . . 8 (-g𝑇) = (-g𝑇)
269, 22, 25ghmsub 17491 . . . . . . 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 1318 . . . . . 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 2610 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
299, 19, 28ghmlin 17488 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
308, 6, 18, 29syl3anc 1318 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3130oveq1d 6564 . . . . . 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 2644 . . . . 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 788 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑉 ∈ (NrmSGrp‘𝑇))
3412, 6ffvelrnd 6268 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹𝑥) ∈ (Base‘𝑇))
3517simprd 478 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹𝑦) ∈ 𝑉)
3610, 28, 25nsgconj 17450 . . . . . 6 ((𝑉 ∈ (NrmSGrp‘𝑇) ∧ (𝐹𝑥) ∈ (Base‘𝑇) ∧ (𝐹𝑦) ∈ 𝑉) → (((𝐹𝑥)(+g𝑇)(𝐹𝑦))(-g𝑇)(𝐹𝑥)) ∈ 𝑉)
3733, 34, 35, 36syl3anc 1318 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (((𝐹𝑥)(+g𝑇)(𝐹𝑦))(-g𝑇)(𝐹𝑥)) ∈ 𝑉)
3832, 37eqeltrd 2688 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹‘((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥)) ∈ 𝑉)
39 elpreima 6245 . . . . 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 959 . . 3 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → ((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (𝐹𝑉))
4241ralrimivva 2954 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (𝐹𝑉)((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (𝐹𝑉))
439, 19, 22isnsg3 17451 . 2 ((𝐹𝑉) ∈ (NrmSGrp‘𝑆) ↔ ((𝐹𝑉) ∈ (SubGrp‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (𝐹𝑉)((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (𝐹𝑉)))
443, 42, 43sylanbrc 695 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (𝐹𝑉) ∈ (NrmSGrp‘𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ◡ccnv 5037   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  Grpcgrp 17245  -gcsg 17247  SubGrpcsubg 17411  NrmSGrpcnsg 17412   GrpHom cghm 17480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-nsg 17415  df-ghm 17481 This theorem is referenced by:  ghmker  17509
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