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Theorem conjnmz 19126
Description: A subgroup is unchanged under conjugation by an element of its normalizer. (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))
conjnmz.1 𝑁 = {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}
Assertion
Ref Expression
conjnmz ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝑆 = ran 𝐹)
Distinct variable groups:   𝑥,𝑦,   𝑥,𝑧, + ,𝑦   𝑥,𝐴,𝑦,𝑧   𝑦,𝐹,𝑧   𝑥,𝑁   𝑥,𝐺,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧
Allowed substitution hints:   𝐹(𝑥)   (𝑧)   𝑁(𝑦,𝑧)

Proof of Theorem conjnmz
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 subgrcl 19011 . . . . . . . . . 10 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
21ad2antrr 725 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝐺 ∈ Grp)
3 conjnmz.1 . . . . . . . . . . . 12 𝑁 = {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}
43ssrab3 4081 . . . . . . . . . . 11 𝑁𝑋
5 simplr 768 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝐴𝑁)
64, 5sselid 3981 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝐴𝑋)
7 conjghm.x . . . . . . . . . . 11 𝑋 = (Base‘𝐺)
8 eqid 2733 . . . . . . . . . . 11 (invg𝐺) = (invg𝐺)
97, 8grpinvcl 18872 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
102, 6, 9syl2anc 585 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((invg𝐺)‘𝐴) ∈ 𝑋)
117subgss 19007 . . . . . . . . . . 11 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
1211adantr 482 . . . . . . . . . 10 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝑆𝑋)
1312sselda 3983 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝑤𝑋)
14 conjghm.p . . . . . . . . . 10 + = (+g𝐺)
157, 14grpass 18828 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝐴) ∈ 𝑋𝑤𝑋𝐴𝑋)) → ((((invg𝐺)‘𝐴) + 𝑤) + 𝐴) = (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)))
162, 10, 13, 6, 15syl13anc 1373 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((((invg𝐺)‘𝐴) + 𝑤) + 𝐴) = (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)))
17 eqid 2733 . . . . . . . . . . . . . 14 (0g𝐺) = (0g𝐺)
187, 14, 17, 8grprinv 18875 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐴 + ((invg𝐺)‘𝐴)) = (0g𝐺))
192, 6, 18syl2anc 585 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐴 + ((invg𝐺)‘𝐴)) = (0g𝐺))
2019oveq1d 7424 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + ((invg𝐺)‘𝐴)) + 𝑤) = ((0g𝐺) + 𝑤))
217, 14grpass 18828 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (𝐴𝑋 ∧ ((invg𝐺)‘𝐴) ∈ 𝑋𝑤𝑋)) → ((𝐴 + ((invg𝐺)‘𝐴)) + 𝑤) = (𝐴 + (((invg𝐺)‘𝐴) + 𝑤)))
222, 6, 10, 13, 21syl13anc 1373 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + ((invg𝐺)‘𝐴)) + 𝑤) = (𝐴 + (((invg𝐺)‘𝐴) + 𝑤)))
237, 14, 17grplid 18852 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑤𝑋) → ((0g𝐺) + 𝑤) = 𝑤)
242, 13, 23syl2anc 585 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((0g𝐺) + 𝑤) = 𝑤)
2520, 22, 243eqtr3d 2781 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐴 + (((invg𝐺)‘𝐴) + 𝑤)) = 𝑤)
26 simpr 486 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝑤𝑆)
2725, 26eqeltrd 2834 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐴 + (((invg𝐺)‘𝐴) + 𝑤)) ∈ 𝑆)
287, 14grpcl 18827 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝐴) ∈ 𝑋𝑤𝑋) → (((invg𝐺)‘𝐴) + 𝑤) ∈ 𝑋)
292, 10, 13, 28syl3anc 1372 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (((invg𝐺)‘𝐴) + 𝑤) ∈ 𝑋)
303nmzbi 19044 . . . . . . . . . 10 ((𝐴𝑁 ∧ (((invg𝐺)‘𝐴) + 𝑤) ∈ 𝑋) → ((𝐴 + (((invg𝐺)‘𝐴) + 𝑤)) ∈ 𝑆 ↔ ((((invg𝐺)‘𝐴) + 𝑤) + 𝐴) ∈ 𝑆))
315, 29, 30syl2anc 585 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + (((invg𝐺)‘𝐴) + 𝑤)) ∈ 𝑆 ↔ ((((invg𝐺)‘𝐴) + 𝑤) + 𝐴) ∈ 𝑆))
3227, 31mpbid 231 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((((invg𝐺)‘𝐴) + 𝑤) + 𝐴) ∈ 𝑆)
3316, 32eqeltrrd 2835 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑆)
34 oveq2 7417 . . . . . . . . 9 (𝑥 = (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)) → (𝐴 + 𝑥) = (𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))))
3534oveq1d 7424 . . . . . . . 8 (𝑥 = (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)) → ((𝐴 + 𝑥) 𝐴) = ((𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) 𝐴))
36 conjsubg.f . . . . . . . 8 𝐹 = (𝑥𝑆 ↦ ((𝐴 + 𝑥) 𝐴))
37 ovex 7442 . . . . . . . 8 ((𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) 𝐴) ∈ V
3835, 36, 37fvmpt 6999 . . . . . . 7 ((((invg𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑆 → (𝐹‘(((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) = ((𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) 𝐴))
3933, 38syl 17 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐹‘(((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) = ((𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) 𝐴))
4019oveq1d 7424 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + ((invg𝐺)‘𝐴)) + (𝑤 + 𝐴)) = ((0g𝐺) + (𝑤 + 𝐴)))
417, 14grpcl 18827 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑤𝑋𝐴𝑋) → (𝑤 + 𝐴) ∈ 𝑋)
422, 13, 6, 41syl3anc 1372 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝑤 + 𝐴) ∈ 𝑋)
437, 14grpass 18828 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝐴𝑋 ∧ ((invg𝐺)‘𝐴) ∈ 𝑋 ∧ (𝑤 + 𝐴) ∈ 𝑋)) → ((𝐴 + ((invg𝐺)‘𝐴)) + (𝑤 + 𝐴)) = (𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))))
442, 6, 10, 42, 43syl13anc 1373 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + ((invg𝐺)‘𝐴)) + (𝑤 + 𝐴)) = (𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))))
457, 14, 17grplid 18852 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑤 + 𝐴) ∈ 𝑋) → ((0g𝐺) + (𝑤 + 𝐴)) = (𝑤 + 𝐴))
462, 42, 45syl2anc 585 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((0g𝐺) + (𝑤 + 𝐴)) = (𝑤 + 𝐴))
4740, 44, 463eqtr3d 2781 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) = (𝑤 + 𝐴))
4847oveq1d 7424 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) 𝐴) = ((𝑤 + 𝐴) 𝐴))
49 conjghm.m . . . . . . . 8 = (-g𝐺)
507, 14, 49grppncan 18914 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑤𝑋𝐴𝑋) → ((𝑤 + 𝐴) 𝐴) = 𝑤)
512, 13, 6, 50syl3anc 1372 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝑤 + 𝐴) 𝐴) = 𝑤)
5239, 48, 513eqtrd 2777 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐹‘(((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) = 𝑤)
53 ovex 7442 . . . . . . 7 ((𝐴 + 𝑥) 𝐴) ∈ V
5453, 36fnmpti 6694 . . . . . 6 𝐹 Fn 𝑆
55 fnfvelrn 7083 . . . . . 6 ((𝐹 Fn 𝑆 ∧ (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑆) → (𝐹‘(((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) ∈ ran 𝐹)
5654, 33, 55sylancr 588 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐹‘(((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) ∈ ran 𝐹)
5752, 56eqeltrrd 2835 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝑤 ∈ ran 𝐹)
5857ex 414 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → (𝑤𝑆𝑤 ∈ ran 𝐹))
5958ssrdv 3989 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝑆 ⊆ ran 𝐹)
601ad2antrr 725 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → 𝐺 ∈ Grp)
61 simplr 768 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → 𝐴𝑁)
624, 61sselid 3981 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → 𝐴𝑋)
6312sselda 3983 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → 𝑥𝑋)
647, 14, 49grpaddsubass 18913 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝐴𝑋𝑥𝑋𝐴𝑋)) → ((𝐴 + 𝑥) 𝐴) = (𝐴 + (𝑥 𝐴)))
6560, 62, 63, 62, 64syl13anc 1373 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → ((𝐴 + 𝑥) 𝐴) = (𝐴 + (𝑥 𝐴)))
667, 14, 49grpnpcan 18915 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → ((𝑥 𝐴) + 𝐴) = 𝑥)
6760, 63, 62, 66syl3anc 1372 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → ((𝑥 𝐴) + 𝐴) = 𝑥)
68 simpr 486 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → 𝑥𝑆)
6967, 68eqeltrd 2834 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → ((𝑥 𝐴) + 𝐴) ∈ 𝑆)
707, 49grpsubcl 18903 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → (𝑥 𝐴) ∈ 𝑋)
7160, 63, 62, 70syl3anc 1372 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → (𝑥 𝐴) ∈ 𝑋)
723nmzbi 19044 . . . . . . 7 ((𝐴𝑁 ∧ (𝑥 𝐴) ∈ 𝑋) → ((𝐴 + (𝑥 𝐴)) ∈ 𝑆 ↔ ((𝑥 𝐴) + 𝐴) ∈ 𝑆))
7361, 71, 72syl2anc 585 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → ((𝐴 + (𝑥 𝐴)) ∈ 𝑆 ↔ ((𝑥 𝐴) + 𝐴) ∈ 𝑆))
7469, 73mpbird 257 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → (𝐴 + (𝑥 𝐴)) ∈ 𝑆)
7565, 74eqeltrd 2834 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → ((𝐴 + 𝑥) 𝐴) ∈ 𝑆)
7675, 36fmptd 7114 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝐹:𝑆𝑆)
7776frnd 6726 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → ran 𝐹𝑆)
7859, 77eqssd 4000 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝑆 = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  {crab 3433  wss 3949  cmpt 5232  ran crn 5678   Fn wfn 6539  cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  0gc0g 17385  Grpcgrp 18819  invgcminusg 18820  -gcsg 18821  SubGrpcsubg 19000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-sbg 18824  df-subg 19003
This theorem is referenced by:  conjnmzb  19127  conjnsg  19128  sylow3lem2  19496
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