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Theorem ssnmz 17459
Description: A subgroup is a subset of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
Hypotheses
Ref Expression
elnmz.1 𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}
nmzsubg.2 𝑋 = (Base‘𝐺)
nmzsubg.3 + = (+g𝐺)
Assertion
Ref Expression
ssnmz (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑁)
Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑆,𝑦   𝑥, + ,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝑁(𝑥,𝑦)

Proof of Theorem ssnmz
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmzsubg.2 . . . . . 6 𝑋 = (Base‘𝐺)
21subgss 17418 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
32sselda 3568 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧𝑆) → 𝑧𝑋)
4 simpll 786 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
5 subgrcl 17422 . . . . . . . . . . . . 13 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
64, 5syl 17 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝐺 ∈ Grp)
74, 2syl 17 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑆𝑋)
8 simplrl 796 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑧𝑆)
97, 8sseldd 3569 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑧𝑋)
10 nmzsubg.3 . . . . . . . . . . . . 13 + = (+g𝐺)
11 eqid 2610 . . . . . . . . . . . . 13 (0g𝐺) = (0g𝐺)
12 eqid 2610 . . . . . . . . . . . . 13 (invg𝐺) = (invg𝐺)
131, 10, 11, 12grplinv 17291 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑧𝑋) → (((invg𝐺)‘𝑧) + 𝑧) = (0g𝐺))
146, 9, 13syl2anc 691 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → (((invg𝐺)‘𝑧) + 𝑧) = (0g𝐺))
1514oveq1d 6564 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = ((0g𝐺) + 𝑤))
1612subginvcl 17426 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧𝑆) → ((invg𝐺)‘𝑧) ∈ 𝑆)
174, 8, 16syl2anc 691 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → ((invg𝐺)‘𝑧) ∈ 𝑆)
187, 17sseldd 3569 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → ((invg𝐺)‘𝑧) ∈ 𝑋)
19 simplrr 797 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑤𝑋)
201, 10grpass 17254 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑧) ∈ 𝑋𝑧𝑋𝑤𝑋)) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)))
216, 18, 9, 19, 20syl13anc 1320 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)))
221, 10, 11grplid 17275 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑤𝑋) → ((0g𝐺) + 𝑤) = 𝑤)
236, 19, 22syl2anc 691 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → ((0g𝐺) + 𝑤) = 𝑤)
2415, 21, 233eqtr3d 2652 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) = 𝑤)
25 simpr 476 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → (𝑧 + 𝑤) ∈ 𝑆)
2610subgcl 17427 . . . . . . . . . 10 ((𝑆 ∈ (SubGrp‘𝐺) ∧ ((invg𝐺)‘𝑧) ∈ 𝑆 ∧ (𝑧 + 𝑤) ∈ 𝑆) → (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ∈ 𝑆)
274, 17, 25, 26syl3anc 1318 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ∈ 𝑆)
2824, 27eqeltrrd 2689 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑤𝑆)
2910subgcl 17427 . . . . . . . 8 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑤𝑆𝑧𝑆) → (𝑤 + 𝑧) ∈ 𝑆)
304, 28, 8, 29syl3anc 1318 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → (𝑤 + 𝑧) ∈ 𝑆)
31 simpll 786 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
32 simplrl 796 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝑧𝑆)
3331, 5syl 17 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝐺 ∈ Grp)
34 simplrr 797 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝑤𝑋)
3531, 32, 3syl2anc 691 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝑧𝑋)
36 eqid 2610 . . . . . . . . . . 11 (-g𝐺) = (-g𝐺)
371, 10, 36grppncan 17329 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑤𝑋𝑧𝑋) → ((𝑤 + 𝑧)(-g𝐺)𝑧) = 𝑤)
3833, 34, 35, 37syl3anc 1318 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → ((𝑤 + 𝑧)(-g𝐺)𝑧) = 𝑤)
39 simpr 476 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → (𝑤 + 𝑧) ∈ 𝑆)
4036subgsubcl 17428 . . . . . . . . . 10 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑤 + 𝑧) ∈ 𝑆𝑧𝑆) → ((𝑤 + 𝑧)(-g𝐺)𝑧) ∈ 𝑆)
4131, 39, 32, 40syl3anc 1318 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → ((𝑤 + 𝑧)(-g𝐺)𝑧) ∈ 𝑆)
4238, 41eqeltrrd 2689 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝑤𝑆)
4310subgcl 17427 . . . . . . . 8 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧𝑆𝑤𝑆) → (𝑧 + 𝑤) ∈ 𝑆)
4431, 32, 42, 43syl3anc 1318 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → (𝑧 + 𝑤) ∈ 𝑆)
4530, 44impbida 873 . . . . . 6 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))
4645anassrs 678 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧𝑆) ∧ 𝑤𝑋) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))
4746ralrimiva 2949 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧𝑆) → ∀𝑤𝑋 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))
48 elnmz.1 . . . . 5 𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}
4948elnmz 17456 . . . 4 (𝑧𝑁 ↔ (𝑧𝑋 ∧ ∀𝑤𝑋 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)))
503, 47, 49sylanbrc 695 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧𝑆) → 𝑧𝑁)
5150ex 449 . 2 (𝑆 ∈ (SubGrp‘𝐺) → (𝑧𝑆𝑧𝑁))
5251ssrdv 3574 1 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  {crab 2900  wss 3540  cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Grpcgrp 17245  invgcminusg 17246  -gcsg 17247  SubGrpcsubg 17411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414
This theorem is referenced by:  nmznsg  17461  sylow3lem6  17870
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