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Theorem conjnsg 19215
Description: A normal subgroup is unchanged under conjugation. (Contributed by Mario Carneiro, 18-Jan-2015.)
Hypotheses
Ref Expression
conjghm.x 𝑋 = (Base‘𝐺)
conjghm.p + = (+g𝐺)
conjghm.m = (-g𝐺)
conjsubg.f 𝐹 = (𝑥𝑆 ↦ ((𝐴 + 𝑥) 𝐴))
Assertion
Ref Expression
conjnsg ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋) → 𝑆 = ran 𝐹)
Distinct variable groups:   𝑥,   𝑥, +   𝑥,𝐴   𝑥,𝐺   𝑥,𝑆   𝑥,𝑋
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem conjnsg
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nsgsubg 19120 . 2 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
2 eqid 2728 . . . . . 6 {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} = {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}
3 conjghm.x . . . . . 6 𝑋 = (Base‘𝐺)
4 conjghm.p . . . . . 6 + = (+g𝐺)
52, 3, 4isnsg4 19129 . . . . 5 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} = 𝑋))
65simprbi 495 . . . 4 (𝑆 ∈ (NrmSGrp‘𝐺) → {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} = 𝑋)
76eleq2d 2815 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) → (𝐴 ∈ {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} ↔ 𝐴𝑋))
87biimpar 476 . 2 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋) → 𝐴 ∈ {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)})
9 conjghm.m . . 3 = (-g𝐺)
10 conjsubg.f . . 3 𝐹 = (𝑥𝑆 ↦ ((𝐴 + 𝑥) 𝐴))
113, 4, 9, 10, 2conjnmz 19213 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}) → 𝑆 = ran 𝐹)
121, 8, 11syl2an2r 683 1 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋) → 𝑆 = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3058  {crab 3430  cmpt 5235  ran crn 5683  cfv 6553  (class class class)co 7426  Basecbs 17187  +gcplusg 17240  -gcsg 18899  SubGrpcsubg 19082  NrmSGrpcnsg 19083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-0g 17430  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-grp 18900  df-minusg 18901  df-sbg 18902  df-subg 19085  df-nsg 19086
This theorem is referenced by:  sylow3lem6  19594
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