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Theorem conjnmzb 19112
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 4078 . . . 4 𝑁𝑋
3 simpr 486 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝐴𝑁)
42, 3sselid 3978 . . 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 19111 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝑆 = ran 𝐹)
104, 9jca 513 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → (𝐴𝑋𝑆 = ran 𝐹))
11 simprl 770 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) → 𝐴𝑋)
12 simplrr 777 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → 𝑆 = ran 𝐹)
1312eleq2d 2820 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝐴 + 𝑤) ∈ ran 𝐹))
14 subgrcl 18996 . . . . . . . . . . . . 13 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
1514ad3antrrr 729 . . . . . . . . . . . 12 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → 𝐺 ∈ Grp)
16 simpllr 775 . . . . . . . . . . . 12 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → 𝐴𝑋)
175subgss 18992 . . . . . . . . . . . . . 14 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
1817ad2antrr 725 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) → 𝑆𝑋)
1918sselda 3980 . . . . . . . . . . . 12 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → 𝑥𝑋)
205, 6, 7grpaddsubass 18900 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (𝐴𝑋𝑥𝑋𝐴𝑋)) → ((𝐴 + 𝑥) 𝐴) = (𝐴 + (𝑥 𝐴)))
2115, 16, 19, 16, 20syl13anc 1373 . . . . . . . . . . 11 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → ((𝐴 + 𝑥) 𝐴) = (𝐴 + (𝑥 𝐴)))
2221eqeq1d 2735 . . . . . . . . . 10 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → (((𝐴 + 𝑥) 𝐴) = (𝐴 + 𝑤) ↔ (𝐴 + (𝑥 𝐴)) = (𝐴 + 𝑤)))
235, 7grpsubcl 18890 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → (𝑥 𝐴) ∈ 𝑋)
2415, 19, 16, 23syl3anc 1372 . . . . . . . . . . 11 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → (𝑥 𝐴) ∈ 𝑋)
25 simplr 768 . . . . . . . . . . 11 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → 𝑤𝑋)
265, 6grplcan 18872 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ ((𝑥 𝐴) ∈ 𝑋𝑤𝑋𝐴𝑋)) → ((𝐴 + (𝑥 𝐴)) = (𝐴 + 𝑤) ↔ (𝑥 𝐴) = 𝑤))
2715, 24, 25, 16, 26syl13anc 1373 . . . . . . . . . 10 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → ((𝐴 + (𝑥 𝐴)) = (𝐴 + 𝑤) ↔ (𝑥 𝐴) = 𝑤))
285, 6, 7grpsubadd 18898 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑥𝑋𝐴𝑋𝑤𝑋)) → ((𝑥 𝐴) = 𝑤 ↔ (𝑤 + 𝐴) = 𝑥))
2915, 19, 16, 25, 28syl13anc 1373 . . . . . . . . . 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 2740 . . . . . . . . 9 ((𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴) ↔ ((𝐴 + 𝑥) 𝐴) = (𝐴 + 𝑤))
32 eqcom 2740 . . . . . . . . 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 3177 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) → (∃𝑥𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴) ↔ ∃𝑥𝑆 𝑥 = (𝑤 + 𝐴)))
3534adantlrr 720 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → (∃𝑥𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴) ↔ ∃𝑥𝑆 𝑥 = (𝑤 + 𝐴)))
36 ovex 7429 . . . . . . 7 (𝐴 + 𝑤) ∈ V
37 eqeq1 2737 . . . . . . . 8 (𝑦 = (𝐴 + 𝑤) → (𝑦 = ((𝐴 + 𝑥) 𝐴) ↔ (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴)))
3837rexbidv 3179 . . . . . . 7 (𝑦 = (𝐴 + 𝑤) → (∃𝑥𝑆 𝑦 = ((𝐴 + 𝑥) 𝐴) ↔ ∃𝑥𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴)))
398rnmpt 5949 . . . . . . 7 ran 𝐹 = {𝑦 ∣ ∃𝑥𝑆 𝑦 = ((𝐴 + 𝑥) 𝐴)}
4036, 38, 39elab2 3670 . . . . . 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 3147 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) → ∀𝑤𝑋 ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆))
451elnmz 19028 . . 3 (𝐴𝑁 ↔ (𝐴𝑋 ∧ ∀𝑤𝑋 ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆)))
4611, 44, 45sylanbrc 584 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) → 𝐴𝑁)
4710, 46impbida 800 1 (𝑆 ∈ (SubGrp‘𝐺) → (𝐴𝑁 ↔ (𝐴𝑋𝑆 = ran 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  {crab 3433  wss 3946  cmpt 5227  ran crn 5673  cfv 6535  (class class class)co 7396  Basecbs 17131  +gcplusg 17184  Grpcgrp 18806  -gcsg 18808  SubGrpcsubg 18985
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-fv 6543  df-riota 7352  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7962  df-2nd 7963  df-0g 17374  df-mgm 18548  df-sgrp 18597  df-mnd 18613  df-grp 18809  df-minusg 18810  df-sbg 18811  df-subg 18988
This theorem is referenced by:  sylow3lem6  19484
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