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Theorem conjnsg 18390
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 18306 . 2 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
2 eqid 2801 . . . . . 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 18315 . . . . 5 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} = 𝑋))
65simprbi 500 . . . 4 (𝑆 ∈ (NrmSGrp‘𝐺) → {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} = 𝑋)
76eleq2d 2878 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) → (𝐴 ∈ {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} ↔ 𝐴𝑋))
87biimpar 481 . 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 18388 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}) → 𝑆 = ran 𝐹)
121, 8, 11syl2an2r 684 1 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋) → 𝑆 = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wral 3109  {crab 3113  cmpt 5113  ran crn 5524  cfv 6328  (class class class)co 7139  Basecbs 16479  +gcplusg 16561  -gcsg 18101  SubGrpcsubg 18269  NrmSGrpcnsg 18270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-0g 16711  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-grp 18102  df-minusg 18103  df-sbg 18104  df-subg 18272  df-nsg 18273
This theorem is referenced by:  sylow3lem6  18753
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