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Theorem conjnmzb 19284
Description: Alternative condition for elementhood in the 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
conjnmzb (𝑆 ∈ (SubGrp‘𝐺) → (𝐴𝑁 ↔ (𝐴𝑋𝑆 = 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 conjnmzb
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 conjnmz.1 . . . . 5 𝑁 = {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}
21ssrab3 4092 . . . 4 𝑁𝑋
3 simpr 484 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝐴𝑁)
42, 3sselid 3993 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝐴𝑋)
5 conjghm.x . . . 4 𝑋 = (Base‘𝐺)
6 conjghm.p . . . 4 + = (+g𝐺)
7 conjghm.m . . . 4 = (-g𝐺)
8 conjsubg.f . . . 4 𝐹 = (𝑥𝑆 ↦ ((𝐴 + 𝑥) 𝐴))
95, 6, 7, 8, 1conjnmz 19283 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝑆 = ran 𝐹)
104, 9jca 511 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → (𝐴𝑋𝑆 = ran 𝐹))
11 simprl 771 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) → 𝐴𝑋)
12 simplrr 778 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → 𝑆 = ran 𝐹)
1312eleq2d 2825 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝐴 + 𝑤) ∈ ran 𝐹))
14 subgrcl 19162 . . . . . . . . . . . . 13 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
1514ad3antrrr 730 . . . . . . . . . . . 12 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → 𝐺 ∈ Grp)
16 simpllr 776 . . . . . . . . . . . 12 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → 𝐴𝑋)
175subgss 19158 . . . . . . . . . . . . . 14 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
1817ad2antrr 726 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) → 𝑆𝑋)
1918sselda 3995 . . . . . . . . . . . 12 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → 𝑥𝑋)
205, 6, 7grpaddsubass 19061 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (𝐴𝑋𝑥𝑋𝐴𝑋)) → ((𝐴 + 𝑥) 𝐴) = (𝐴 + (𝑥 𝐴)))
2115, 16, 19, 16, 20syl13anc 1371 . . . . . . . . . . 11 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → ((𝐴 + 𝑥) 𝐴) = (𝐴 + (𝑥 𝐴)))
2221eqeq1d 2737 . . . . . . . . . 10 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → (((𝐴 + 𝑥) 𝐴) = (𝐴 + 𝑤) ↔ (𝐴 + (𝑥 𝐴)) = (𝐴 + 𝑤)))
235, 7grpsubcl 19051 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → (𝑥 𝐴) ∈ 𝑋)
2415, 19, 16, 23syl3anc 1370 . . . . . . . . . . 11 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → (𝑥 𝐴) ∈ 𝑋)
25 simplr 769 . . . . . . . . . . 11 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → 𝑤𝑋)
265, 6grplcan 19031 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ ((𝑥 𝐴) ∈ 𝑋𝑤𝑋𝐴𝑋)) → ((𝐴 + (𝑥 𝐴)) = (𝐴 + 𝑤) ↔ (𝑥 𝐴) = 𝑤))
2715, 24, 25, 16, 26syl13anc 1371 . . . . . . . . . 10 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → ((𝐴 + (𝑥 𝐴)) = (𝐴 + 𝑤) ↔ (𝑥 𝐴) = 𝑤))
285, 6, 7grpsubadd 19059 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑥𝑋𝐴𝑋𝑤𝑋)) → ((𝑥 𝐴) = 𝑤 ↔ (𝑤 + 𝐴) = 𝑥))
2915, 19, 16, 25, 28syl13anc 1371 . . . . . . . . . 10 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → ((𝑥 𝐴) = 𝑤 ↔ (𝑤 + 𝐴) = 𝑥))
3022, 27, 293bitrd 305 . . . . . . . . 9 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → (((𝐴 + 𝑥) 𝐴) = (𝐴 + 𝑤) ↔ (𝑤 + 𝐴) = 𝑥))
31 eqcom 2742 . . . . . . . . 9 ((𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴) ↔ ((𝐴 + 𝑥) 𝐴) = (𝐴 + 𝑤))
32 eqcom 2742 . . . . . . . . 9 (𝑥 = (𝑤 + 𝐴) ↔ (𝑤 + 𝐴) = 𝑥)
3330, 31, 323bitr4g 314 . . . . . . . 8 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → ((𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴) ↔ 𝑥 = (𝑤 + 𝐴)))
3433rexbidva 3175 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) → (∃𝑥𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴) ↔ ∃𝑥𝑆 𝑥 = (𝑤 + 𝐴)))
3534adantlrr 721 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → (∃𝑥𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴) ↔ ∃𝑥𝑆 𝑥 = (𝑤 + 𝐴)))
36 ovex 7464 . . . . . . 7 (𝐴 + 𝑤) ∈ V
37 eqeq1 2739 . . . . . . . 8 (𝑦 = (𝐴 + 𝑤) → (𝑦 = ((𝐴 + 𝑥) 𝐴) ↔ (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴)))
3837rexbidv 3177 . . . . . . 7 (𝑦 = (𝐴 + 𝑤) → (∃𝑥𝑆 𝑦 = ((𝐴 + 𝑥) 𝐴) ↔ ∃𝑥𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴)))
398rnmpt 5971 . . . . . . 7 ran 𝐹 = {𝑦 ∣ ∃𝑥𝑆 𝑦 = ((𝐴 + 𝑥) 𝐴)}
4036, 38, 39elab2 3685 . . . . . 6 ((𝐴 + 𝑤) ∈ ran 𝐹 ↔ ∃𝑥𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴))
41 risset 3231 . . . . . 6 ((𝑤 + 𝐴) ∈ 𝑆 ↔ ∃𝑥𝑆 𝑥 = (𝑤 + 𝐴))
4235, 40, 413bitr4g 314 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → ((𝐴 + 𝑤) ∈ ran 𝐹 ↔ (𝑤 + 𝐴) ∈ 𝑆))
4313, 42bitrd 279 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆))
4443ralrimiva 3144 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) → ∀𝑤𝑋 ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆))
451elnmz 19194 . . 3 (𝐴𝑁 ↔ (𝐴𝑋 ∧ ∀𝑤𝑋 ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆)))
4611, 44, 45sylanbrc 583 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) → 𝐴𝑁)
4710, 46impbida 801 1 (𝑆 ∈ (SubGrp‘𝐺) → (𝐴𝑁 ↔ (𝐴𝑋𝑆 = ran 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  {crab 3433  wss 3963  cmpt 5231  ran crn 5690  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Grpcgrp 18964  -gcsg 18966  SubGrpcsubg 19151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154
This theorem is referenced by:  sylow3lem6  19665
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