MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ghmnsgpreima Structured version   Visualization version   GIF version

Theorem ghmnsgpreima 17998
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 17939 . . 3 (𝑉 ∈ (NrmSGrp‘𝑇) → 𝑉 ∈ (SubGrp‘𝑇))
2 ghmpreima 17995 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹𝑉) ∈ (SubGrp‘𝑆))
31, 2sylan2 587 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (𝐹𝑉) ∈ (SubGrp‘𝑆))
4 ghmgrp1 17975 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
54ad2antrr 718 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑆 ∈ Grp)
6 simprl 788 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑥 ∈ (Base‘𝑆))
7 simprr 790 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑦 ∈ (𝐹𝑉))
8 simpll 784 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
9 eqid 2799 . . . . . . . . . . . 12 (Base‘𝑆) = (Base‘𝑆)
10 eqid 2799 . . . . . . . . . . . 12 (Base‘𝑇) = (Base‘𝑇)
119, 10ghmf 17977 . . . . . . . . . . 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 6257 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝐹 Fn (Base‘𝑆))
14 elpreima 6563 . . . . . . . . 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 224 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) ∈ 𝑉))
1716simpld 489 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑦 ∈ (Base‘𝑆))
18 eqid 2799 . . . . . . 7 (+g𝑆) = (+g𝑆)
199, 18grpcl 17746 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
205, 6, 17, 19syl3anc 1491 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
21 eqid 2799 . . . . . 6 (-g𝑆) = (-g𝑆)
229, 21grpsubcl 17811 . . . . 5 ((𝑆 ∈ Grp ∧ (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (Base‘𝑆))
235, 20, 6, 22syl3anc 1491 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → ((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (Base‘𝑆))
24 eqid 2799 . . . . . . . 8 (-g𝑇) = (-g𝑇)
259, 21, 24ghmsub 17981 . . . . . . 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 1491 . . . . . 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 2799 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
289, 18, 27ghmlin 17978 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
298, 6, 17, 28syl3anc 1491 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3029oveq1d 6893 . . . . . 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 2833 . . . . 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 786 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → 𝑉 ∈ (NrmSGrp‘𝑇))
3312, 6ffvelrnd 6586 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹𝑥) ∈ (Base‘𝑇))
3416simprd 490 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹𝑦) ∈ 𝑉)
3510, 27, 24nsgconj 17940 . . . . . 6 ((𝑉 ∈ (NrmSGrp‘𝑇) ∧ (𝐹𝑥) ∈ (Base‘𝑇) ∧ (𝐹𝑦) ∈ 𝑉) → (((𝐹𝑥)(+g𝑇)(𝐹𝑦))(-g𝑇)(𝐹𝑥)) ∈ 𝑉)
3632, 33, 34, 35syl3anc 1491 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (((𝐹𝑥)(+g𝑇)(𝐹𝑦))(-g𝑇)(𝐹𝑥)) ∈ 𝑉)
3731, 36eqeltrd 2878 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → (𝐹‘((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥)) ∈ 𝑉)
38 elpreima 6563 . . . . 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 705 . . 3 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝐹𝑉))) → ((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (𝐹𝑉))
4140ralrimivva 3152 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (𝐹𝑉)((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (𝐹𝑉))
429, 18, 21isnsg3 17941 . 2 ((𝐹𝑉) ∈ (NrmSGrp‘𝑆) ↔ ((𝐹𝑉) ∈ (SubGrp‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (𝐹𝑉)((𝑥(+g𝑆)𝑦)(-g𝑆)𝑥) ∈ (𝐹𝑉)))
433, 41, 42sylanbrc 579 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (𝐹𝑉) ∈ (NrmSGrp‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089  ccnv 5311  cima 5315   Fn wfn 6096  wf 6097  cfv 6101  (class class class)co 6878  Basecbs 16184  +gcplusg 16267  Grpcgrp 17738  -gcsg 17740  SubGrpcsubg 17901  NrmSGrpcnsg 17902   GrpHom cghm 17970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-2 11376  df-ndx 16187  df-slot 16188  df-base 16190  df-sets 16191  df-ress 16192  df-plusg 16280  df-0g 16417  df-mgm 17557  df-sgrp 17599  df-mnd 17610  df-grp 17741  df-minusg 17742  df-sbg 17743  df-subg 17904  df-nsg 17905  df-ghm 17971
This theorem is referenced by:  ghmker  17999
  Copyright terms: Public domain W3C validator