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Theorem conjnmzb 17959
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 3847 . . . . 5 {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} ⊆ 𝑋
31, 2eqsstri 3795 . . . 4 𝑁𝑋
4 simpr 477 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝐴𝑁)
53, 4sseldi 3759 . . 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 17958 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝑆 = ran 𝐹)
115, 10jca 507 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → (𝐴𝑋𝑆 = ran 𝐹))
12 simprl 787 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) → 𝐴𝑋)
13 simplrr 796 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → 𝑆 = ran 𝐹)
1413eleq2d 2830 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝐴 + 𝑤) ∈ ran 𝐹))
15 subgrcl 17863 . . . . . . . . . . . . 13 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
1615ad3antrrr 721 . . . . . . . . . . . 12 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → 𝐺 ∈ Grp)
17 simpllr 793 . . . . . . . . . . . 12 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → 𝐴𝑋)
186subgss 17859 . . . . . . . . . . . . . 14 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
1918ad2antrr 717 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) → 𝑆𝑋)
2019sselda 3761 . . . . . . . . . . . 12 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → 𝑥𝑋)
216, 7, 8grpaddsubass 17772 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (𝐴𝑋𝑥𝑋𝐴𝑋)) → ((𝐴 + 𝑥) 𝐴) = (𝐴 + (𝑥 𝐴)))
2216, 17, 20, 17, 21syl13anc 1491 . . . . . . . . . . 11 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → ((𝐴 + 𝑥) 𝐴) = (𝐴 + (𝑥 𝐴)))
2322eqeq1d 2767 . . . . . . . . . 10 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → (((𝐴 + 𝑥) 𝐴) = (𝐴 + 𝑤) ↔ (𝐴 + (𝑥 𝐴)) = (𝐴 + 𝑤)))
246, 8grpsubcl 17762 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → (𝑥 𝐴) ∈ 𝑋)
2516, 20, 17, 24syl3anc 1490 . . . . . . . . . . 11 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → (𝑥 𝐴) ∈ 𝑋)
26 simplr 785 . . . . . . . . . . 11 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → 𝑤𝑋)
276, 7grplcan 17744 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ ((𝑥 𝐴) ∈ 𝑋𝑤𝑋𝐴𝑋)) → ((𝐴 + (𝑥 𝐴)) = (𝐴 + 𝑤) ↔ (𝑥 𝐴) = 𝑤))
2816, 25, 26, 17, 27syl13anc 1491 . . . . . . . . . 10 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → ((𝐴 + (𝑥 𝐴)) = (𝐴 + 𝑤) ↔ (𝑥 𝐴) = 𝑤))
296, 7, 8grpsubadd 17770 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑥𝑋𝐴𝑋𝑤𝑋)) → ((𝑥 𝐴) = 𝑤 ↔ (𝑤 + 𝐴) = 𝑥))
3016, 20, 17, 26, 29syl13anc 1491 . . . . . . . . . 10 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → ((𝑥 𝐴) = 𝑤 ↔ (𝑤 + 𝐴) = 𝑥))
3123, 28, 303bitrd 296 . . . . . . . . 9 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → (((𝐴 + 𝑥) 𝐴) = (𝐴 + 𝑤) ↔ (𝑤 + 𝐴) = 𝑥))
32 eqcom 2772 . . . . . . . . 9 ((𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴) ↔ ((𝐴 + 𝑥) 𝐴) = (𝐴 + 𝑤))
33 eqcom 2772 . . . . . . . . 9 (𝑥 = (𝑤 + 𝐴) ↔ (𝑤 + 𝐴) = 𝑥)
3431, 32, 333bitr4g 305 . . . . . . . 8 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) ∧ 𝑥𝑆) → ((𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴) ↔ 𝑥 = (𝑤 + 𝐴)))
3534rexbidva 3196 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) ∧ 𝑤𝑋) → (∃𝑥𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴) ↔ ∃𝑥𝑆 𝑥 = (𝑤 + 𝐴)))
3635adantlrr 712 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → (∃𝑥𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴) ↔ ∃𝑥𝑆 𝑥 = (𝑤 + 𝐴)))
37 ovex 6874 . . . . . . 7 (𝐴 + 𝑤) ∈ V
38 eqeq1 2769 . . . . . . . 8 (𝑦 = (𝐴 + 𝑤) → (𝑦 = ((𝐴 + 𝑥) 𝐴) ↔ (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴)))
3938rexbidv 3199 . . . . . . 7 (𝑦 = (𝐴 + 𝑤) → (∃𝑥𝑆 𝑦 = ((𝐴 + 𝑥) 𝐴) ↔ ∃𝑥𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴)))
409rnmpt 5540 . . . . . . 7 ran 𝐹 = {𝑦 ∣ ∃𝑥𝑆 𝑦 = ((𝐴 + 𝑥) 𝐴)}
4137, 39, 40elab2 3509 . . . . . 6 ((𝐴 + 𝑤) ∈ ran 𝐹 ↔ ∃𝑥𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) 𝐴))
42 risset 3209 . . . . . 6 ((𝑤 + 𝐴) ∈ 𝑆 ↔ ∃𝑥𝑆 𝑥 = (𝑤 + 𝐴))
4336, 41, 423bitr4g 305 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → ((𝐴 + 𝑤) ∈ ran 𝐹 ↔ (𝑤 + 𝐴) ∈ 𝑆))
4414, 43bitrd 270 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) ∧ 𝑤𝑋) → ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆))
4544ralrimiva 3113 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) → ∀𝑤𝑋 ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆))
461elnmz 17897 . . 3 (𝐴𝑁 ↔ (𝐴𝑋 ∧ ∀𝑤𝑋 ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆)))
4712, 45, 46sylanbrc 578 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴𝑋𝑆 = ran 𝐹)) → 𝐴𝑁)
4811, 47impbida 835 1 (𝑆 ∈ (SubGrp‘𝐺) → (𝐴𝑁 ↔ (𝐴𝑋𝑆 = ran 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  {crab 3059  wss 3732  cmpt 4888  ran crn 5278  cfv 6068  (class class class)co 6842  Basecbs 16130  +gcplusg 16214  Grpcgrp 17689  -gcsg 17691  SubGrpcsubg 17852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-0g 16368  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-grp 17692  df-minusg 17693  df-sbg 17694  df-subg 17855
This theorem is referenced by:  sylow3lem6  18311
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