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Theorem cntrsubgnsg 19374
Description: A central subgroup is normal. (Contributed by Stefan O'Rear, 6-Sep-2015.)
Hypothesis
Ref Expression
cntrnsg.z 𝑍 = (Cntr‘𝑀)
Assertion
Ref Expression
cntrsubgnsg ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) → 𝑋 ∈ (NrmSGrp‘𝑀))

Proof of Theorem cntrsubgnsg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 482 . 2 ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) → 𝑋 ∈ (SubGrp‘𝑀))
2 simplr 769 . . . . . . . . 9 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑋𝑍)
3 simprr 773 . . . . . . . . 9 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑦𝑋)
42, 3sseldd 3996 . . . . . . . 8 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑦𝑍)
5 eqid 2735 . . . . . . . . . 10 (Base‘𝑀) = (Base‘𝑀)
6 eqid 2735 . . . . . . . . . 10 (Cntz‘𝑀) = (Cntz‘𝑀)
75, 6cntrval 19350 . . . . . . . . 9 ((Cntz‘𝑀)‘(Base‘𝑀)) = (Cntr‘𝑀)
8 cntrnsg.z . . . . . . . . 9 𝑍 = (Cntr‘𝑀)
97, 8eqtr4i 2766 . . . . . . . 8 ((Cntz‘𝑀)‘(Base‘𝑀)) = 𝑍
104, 9eleqtrrdi 2850 . . . . . . 7 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑦 ∈ ((Cntz‘𝑀)‘(Base‘𝑀)))
11 simprl 771 . . . . . . 7 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑥 ∈ (Base‘𝑀))
12 eqid 2735 . . . . . . . 8 (+g𝑀) = (+g𝑀)
1312, 6cntzi 19360 . . . . . . 7 ((𝑦 ∈ ((Cntz‘𝑀)‘(Base‘𝑀)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
1410, 11, 13syl2anc 584 . . . . . 6 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
1514oveq1d 7446 . . . . 5 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → ((𝑦(+g𝑀)𝑥)(-g𝑀)𝑥) = ((𝑥(+g𝑀)𝑦)(-g𝑀)𝑥))
16 subgrcl 19162 . . . . . . 7 (𝑋 ∈ (SubGrp‘𝑀) → 𝑀 ∈ Grp)
1716ad2antrr 726 . . . . . 6 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑀 ∈ Grp)
185subgss 19158 . . . . . . . 8 (𝑋 ∈ (SubGrp‘𝑀) → 𝑋 ⊆ (Base‘𝑀))
1918ad2antrr 726 . . . . . . 7 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑋 ⊆ (Base‘𝑀))
2019, 3sseldd 3996 . . . . . 6 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑦 ∈ (Base‘𝑀))
21 eqid 2735 . . . . . . 7 (-g𝑀) = (-g𝑀)
225, 12, 21grppncan 19062 . . . . . 6 ((𝑀 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → ((𝑦(+g𝑀)𝑥)(-g𝑀)𝑥) = 𝑦)
2317, 20, 11, 22syl3anc 1370 . . . . 5 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → ((𝑦(+g𝑀)𝑥)(-g𝑀)𝑥) = 𝑦)
2415, 23eqtr3d 2777 . . . 4 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → ((𝑥(+g𝑀)𝑦)(-g𝑀)𝑥) = 𝑦)
2524, 3eqeltrd 2839 . . 3 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → ((𝑥(+g𝑀)𝑦)(-g𝑀)𝑥) ∈ 𝑋)
2625ralrimivva 3200 . 2 ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) → ∀𝑥 ∈ (Base‘𝑀)∀𝑦𝑋 ((𝑥(+g𝑀)𝑦)(-g𝑀)𝑥) ∈ 𝑋)
275, 12, 21isnsg3 19191 . 2 (𝑋 ∈ (NrmSGrp‘𝑀) ↔ (𝑋 ∈ (SubGrp‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝑀)∀𝑦𝑋 ((𝑥(+g𝑀)𝑦)(-g𝑀)𝑥) ∈ 𝑋))
281, 26, 27sylanbrc 583 1 ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) → 𝑋 ∈ (NrmSGrp‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  wss 3963  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Grpcgrp 18964  -gcsg 18966  SubGrpcsubg 19151  NrmSGrpcnsg 19152  Cntzccntz 19346  Cntrccntr 19347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-nsg 19155  df-cntz 19348  df-cntr 19349
This theorem is referenced by:  cntrnsg  19375
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