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Theorem conjnmzb 18869
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 4015 . . . 4 𝑁𝑋
3 simpr 485 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝐴𝑁)
42, 3sselid 3919 . . 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 18868 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝑆 = ran 𝐹)
104, 9jca 512 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → (𝐴𝑋𝑆 = ran 𝐹))
11 simprl 768 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) → 𝐴𝑋)
12 simplrr 775 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → 𝑆 = ran 𝐹)
1312eleq2d 2824 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝐴 + 𝑤) ∈ ran 𝐹))
14 subgrcl 18760 . . . . . . . . . . . . 13 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
1514ad3antrrr 727 . . . . . . . . . . . 12 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → 𝐺 ∈ Grp)
16 simpllr 773 . . . . . . . . . . . 12 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → 𝐴𝑋)
175subgss 18756 . . . . . . . . . . . . . 14 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
1817ad2antrr 723 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) → 𝑆𝑋)
1918sselda 3921 . . . . . . . . . . . 12 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → 𝑥𝑋)
205, 6, 7grpaddsubass 18665 . . . . . . . . . . . 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 2740 . . . . . . . . . 10 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → (((𝐴 + 𝑥) 𝐴) = (𝐴 + 𝑤) ↔ (𝐴 + (𝑥 𝐴)) = (𝐴 + 𝑤)))
235, 7grpsubcl 18655 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → (𝑥 𝐴) ∈ 𝑋)
2415, 19, 16, 23syl3anc 1370 . . . . . . . . . . 11 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → (𝑥 𝐴) ∈ 𝑋)
25 simplr 766 . . . . . . . . . . 11 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → 𝑤𝑋)
265, 6grplcan 18637 . . . . . . . . . . 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 18663 . . . . . . . . . . 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 2745 . . . . . . . . 9 ((𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴) ↔ ((𝐴 + 𝑥) 𝐴) = (𝐴 + 𝑤))
32 eqcom 2745 . . . . . . . . 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 3225 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) → (∃𝑥𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴) ↔ ∃𝑥𝑆 𝑥 = (𝑤 + 𝐴)))
3534adantlrr 718 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → (∃𝑥𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴) ↔ ∃𝑥𝑆 𝑥 = (𝑤 + 𝐴)))
36 ovex 7308 . . . . . . 7 (𝐴 + 𝑤) ∈ V
37 eqeq1 2742 . . . . . . . 8 (𝑦 = (𝐴 + 𝑤) → (𝑦 = ((𝐴 + 𝑥) 𝐴) ↔ (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴)))
3837rexbidv 3226 . . . . . . 7 (𝑦 = (𝐴 + 𝑤) → (∃𝑥𝑆 𝑦 = ((𝐴 + 𝑥) 𝐴) ↔ ∃𝑥𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴)))
398rnmpt 5864 . . . . . . 7 ran 𝐹 = {𝑦 ∣ ∃𝑥𝑆 𝑦 = ((𝐴 + 𝑥) 𝐴)}
4036, 38, 39elab2 3613 . . . . . 6 ((𝐴 + 𝑤) ∈ ran 𝐹 ↔ ∃𝑥𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴))
41 risset 3194 . . . . . 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 278 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆))
4443ralrimiva 3103 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) → ∀𝑤𝑋 ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆))
451elnmz 18791 . . 3 (𝐴𝑁 ↔ (𝐴𝑋 ∧ ∀𝑤𝑋 ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆)))
4611, 44, 45sylanbrc 583 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) → 𝐴𝑁)
4710, 46impbida 798 1 (𝑆 ∈ (SubGrp‘𝐺) → (𝐴𝑁 ↔ (𝐴𝑋𝑆 = ran 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wral 3064  wrex 3065  {crab 3068  wss 3887  cmpt 5157  ran crn 5590  cfv 6433  (class class class)co 7275  Basecbs 16912  +gcplusg 16962  Grpcgrp 18577  -gcsg 18579  SubGrpcsubg 18749
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-minusg 18581  df-sbg 18582  df-subg 18752
This theorem is referenced by:  sylow3lem6  19237
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