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Theorem cntrsubgnsg 17689
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 473 . 2 ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) → 𝑋 ∈ (SubGrp‘𝑀))
2 simplr 791 . . . . . . . . 9 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑋𝑍)
3 simprr 795 . . . . . . . . 9 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑦𝑋)
42, 3sseldd 3589 . . . . . . . 8 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑦𝑍)
5 eqid 2626 . . . . . . . . . 10 (Base‘𝑀) = (Base‘𝑀)
6 eqid 2626 . . . . . . . . . 10 (Cntz‘𝑀) = (Cntz‘𝑀)
75, 6cntrval 17668 . . . . . . . . 9 ((Cntz‘𝑀)‘(Base‘𝑀)) = (Cntr‘𝑀)
8 cntrnsg.z . . . . . . . . 9 𝑍 = (Cntr‘𝑀)
97, 8eqtr4i 2651 . . . . . . . 8 ((Cntz‘𝑀)‘(Base‘𝑀)) = 𝑍
104, 9syl6eleqr 2715 . . . . . . 7 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑦 ∈ ((Cntz‘𝑀)‘(Base‘𝑀)))
11 simprl 793 . . . . . . 7 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑥 ∈ (Base‘𝑀))
12 eqid 2626 . . . . . . . 8 (+g𝑀) = (+g𝑀)
1312, 6cntzi 17678 . . . . . . 7 ((𝑦 ∈ ((Cntz‘𝑀)‘(Base‘𝑀)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
1410, 11, 13syl2anc 692 . . . . . 6 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
1514oveq1d 6620 . . . . 5 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → ((𝑦(+g𝑀)𝑥)(-g𝑀)𝑥) = ((𝑥(+g𝑀)𝑦)(-g𝑀)𝑥))
16 subgrcl 17515 . . . . . . 7 (𝑋 ∈ (SubGrp‘𝑀) → 𝑀 ∈ Grp)
1716ad2antrr 761 . . . . . 6 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑀 ∈ Grp)
185subgss 17511 . . . . . . . 8 (𝑋 ∈ (SubGrp‘𝑀) → 𝑋 ⊆ (Base‘𝑀))
1918ad2antrr 761 . . . . . . 7 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑋 ⊆ (Base‘𝑀))
2019, 3sseldd 3589 . . . . . 6 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑦 ∈ (Base‘𝑀))
21 eqid 2626 . . . . . . 7 (-g𝑀) = (-g𝑀)
225, 12, 21grppncan 17422 . . . . . 6 ((𝑀 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → ((𝑦(+g𝑀)𝑥)(-g𝑀)𝑥) = 𝑦)
2317, 20, 11, 22syl3anc 1323 . . . . 5 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → ((𝑦(+g𝑀)𝑥)(-g𝑀)𝑥) = 𝑦)
2415, 23eqtr3d 2662 . . . 4 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → ((𝑥(+g𝑀)𝑦)(-g𝑀)𝑥) = 𝑦)
2524, 3eqeltrd 2704 . . 3 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → ((𝑥(+g𝑀)𝑦)(-g𝑀)𝑥) ∈ 𝑋)
2625ralrimivva 2970 . 2 ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) → ∀𝑥 ∈ (Base‘𝑀)∀𝑦𝑋 ((𝑥(+g𝑀)𝑦)(-g𝑀)𝑥) ∈ 𝑋)
275, 12, 21isnsg3 17544 . 2 (𝑋 ∈ (NrmSGrp‘𝑀) ↔ (𝑋 ∈ (SubGrp‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝑀)∀𝑦𝑋 ((𝑥(+g𝑀)𝑦)(-g𝑀)𝑥) ∈ 𝑋))
281, 26, 27sylanbrc 697 1 ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) → 𝑋 ∈ (NrmSGrp‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wral 2912  wss 3560  cfv 5850  (class class class)co 6605  Basecbs 15776  +gcplusg 15857  Grpcgrp 17338  -gcsg 17340  SubGrpcsubg 17504  NrmSGrpcnsg 17505  Cntzccntz 17664  Cntrccntr 17665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-0g 16018  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-grp 17341  df-minusg 17342  df-sbg 17343  df-subg 17507  df-nsg 17508  df-cntz 17666  df-cntr 17667
This theorem is referenced by:  cntrnsg  17690
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