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Theorem cntrsubgnsg 18462
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 486 . 2 ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) → 𝑋 ∈ (SubGrp‘𝑀))
2 simplr 768 . . . . . . . . 9 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑋𝑍)
3 simprr 772 . . . . . . . . 9 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑦𝑋)
42, 3sseldd 3943 . . . . . . . 8 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑦𝑍)
5 eqid 2822 . . . . . . . . . 10 (Base‘𝑀) = (Base‘𝑀)
6 eqid 2822 . . . . . . . . . 10 (Cntz‘𝑀) = (Cntz‘𝑀)
75, 6cntrval 18440 . . . . . . . . 9 ((Cntz‘𝑀)‘(Base‘𝑀)) = (Cntr‘𝑀)
8 cntrnsg.z . . . . . . . . 9 𝑍 = (Cntr‘𝑀)
97, 8eqtr4i 2848 . . . . . . . 8 ((Cntz‘𝑀)‘(Base‘𝑀)) = 𝑍
104, 9eleqtrrdi 2925 . . . . . . 7 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑦 ∈ ((Cntz‘𝑀)‘(Base‘𝑀)))
11 simprl 770 . . . . . . 7 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑥 ∈ (Base‘𝑀))
12 eqid 2822 . . . . . . . 8 (+g𝑀) = (+g𝑀)
1312, 6cntzi 18450 . . . . . . 7 ((𝑦 ∈ ((Cntz‘𝑀)‘(Base‘𝑀)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
1410, 11, 13syl2anc 587 . . . . . 6 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
1514oveq1d 7155 . . . . 5 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → ((𝑦(+g𝑀)𝑥)(-g𝑀)𝑥) = ((𝑥(+g𝑀)𝑦)(-g𝑀)𝑥))
16 subgrcl 18275 . . . . . . 7 (𝑋 ∈ (SubGrp‘𝑀) → 𝑀 ∈ Grp)
1716ad2antrr 725 . . . . . 6 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑀 ∈ Grp)
185subgss 18271 . . . . . . . 8 (𝑋 ∈ (SubGrp‘𝑀) → 𝑋 ⊆ (Base‘𝑀))
1918ad2antrr 725 . . . . . . 7 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑋 ⊆ (Base‘𝑀))
2019, 3sseldd 3943 . . . . . 6 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑦 ∈ (Base‘𝑀))
21 eqid 2822 . . . . . . 7 (-g𝑀) = (-g𝑀)
225, 12, 21grppncan 18181 . . . . . 6 ((𝑀 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → ((𝑦(+g𝑀)𝑥)(-g𝑀)𝑥) = 𝑦)
2317, 20, 11, 22syl3anc 1368 . . . . 5 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → ((𝑦(+g𝑀)𝑥)(-g𝑀)𝑥) = 𝑦)
2415, 23eqtr3d 2859 . . . 4 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → ((𝑥(+g𝑀)𝑦)(-g𝑀)𝑥) = 𝑦)
2524, 3eqeltrd 2914 . . 3 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → ((𝑥(+g𝑀)𝑦)(-g𝑀)𝑥) ∈ 𝑋)
2625ralrimivva 3181 . 2 ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) → ∀𝑥 ∈ (Base‘𝑀)∀𝑦𝑋 ((𝑥(+g𝑀)𝑦)(-g𝑀)𝑥) ∈ 𝑋)
275, 12, 21isnsg3 18303 . 2 (𝑋 ∈ (NrmSGrp‘𝑀) ↔ (𝑋 ∈ (SubGrp‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝑀)∀𝑦𝑋 ((𝑥(+g𝑀)𝑦)(-g𝑀)𝑥) ∈ 𝑋))
281, 26, 27sylanbrc 586 1 ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) → 𝑋 ∈ (NrmSGrp‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wral 3130  wss 3908  cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  Grpcgrp 18094  -gcsg 18096  SubGrpcsubg 18264  NrmSGrpcnsg 18265  Cntzccntz 18436  Cntrccntr 18437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-0g 16706  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-grp 18097  df-minusg 18098  df-sbg 18099  df-subg 18267  df-nsg 18268  df-cntz 18438  df-cntr 18439
This theorem is referenced by:  cntrnsg  18463
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