ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nmznsg GIF version

Theorem nmznsg 13004
Description: Any subgroup is a normal subgroup of its normalizer. (Contributed by Mario Carneiro, 19-Jan-2015.)
Hypotheses
Ref Expression
elnmz.1 𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}
nmzsubg.2 𝑋 = (Base‘𝐺)
nmzsubg.3 + = (+g𝐺)
nmznsg.4 𝐻 = (𝐺s 𝑁)
Assertion
Ref Expression
nmznsg (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (NrmSGrp‘𝐻))
Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑆,𝑦   𝑥, + ,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐻(𝑥,𝑦)   𝑁(𝑥,𝑦)

Proof of Theorem nmznsg
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
2 elnmz.1 . . . 4 𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}
3 nmzsubg.2 . . . 4 𝑋 = (Base‘𝐺)
4 nmzsubg.3 . . . 4 + = (+g𝐺)
52, 3, 4ssnmz 13002 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑁)
6 subgrcl 12970 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
72, 3, 4nmzsubg 13001 . . . . 5 (𝐺 ∈ Grp → 𝑁 ∈ (SubGrp‘𝐺))
86, 7syl 14 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
9 nmznsg.4 . . . . 5 𝐻 = (𝐺s 𝑁)
109subsubg 12988 . . . 4 (𝑁 ∈ (SubGrp‘𝐺) → (𝑆 ∈ (SubGrp‘𝐻) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝑁)))
118, 10syl 14 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → (𝑆 ∈ (SubGrp‘𝐻) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝑁)))
121, 5, 11mpbir2and 944 . 2 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐻))
132ssrab3 3241 . . . . . 6 𝑁𝑋
1413sseli 3151 . . . . 5 (𝑤𝑁𝑤𝑋)
152nmzbi 13000 . . . . 5 ((𝑧𝑁𝑤𝑋) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))
1614, 15sylan2 286 . . . 4 ((𝑧𝑁𝑤𝑁) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))
1716rgen2 2563 . . 3 𝑧𝑁𝑤𝑁 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)
189subgbas 12969 . . . . 5 (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 = (Base‘𝐻))
198, 18syl 14 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝑁 = (Base‘𝐻))
2019raleqdv 2678 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → (∀𝑤𝑁 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆) ↔ ∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)))
2119, 20raleqbidv 2684 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → (∀𝑧𝑁𝑤𝑁 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆) ↔ ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)))
2217, 21mpbii 148 . 2 (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))
23 eqid 2177 . . . 4 (Base‘𝐻) = (Base‘𝐻)
24 eqid 2177 . . . 4 (+g𝐻) = (+g𝐻)
2523, 24isnsg 12993 . . 3 (𝑆 ∈ (NrmSGrp‘𝐻) ↔ (𝑆 ∈ (SubGrp‘𝐻) ∧ ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧(+g𝐻)𝑤) ∈ 𝑆 ↔ (𝑤(+g𝐻)𝑧) ∈ 𝑆)))
269a1i 9 . . . . . . . . 9 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 = (𝐺s 𝑁))
274a1i 9 . . . . . . . . 9 (𝑆 ∈ (SubGrp‘𝐺) → + = (+g𝐺))
2826, 27, 8, 6ressplusgd 12579 . . . . . . . 8 (𝑆 ∈ (SubGrp‘𝐺) → + = (+g𝐻))
2928oveqd 5889 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → (𝑧 + 𝑤) = (𝑧(+g𝐻)𝑤))
3029eleq1d 2246 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑧(+g𝐻)𝑤) ∈ 𝑆))
3128oveqd 5889 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → (𝑤 + 𝑧) = (𝑤(+g𝐻)𝑧))
3231eleq1d 2246 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → ((𝑤 + 𝑧) ∈ 𝑆 ↔ (𝑤(+g𝐻)𝑧) ∈ 𝑆))
3330, 32bibi12d 235 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆) ↔ ((𝑧(+g𝐻)𝑤) ∈ 𝑆 ↔ (𝑤(+g𝐻)𝑧) ∈ 𝑆)))
34332ralbidv 2501 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → (∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆) ↔ ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧(+g𝐻)𝑤) ∈ 𝑆 ↔ (𝑤(+g𝐻)𝑧) ∈ 𝑆)))
3534anbi2d 464 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → ((𝑆 ∈ (SubGrp‘𝐻) ∧ ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)) ↔ (𝑆 ∈ (SubGrp‘𝐻) ∧ ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧(+g𝐻)𝑤) ∈ 𝑆 ↔ (𝑤(+g𝐻)𝑧) ∈ 𝑆))))
3625, 35bitr4id 199 . 2 (𝑆 ∈ (SubGrp‘𝐺) → (𝑆 ∈ (NrmSGrp‘𝐻) ↔ (𝑆 ∈ (SubGrp‘𝐻) ∧ ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))))
3712, 22, 36mpbir2and 944 1 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (NrmSGrp‘𝐻))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  {crab 2459  wss 3129  cfv 5215  (class class class)co 5872  Basecbs 12454  s cress 12455  +gcplusg 12528  Grpcgrp 12809  SubGrpcsubg 12958  NrmSGrpcnsg 12959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-ltadd 7924
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-sbg 12814  df-subg 12961  df-nsg 12962
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator