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Theorem nmznsg 13026
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 13024 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑁)
6 subgrcl 12992 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
72, 3, 4nmzsubg 13023 . . . . 5 (𝐺 ∈ Grp → 𝑁 ∈ (SubGrp‘𝐺))
86, 7syl 14 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
9 nmznsg.4 . . . . 5 𝐻 = (𝐺s 𝑁)
109subsubg 13010 . . . 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 13022 . . . . 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 12991 . . . . 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 13015 . . 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 12581 . . . . . . . 8 (𝑆 ∈ (SubGrp‘𝐺) → + = (+g𝐻))
2928oveqd 5891 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → (𝑧 + 𝑤) = (𝑧(+g𝐻)𝑤))
3029eleq1d 2246 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑧(+g𝐻)𝑤) ∈ 𝑆))
3128oveqd 5891 . . . . . . 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 5216  (class class class)co 5874  Basecbs 12456  s cress 12457  +gcplusg 12530  Grpcgrp 12831  SubGrpcsubg 12980  NrmSGrpcnsg 12981
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 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-pre-ltirr 7922  ax-pre-ltadd 7926
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 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-inn 8918  df-2 8976  df-ndx 12459  df-slot 12460  df-base 12462  df-sets 12463  df-iress 12464  df-plusg 12543  df-0g 12697  df-mgm 12729  df-sgrp 12762  df-mnd 12772  df-grp 12834  df-minusg 12835  df-sbg 12836  df-subg 12983  df-nsg 12984
This theorem is referenced by: (None)
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