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Theorem conjnsg 19176
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 19082 . 2 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
2 eqid 2726 . . . . . 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 19091 . . . . 5 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} = 𝑋))
65simprbi 496 . . . 4 (𝑆 ∈ (NrmSGrp‘𝐺) → {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} = 𝑋)
76eleq2d 2813 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) → (𝐴 ∈ {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} ↔ 𝐴𝑋))
87biimpar 477 . 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 19174 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}) → 𝑆 = ran 𝐹)
121, 8, 11syl2an2r 682 1 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋) → 𝑆 = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wral 3055  {crab 3426  cmpt 5224  ran crn 5670  cfv 6536  (class class class)co 7404  Basecbs 17150  +gcplusg 17203  -gcsg 18862  SubGrpcsubg 19044  NrmSGrpcnsg 19045
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-0g 17393  df-mgm 18570  df-sgrp 18649  df-mnd 18665  df-grp 18863  df-minusg 18864  df-sbg 18865  df-subg 19047  df-nsg 19048
This theorem is referenced by:  sylow3lem6  19549
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