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Theorem conjnmz 17895
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 17800 . . . . . . . . . 10 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
21ad2antrr 764 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝐺 ∈ Grp)
3 conjnmz.1 . . . . . . . . . . . 12 𝑁 = {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}
4 ssrab2 3828 . . . . . . . . . . . 12 {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} ⊆ 𝑋
53, 4eqsstri 3776 . . . . . . . . . . 11 𝑁𝑋
6 simplr 809 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝐴𝑁)
75, 6sseldi 3742 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝐴𝑋)
8 conjghm.x . . . . . . . . . . 11 𝑋 = (Base‘𝐺)
9 eqid 2760 . . . . . . . . . . 11 (invg𝐺) = (invg𝐺)
108, 9grpinvcl 17668 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
112, 7, 10syl2anc 696 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((invg𝐺)‘𝐴) ∈ 𝑋)
128subgss 17796 . . . . . . . . . . 11 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
1312adantr 472 . . . . . . . . . 10 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝑆𝑋)
1413sselda 3744 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝑤𝑋)
15 conjghm.p . . . . . . . . . 10 + = (+g𝐺)
168, 15grpass 17632 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝐴) ∈ 𝑋𝑤𝑋𝐴𝑋)) → ((((invg𝐺)‘𝐴) + 𝑤) + 𝐴) = (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)))
172, 11, 14, 7, 16syl13anc 1479 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((((invg𝐺)‘𝐴) + 𝑤) + 𝐴) = (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)))
18 eqid 2760 . . . . . . . . . . . . . 14 (0g𝐺) = (0g𝐺)
198, 15, 18, 9grprinv 17670 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐴 + ((invg𝐺)‘𝐴)) = (0g𝐺))
202, 7, 19syl2anc 696 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐴 + ((invg𝐺)‘𝐴)) = (0g𝐺))
2120oveq1d 6828 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + ((invg𝐺)‘𝐴)) + 𝑤) = ((0g𝐺) + 𝑤))
228, 15grpass 17632 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (𝐴𝑋 ∧ ((invg𝐺)‘𝐴) ∈ 𝑋𝑤𝑋)) → ((𝐴 + ((invg𝐺)‘𝐴)) + 𝑤) = (𝐴 + (((invg𝐺)‘𝐴) + 𝑤)))
232, 7, 11, 14, 22syl13anc 1479 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + ((invg𝐺)‘𝐴)) + 𝑤) = (𝐴 + (((invg𝐺)‘𝐴) + 𝑤)))
248, 15, 18grplid 17653 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑤𝑋) → ((0g𝐺) + 𝑤) = 𝑤)
252, 14, 24syl2anc 696 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((0g𝐺) + 𝑤) = 𝑤)
2621, 23, 253eqtr3d 2802 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐴 + (((invg𝐺)‘𝐴) + 𝑤)) = 𝑤)
27 simpr 479 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝑤𝑆)
2826, 27eqeltrd 2839 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐴 + (((invg𝐺)‘𝐴) + 𝑤)) ∈ 𝑆)
298, 15grpcl 17631 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝐴) ∈ 𝑋𝑤𝑋) → (((invg𝐺)‘𝐴) + 𝑤) ∈ 𝑋)
302, 11, 14, 29syl3anc 1477 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (((invg𝐺)‘𝐴) + 𝑤) ∈ 𝑋)
313nmzbi 17835 . . . . . . . . . 10 ((𝐴𝑁 ∧ (((invg𝐺)‘𝐴) + 𝑤) ∈ 𝑋) → ((𝐴 + (((invg𝐺)‘𝐴) + 𝑤)) ∈ 𝑆 ↔ ((((invg𝐺)‘𝐴) + 𝑤) + 𝐴) ∈ 𝑆))
326, 30, 31syl2anc 696 . . . . . . . . 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 2840 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑆)
35 oveq2 6821 . . . . . . . . 9 (𝑥 = (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)) → (𝐴 + 𝑥) = (𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))))
3635oveq1d 6828 . . . . . . . 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 6841 . . . . . . . 8 ((𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) 𝐴) ∈ V
3936, 37, 38fvmpt 6444 . . . . . . 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 6828 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + ((invg𝐺)‘𝐴)) + (𝑤 + 𝐴)) = ((0g𝐺) + (𝑤 + 𝐴)))
428, 15grpcl 17631 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑤𝑋𝐴𝑋) → (𝑤 + 𝐴) ∈ 𝑋)
432, 14, 7, 42syl3anc 1477 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝑤 + 𝐴) ∈ 𝑋)
448, 15grpass 17632 . . . . . . . . 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 1479 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + ((invg𝐺)‘𝐴)) + (𝑤 + 𝐴)) = (𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))))
468, 15, 18grplid 17653 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑤 + 𝐴) ∈ 𝑋) → ((0g𝐺) + (𝑤 + 𝐴)) = (𝑤 + 𝐴))
472, 43, 46syl2anc 696 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((0g𝐺) + (𝑤 + 𝐴)) = (𝑤 + 𝐴))
4841, 45, 473eqtr3d 2802 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) = (𝑤 + 𝐴))
4948oveq1d 6828 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝐴 + (((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) 𝐴) = ((𝑤 + 𝐴) 𝐴))
50 conjghm.m . . . . . . . 8 = (-g𝐺)
518, 15, 50grppncan 17707 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑤𝑋𝐴𝑋) → ((𝑤 + 𝐴) 𝐴) = 𝑤)
522, 14, 7, 51syl3anc 1477 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → ((𝑤 + 𝐴) 𝐴) = 𝑤)
5340, 49, 523eqtrd 2798 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐹‘(((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) = 𝑤)
54 ovex 6841 . . . . . . 7 ((𝐴 + 𝑥) 𝐴) ∈ V
5554, 37fnmpti 6183 . . . . . 6 𝐹 Fn 𝑆
56 fnfvelrn 6519 . . . . . 6 ((𝐹 Fn 𝑆 ∧ (((invg𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑆) → (𝐹‘(((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) ∈ ran 𝐹)
5755, 34, 56sylancr 698 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → (𝐹‘(((invg𝐺)‘𝐴) + (𝑤 + 𝐴))) ∈ ran 𝐹)
5853, 57eqeltrrd 2840 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑤𝑆) → 𝑤 ∈ ran 𝐹)
5958ex 449 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → (𝑤𝑆𝑤 ∈ ran 𝐹))
6059ssrdv 3750 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝑆 ⊆ ran 𝐹)
611ad2antrr 764 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → 𝐺 ∈ Grp)
62 simplr 809 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → 𝐴𝑁)
635, 62sseldi 3742 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → 𝐴𝑋)
6413sselda 3744 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → 𝑥𝑋)
658, 15, 50grpaddsubass 17706 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝐴𝑋𝑥𝑋𝐴𝑋)) → ((𝐴 + 𝑥) 𝐴) = (𝐴 + (𝑥 𝐴)))
6661, 63, 64, 63, 65syl13anc 1479 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → ((𝐴 + 𝑥) 𝐴) = (𝐴 + (𝑥 𝐴)))
678, 15, 50grpnpcan 17708 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → ((𝑥 𝐴) + 𝐴) = 𝑥)
6861, 64, 63, 67syl3anc 1477 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → ((𝑥 𝐴) + 𝐴) = 𝑥)
69 simpr 479 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → 𝑥𝑆)
7068, 69eqeltrd 2839 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → ((𝑥 𝐴) + 𝐴) ∈ 𝑆)
718, 50grpsubcl 17696 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → (𝑥 𝐴) ∈ 𝑋)
7261, 64, 63, 71syl3anc 1477 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → (𝑥 𝐴) ∈ 𝑋)
733nmzbi 17835 . . . . . . 7 ((𝐴𝑁 ∧ (𝑥 𝐴) ∈ 𝑋) → ((𝐴 + (𝑥 𝐴)) ∈ 𝑆 ↔ ((𝑥 𝐴) + 𝐴) ∈ 𝑆))
7462, 72, 73syl2anc 696 . . . . . 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 2839 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) ∧ 𝑥𝑆) → ((𝐴 + 𝑥) 𝐴) ∈ 𝑆)
7776, 37fmptd 6548 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝐹:𝑆𝑆)
78 frn 6214 . . 3 (𝐹:𝑆𝑆 → ran 𝐹𝑆)
7977, 78syl 17 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → ran 𝐹𝑆)
8060, 79eqssd 3761 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝑆 = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  {crab 3054  wss 3715  cmpt 4881  ran crn 5267   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6813  Basecbs 16059  +gcplusg 16143  0gc0g 16302  Grpcgrp 17623  invgcminusg 17624  -gcsg 17625  SubGrpcsubg 17789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-0g 16304  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-grp 17626  df-minusg 17627  df-sbg 17628  df-subg 17792
This theorem is referenced by:  conjnmzb  17896  conjnsg  17897  sylow3lem2  18243
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