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Theorem conjnmzb 17742
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)}
2 ssrab2 3720 . . . . 5 {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} ⊆ 𝑋
31, 2eqsstri 3668 . . . 4 𝑁𝑋
4 simpr 476 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝐴𝑁)
53, 4sseldi 3634 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝐴𝑋)
6 conjghm.x . . . 4 𝑋 = (Base‘𝐺)
7 conjghm.p . . . 4 + = (+g𝐺)
8 conjghm.m . . . 4 = (-g𝐺)
9 conjsubg.f . . . 4 𝐹 = (𝑥𝑆 ↦ ((𝐴 + 𝑥) 𝐴))
106, 7, 8, 9, 1conjnmz 17741 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝑆 = ran 𝐹)
115, 10jca 553 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → (𝐴𝑋𝑆 = ran 𝐹))
12 simprl 809 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) → 𝐴𝑋)
13 simplrr 818 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → 𝑆 = ran 𝐹)
1413eleq2d 2716 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝐴 + 𝑤) ∈ ran 𝐹))
15 subgrcl 17646 . . . . . . . . . . . . 13 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
1615ad3antrrr 766 . . . . . . . . . . . 12 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → 𝐺 ∈ Grp)
17 simpllr 815 . . . . . . . . . . . 12 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → 𝐴𝑋)
186subgss 17642 . . . . . . . . . . . . . 14 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
1918ad2antrr 762 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) → 𝑆𝑋)
2019sselda 3636 . . . . . . . . . . . 12 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → 𝑥𝑋)
216, 7, 8grpaddsubass 17552 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (𝐴𝑋𝑥𝑋𝐴𝑋)) → ((𝐴 + 𝑥) 𝐴) = (𝐴 + (𝑥 𝐴)))
2216, 17, 20, 17, 21syl13anc 1368 . . . . . . . . . . 11 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → ((𝐴 + 𝑥) 𝐴) = (𝐴 + (𝑥 𝐴)))
2322eqeq1d 2653 . . . . . . . . . 10 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → (((𝐴 + 𝑥) 𝐴) = (𝐴 + 𝑤) ↔ (𝐴 + (𝑥 𝐴)) = (𝐴 + 𝑤)))
246, 8grpsubcl 17542 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → (𝑥 𝐴) ∈ 𝑋)
2516, 20, 17, 24syl3anc 1366 . . . . . . . . . . 11 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → (𝑥 𝐴) ∈ 𝑋)
26 simplr 807 . . . . . . . . . . 11 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → 𝑤𝑋)
276, 7grplcan 17524 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ ((𝑥 𝐴) ∈ 𝑋𝑤𝑋𝐴𝑋)) → ((𝐴 + (𝑥 𝐴)) = (𝐴 + 𝑤) ↔ (𝑥 𝐴) = 𝑤))
2816, 25, 26, 17, 27syl13anc 1368 . . . . . . . . . 10 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → ((𝐴 + (𝑥 𝐴)) = (𝐴 + 𝑤) ↔ (𝑥 𝐴) = 𝑤))
296, 7, 8grpsubadd 17550 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑥𝑋𝐴𝑋𝑤𝑋)) → ((𝑥 𝐴) = 𝑤 ↔ (𝑤 + 𝐴) = 𝑥))
3016, 20, 17, 26, 29syl13anc 1368 . . . . . . . . . 10 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → ((𝑥 𝐴) = 𝑤 ↔ (𝑤 + 𝐴) = 𝑥))
3123, 28, 303bitrd 294 . . . . . . . . 9 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → (((𝐴 + 𝑥) 𝐴) = (𝐴 + 𝑤) ↔ (𝑤 + 𝐴) = 𝑥))
32 eqcom 2658 . . . . . . . . 9 ((𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴) ↔ ((𝐴 + 𝑥) 𝐴) = (𝐴 + 𝑤))
33 eqcom 2658 . . . . . . . . 9 (𝑥 = (𝑤 + 𝐴) ↔ (𝑤 + 𝐴) = 𝑥)
3431, 32, 333bitr4g 303 . . . . . . . 8 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → ((𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴) ↔ 𝑥 = (𝑤 + 𝐴)))
3534rexbidva 3078 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) → (∃𝑥𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴) ↔ ∃𝑥𝑆 𝑥 = (𝑤 + 𝐴)))
3635adantlrr 757 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → (∃𝑥𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴) ↔ ∃𝑥𝑆 𝑥 = (𝑤 + 𝐴)))
37 ovex 6718 . . . . . . 7 (𝐴 + 𝑤) ∈ V
38 eqeq1 2655 . . . . . . . 8 (𝑦 = (𝐴 + 𝑤) → (𝑦 = ((𝐴 + 𝑥) 𝐴) ↔ (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴)))
3938rexbidv 3081 . . . . . . 7 (𝑦 = (𝐴 + 𝑤) → (∃𝑥𝑆 𝑦 = ((𝐴 + 𝑥) 𝐴) ↔ ∃𝑥𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴)))
409rnmpt 5403 . . . . . . 7 ran 𝐹 = {𝑦 ∣ ∃𝑥𝑆 𝑦 = ((𝐴 + 𝑥) 𝐴)}
4137, 39, 40elab2 3386 . . . . . 6 ((𝐴 + 𝑤) ∈ ran 𝐹 ↔ ∃𝑥𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴))
42 risset 3091 . . . . . 6 ((𝑤 + 𝐴) ∈ 𝑆 ↔ ∃𝑥𝑆 𝑥 = (𝑤 + 𝐴))
4336, 41, 423bitr4g 303 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → ((𝐴 + 𝑤) ∈ ran 𝐹 ↔ (𝑤 + 𝐴) ∈ 𝑆))
4414, 43bitrd 268 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆))
4544ralrimiva 2995 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) → ∀𝑤𝑋 ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆))
461elnmz 17680 . . 3 (𝐴𝑁 ↔ (𝐴𝑋 ∧ ∀𝑤𝑋 ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆)))
4712, 45, 46sylanbrc 699 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) → 𝐴𝑁)
4811, 47impbida 895 1 (𝑆 ∈ (SubGrp‘𝐺) → (𝐴𝑁 ↔ (𝐴𝑋𝑆 = ran 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  wss 3607  cmpt 4762  ran crn 5144  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  Grpcgrp 17469  -gcsg 17471  SubGrpcsubg 17635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-sbg 17474  df-subg 17638
This theorem is referenced by:  sylow3lem6  18093
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