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Theorem conjnmz 17902
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 17807 . . . . . . . . . 10 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
21ad2antrr 705 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝐺 ∈ Grp)
3 conjnmz.1 . . . . . . . . . . . 12 𝑁 = {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}
4 ssrab2 3836 . . . . . . . . . . . 12 {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} ⊆ 𝑋
53, 4eqsstri 3784 . . . . . . . . . . 11 𝑁𝑋
6 simplr 752 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝐴𝑁)
75, 6sseldi 3750 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝐴𝑋)
8 conjghm.x . . . . . . . . . . 11 𝑋 = (Base‘𝐺)
9 eqid 2771 . . . . . . . . . . 11 (invg𝐺) = (invg𝐺)
108, 9grpinvcl 17675 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
112, 7, 10syl2anc 573 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((invg𝐺)‘𝐴) ∈ 𝑋)
128subgss 17803 . . . . . . . . . . 11 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
1312adantr 466 . . . . . . . . . 10 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝑆𝑋)
1413sselda 3752 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝑤𝑋)
15 conjghm.p . . . . . . . . . 10 + = (+g𝐺)
168, 15grpass 17639 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝐴) ∈ 𝑋𝑤𝑋𝐴𝑋)) → ((((invg𝐺)‘𝐴) + 𝑤) + 𝐴) = (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)))
172, 11, 14, 7, 16syl13anc 1478 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((((invg𝐺)‘𝐴) + 𝑤) + 𝐴) = (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)))
18 eqid 2771 . . . . . . . . . . . . . 14 (0g𝐺) = (0g𝐺)
198, 15, 18, 9grprinv 17677 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐴 + ((invg𝐺)‘𝐴)) = (0g𝐺))
202, 7, 19syl2anc 573 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐴 + ((invg𝐺)‘𝐴)) = (0g𝐺))
2120oveq1d 6808 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + ((invg𝐺)‘𝐴)) + 𝑤) = ((0g𝐺) + 𝑤))
228, 15grpass 17639 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (𝐴𝑋 ∧ ((invg𝐺)‘𝐴) ∈ 𝑋𝑤𝑋)) → ((𝐴 + ((invg𝐺)‘𝐴)) + 𝑤) = (𝐴 + (((invg𝐺)‘𝐴) + 𝑤)))
232, 7, 11, 14, 22syl13anc 1478 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + ((invg𝐺)‘𝐴)) + 𝑤) = (𝐴 + (((invg𝐺)‘𝐴) + 𝑤)))
248, 15, 18grplid 17660 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑤𝑋) → ((0g𝐺) + 𝑤) = 𝑤)
252, 14, 24syl2anc 573 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((0g𝐺) + 𝑤) = 𝑤)
2621, 23, 253eqtr3d 2813 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐴 + (((invg𝐺)‘𝐴) + 𝑤)) = 𝑤)
27 simpr 471 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝑤𝑆)
2826, 27eqeltrd 2850 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐴 + (((invg𝐺)‘𝐴) + 𝑤)) ∈ 𝑆)
298, 15grpcl 17638 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝐴) ∈ 𝑋𝑤𝑋) → (((invg𝐺)‘𝐴) + 𝑤) ∈ 𝑋)
302, 11, 14, 29syl3anc 1476 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (((invg𝐺)‘𝐴) + 𝑤) ∈ 𝑋)
313nmzbi 17842 . . . . . . . . . 10 ((𝐴𝑁 ∧ (((invg𝐺)‘𝐴) + 𝑤) ∈ 𝑋) → ((𝐴 + (((invg𝐺)‘𝐴) + 𝑤)) ∈ 𝑆 ↔ ((((invg𝐺)‘𝐴) + 𝑤) + 𝐴) ∈ 𝑆))
326, 30, 31syl2anc 573 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + (((invg𝐺)‘𝐴) + 𝑤)) ∈ 𝑆 ↔ ((((invg𝐺)‘𝐴) + 𝑤) + 𝐴) ∈ 𝑆))
3328, 32mpbid 222 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((((invg𝐺)‘𝐴) + 𝑤) + 𝐴) ∈ 𝑆)
3417, 33eqeltrrd 2851 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑆)
35 oveq2 6801 . . . . . . . . 9 (𝑥 = (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)) → (𝐴 + 𝑥) = (𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))))
3635oveq1d 6808 . . . . . . . 8 (𝑥 = (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)) → ((𝐴 + 𝑥) 𝐴) = ((𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) 𝐴))
37 conjsubg.f . . . . . . . 8 𝐹 = (𝑥𝑆 ↦ ((𝐴 + 𝑥) 𝐴))
38 ovex 6823 . . . . . . . 8 ((𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) 𝐴) ∈ V
3936, 37, 38fvmpt 6424 . . . . . . 7 ((((invg𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑆 → (𝐹‘(((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) = ((𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) 𝐴))
4034, 39syl 17 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐹‘(((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) = ((𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) 𝐴))
4120oveq1d 6808 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + ((invg𝐺)‘𝐴)) + (𝑤 + 𝐴)) = ((0g𝐺) + (𝑤 + 𝐴)))
428, 15grpcl 17638 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑤𝑋𝐴𝑋) → (𝑤 + 𝐴) ∈ 𝑋)
432, 14, 7, 42syl3anc 1476 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝑤 + 𝐴) ∈ 𝑋)
448, 15grpass 17639 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝐴𝑋 ∧ ((invg𝐺)‘𝐴) ∈ 𝑋 ∧ (𝑤 + 𝐴) ∈ 𝑋)) → ((𝐴 + ((invg𝐺)‘𝐴)) + (𝑤 + 𝐴)) = (𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))))
452, 7, 11, 43, 44syl13anc 1478 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + ((invg𝐺)‘𝐴)) + (𝑤 + 𝐴)) = (𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))))
468, 15, 18grplid 17660 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑤 + 𝐴) ∈ 𝑋) → ((0g𝐺) + (𝑤 + 𝐴)) = (𝑤 + 𝐴))
472, 43, 46syl2anc 573 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((0g𝐺) + (𝑤 + 𝐴)) = (𝑤 + 𝐴))
4841, 45, 473eqtr3d 2813 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) = (𝑤 + 𝐴))
4948oveq1d 6808 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) 𝐴) = ((𝑤 + 𝐴) 𝐴))
50 conjghm.m . . . . . . . 8 = (-g𝐺)
518, 15, 50grppncan 17714 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑤𝑋𝐴𝑋) → ((𝑤 + 𝐴) 𝐴) = 𝑤)
522, 14, 7, 51syl3anc 1476 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝑤 + 𝐴) 𝐴) = 𝑤)
5340, 49, 523eqtrd 2809 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐹‘(((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) = 𝑤)
54 ovex 6823 . . . . . . 7 ((𝐴 + 𝑥) 𝐴) ∈ V
5554, 37fnmpti 6162 . . . . . 6 𝐹 Fn 𝑆
56 fnfvelrn 6499 . . . . . 6 ((𝐹 Fn 𝑆 ∧ (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑆) → (𝐹‘(((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) ∈ ran 𝐹)
5755, 34, 56sylancr 575 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐹‘(((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) ∈ ran 𝐹)
5853, 57eqeltrrd 2851 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝑤 ∈ ran 𝐹)
5958ex 397 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → (𝑤𝑆𝑤 ∈ ran 𝐹))
6059ssrdv 3758 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝑆 ⊆ ran 𝐹)
611ad2antrr 705 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → 𝐺 ∈ Grp)
62 simplr 752 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → 𝐴𝑁)
635, 62sseldi 3750 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → 𝐴𝑋)
6413sselda 3752 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → 𝑥𝑋)
658, 15, 50grpaddsubass 17713 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝐴𝑋𝑥𝑋𝐴𝑋)) → ((𝐴 + 𝑥) 𝐴) = (𝐴 + (𝑥 𝐴)))
6661, 63, 64, 63, 65syl13anc 1478 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → ((𝐴 + 𝑥) 𝐴) = (𝐴 + (𝑥 𝐴)))
678, 15, 50grpnpcan 17715 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → ((𝑥 𝐴) + 𝐴) = 𝑥)
6861, 64, 63, 67syl3anc 1476 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → ((𝑥 𝐴) + 𝐴) = 𝑥)
69 simpr 471 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → 𝑥𝑆)
7068, 69eqeltrd 2850 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → ((𝑥 𝐴) + 𝐴) ∈ 𝑆)
718, 50grpsubcl 17703 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → (𝑥 𝐴) ∈ 𝑋)
7261, 64, 63, 71syl3anc 1476 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → (𝑥 𝐴) ∈ 𝑋)
733nmzbi 17842 . . . . . . 7 ((𝐴𝑁 ∧ (𝑥 𝐴) ∈ 𝑋) → ((𝐴 + (𝑥 𝐴)) ∈ 𝑆 ↔ ((𝑥 𝐴) + 𝐴) ∈ 𝑆))
7462, 72, 73syl2anc 573 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → ((𝐴 + (𝑥 𝐴)) ∈ 𝑆 ↔ ((𝑥 𝐴) + 𝐴) ∈ 𝑆))
7570, 74mpbird 247 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → (𝐴 + (𝑥 𝐴)) ∈ 𝑆)
7666, 75eqeltrd 2850 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → ((𝐴 + 𝑥) 𝐴) ∈ 𝑆)
7776, 37fmptd 6527 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝐹:𝑆𝑆)
78 frn 6193 . . 3 (𝐹:𝑆𝑆 → ran 𝐹𝑆)
7977, 78syl 17 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → ran 𝐹𝑆)
8060, 79eqssd 3769 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝑆 = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  wss 3723  cmpt 4863  ran crn 5250   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  0gc0g 16308  Grpcgrp 17630  invgcminusg 17631  -gcsg 17632  SubGrpcsubg 17796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-grp 17633  df-minusg 17634  df-sbg 17635  df-subg 17799
This theorem is referenced by:  conjnmzb  17903  conjnsg  17904  sylow3lem2  18250
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