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Theorem conjnmz 18394
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 18286 . . . . . . . . . 10 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
21ad2antrr 724 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝐺 ∈ Grp)
3 conjnmz.1 . . . . . . . . . . . 12 𝑁 = {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}
43ssrab3 4059 . . . . . . . . . . 11 𝑁𝑋
5 simplr 767 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝐴𝑁)
64, 5sseldi 3967 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝐴𝑋)
7 conjghm.x . . . . . . . . . . 11 𝑋 = (Base‘𝐺)
8 eqid 2823 . . . . . . . . . . 11 (invg𝐺) = (invg𝐺)
97, 8grpinvcl 18153 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
102, 6, 9syl2anc 586 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((invg𝐺)‘𝐴) ∈ 𝑋)
117subgss 18282 . . . . . . . . . . 11 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
1211adantr 483 . . . . . . . . . 10 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝑆𝑋)
1312sselda 3969 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝑤𝑋)
14 conjghm.p . . . . . . . . . 10 + = (+g𝐺)
157, 14grpass 18114 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝐴) ∈ 𝑋𝑤𝑋𝐴𝑋)) → ((((invg𝐺)‘𝐴) + 𝑤) + 𝐴) = (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)))
162, 10, 13, 6, 15syl13anc 1368 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((((invg𝐺)‘𝐴) + 𝑤) + 𝐴) = (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)))
17 eqid 2823 . . . . . . . . . . . . . 14 (0g𝐺) = (0g𝐺)
187, 14, 17, 8grprinv 18155 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐴 + ((invg𝐺)‘𝐴)) = (0g𝐺))
192, 6, 18syl2anc 586 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐴 + ((invg𝐺)‘𝐴)) = (0g𝐺))
2019oveq1d 7173 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + ((invg𝐺)‘𝐴)) + 𝑤) = ((0g𝐺) + 𝑤))
217, 14grpass 18114 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (𝐴𝑋 ∧ ((invg𝐺)‘𝐴) ∈ 𝑋𝑤𝑋)) → ((𝐴 + ((invg𝐺)‘𝐴)) + 𝑤) = (𝐴 + (((invg𝐺)‘𝐴) + 𝑤)))
222, 6, 10, 13, 21syl13anc 1368 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + ((invg𝐺)‘𝐴)) + 𝑤) = (𝐴 + (((invg𝐺)‘𝐴) + 𝑤)))
237, 14, 17grplid 18135 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑤𝑋) → ((0g𝐺) + 𝑤) = 𝑤)
242, 13, 23syl2anc 586 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((0g𝐺) + 𝑤) = 𝑤)
2520, 22, 243eqtr3d 2866 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐴 + (((invg𝐺)‘𝐴) + 𝑤)) = 𝑤)
26 simpr 487 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝑤𝑆)
2725, 26eqeltrd 2915 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐴 + (((invg𝐺)‘𝐴) + 𝑤)) ∈ 𝑆)
287, 14grpcl 18113 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝐴) ∈ 𝑋𝑤𝑋) → (((invg𝐺)‘𝐴) + 𝑤) ∈ 𝑋)
292, 10, 13, 28syl3anc 1367 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (((invg𝐺)‘𝐴) + 𝑤) ∈ 𝑋)
303nmzbi 18318 . . . . . . . . . 10 ((𝐴𝑁 ∧ (((invg𝐺)‘𝐴) + 𝑤) ∈ 𝑋) → ((𝐴 + (((invg𝐺)‘𝐴) + 𝑤)) ∈ 𝑆 ↔ ((((invg𝐺)‘𝐴) + 𝑤) + 𝐴) ∈ 𝑆))
315, 29, 30syl2anc 586 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + (((invg𝐺)‘𝐴) + 𝑤)) ∈ 𝑆 ↔ ((((invg𝐺)‘𝐴) + 𝑤) + 𝐴) ∈ 𝑆))
3227, 31mpbid 234 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((((invg𝐺)‘𝐴) + 𝑤) + 𝐴) ∈ 𝑆)
3316, 32eqeltrrd 2916 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑆)
34 oveq2 7166 . . . . . . . . 9 (𝑥 = (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)) → (𝐴 + 𝑥) = (𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))))
3534oveq1d 7173 . . . . . . . 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 7191 . . . . . . . 8 ((𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) 𝐴) ∈ V
3835, 36, 37fvmpt 6770 . . . . . . 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 7173 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + ((invg𝐺)‘𝐴)) + (𝑤 + 𝐴)) = ((0g𝐺) + (𝑤 + 𝐴)))
417, 14grpcl 18113 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑤𝑋𝐴𝑋) → (𝑤 + 𝐴) ∈ 𝑋)
422, 13, 6, 41syl3anc 1367 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝑤 + 𝐴) ∈ 𝑋)
437, 14grpass 18114 . . . . . . . . 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 1368 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + ((invg𝐺)‘𝐴)) + (𝑤 + 𝐴)) = (𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))))
457, 14, 17grplid 18135 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑤 + 𝐴) ∈ 𝑋) → ((0g𝐺) + (𝑤 + 𝐴)) = (𝑤 + 𝐴))
462, 42, 45syl2anc 586 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((0g𝐺) + (𝑤 + 𝐴)) = (𝑤 + 𝐴))
4740, 44, 463eqtr3d 2866 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) = (𝑤 + 𝐴))
4847oveq1d 7173 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) 𝐴) = ((𝑤 + 𝐴) 𝐴))
49 conjghm.m . . . . . . . 8 = (-g𝐺)
507, 14, 49grppncan 18192 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑤𝑋𝐴𝑋) → ((𝑤 + 𝐴) 𝐴) = 𝑤)
512, 13, 6, 50syl3anc 1367 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝑤 + 𝐴) 𝐴) = 𝑤)
5239, 48, 513eqtrd 2862 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐹‘(((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) = 𝑤)
53 ovex 7191 . . . . . . 7 ((𝐴 + 𝑥) 𝐴) ∈ V
5453, 36fnmpti 6493 . . . . . 6 𝐹 Fn 𝑆
55 fnfvelrn 6850 . . . . . 6 ((𝐹 Fn 𝑆 ∧ (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑆) → (𝐹‘(((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) ∈ ran 𝐹)
5654, 33, 55sylancr 589 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐹‘(((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) ∈ ran 𝐹)
5752, 56eqeltrrd 2916 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝑤 ∈ ran 𝐹)
5857ex 415 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → (𝑤𝑆𝑤 ∈ ran 𝐹))
5958ssrdv 3975 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝑆 ⊆ ran 𝐹)
601ad2antrr 724 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → 𝐺 ∈ Grp)
61 simplr 767 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → 𝐴𝑁)
624, 61sseldi 3967 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → 𝐴𝑋)
6312sselda 3969 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → 𝑥𝑋)
647, 14, 49grpaddsubass 18191 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝐴𝑋𝑥𝑋𝐴𝑋)) → ((𝐴 + 𝑥) 𝐴) = (𝐴 + (𝑥 𝐴)))
6560, 62, 63, 62, 64syl13anc 1368 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → ((𝐴 + 𝑥) 𝐴) = (𝐴 + (𝑥 𝐴)))
667, 14, 49grpnpcan 18193 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → ((𝑥 𝐴) + 𝐴) = 𝑥)
6760, 63, 62, 66syl3anc 1367 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → ((𝑥 𝐴) + 𝐴) = 𝑥)
68 simpr 487 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → 𝑥𝑆)
6967, 68eqeltrd 2915 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → ((𝑥 𝐴) + 𝐴) ∈ 𝑆)
707, 49grpsubcl 18181 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → (𝑥 𝐴) ∈ 𝑋)
7160, 63, 62, 70syl3anc 1367 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → (𝑥 𝐴) ∈ 𝑋)
723nmzbi 18318 . . . . . . 7 ((𝐴𝑁 ∧ (𝑥 𝐴) ∈ 𝑋) → ((𝐴 + (𝑥 𝐴)) ∈ 𝑆 ↔ ((𝑥 𝐴) + 𝐴) ∈ 𝑆))
7361, 71, 72syl2anc 586 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → ((𝐴 + (𝑥 𝐴)) ∈ 𝑆 ↔ ((𝑥 𝐴) + 𝐴) ∈ 𝑆))
7469, 73mpbird 259 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → (𝐴 + (𝑥 𝐴)) ∈ 𝑆)
7565, 74eqeltrd 2915 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → ((𝐴 + 𝑥) 𝐴) ∈ 𝑆)
7675, 36fmptd 6880 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝐹:𝑆𝑆)
7776frnd 6523 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → ran 𝐹𝑆)
7859, 77eqssd 3986 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝑆 = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wcel 2114  wral 3140  {crab 3144  wss 3938  cmpt 5148  ran crn 5558   Fn wfn 6352  cfv 6357  (class class class)co 7158  Basecbs 16485  +gcplusg 16567  0gc0g 16715  Grpcgrp 18105  invgcminusg 18106  -gcsg 18107  SubGrpcsubg 18275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-1st 7691  df-2nd 7692  df-0g 16717  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-grp 18108  df-minusg 18109  df-sbg 18110  df-subg 18278
This theorem is referenced by:  conjnmzb  18395  conjnsg  18396  sylow3lem2  18755
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