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Theorem cntrsubgnsg 17980
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 468 . 2 ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) → 𝑋 ∈ (SubGrp‘𝑀))
2 simplr 752 . . . . . . . . 9 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑋𝑍)
3 simprr 756 . . . . . . . . 9 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑦𝑋)
42, 3sseldd 3753 . . . . . . . 8 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑦𝑍)
5 eqid 2771 . . . . . . . . . 10 (Base‘𝑀) = (Base‘𝑀)
6 eqid 2771 . . . . . . . . . 10 (Cntz‘𝑀) = (Cntz‘𝑀)
75, 6cntrval 17959 . . . . . . . . 9 ((Cntz‘𝑀)‘(Base‘𝑀)) = (Cntr‘𝑀)
8 cntrnsg.z . . . . . . . . 9 𝑍 = (Cntr‘𝑀)
97, 8eqtr4i 2796 . . . . . . . 8 ((Cntz‘𝑀)‘(Base‘𝑀)) = 𝑍
104, 9syl6eleqr 2861 . . . . . . 7 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑦 ∈ ((Cntz‘𝑀)‘(Base‘𝑀)))
11 simprl 754 . . . . . . 7 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑥 ∈ (Base‘𝑀))
12 eqid 2771 . . . . . . . 8 (+g𝑀) = (+g𝑀)
1312, 6cntzi 17969 . . . . . . 7 ((𝑦 ∈ ((Cntz‘𝑀)‘(Base‘𝑀)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
1410, 11, 13syl2anc 573 . . . . . 6 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
1514oveq1d 6811 . . . . 5 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → ((𝑦(+g𝑀)𝑥)(-g𝑀)𝑥) = ((𝑥(+g𝑀)𝑦)(-g𝑀)𝑥))
16 subgrcl 17807 . . . . . . 7 (𝑋 ∈ (SubGrp‘𝑀) → 𝑀 ∈ Grp)
1716ad2antrr 705 . . . . . 6 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑀 ∈ Grp)
185subgss 17803 . . . . . . . 8 (𝑋 ∈ (SubGrp‘𝑀) → 𝑋 ⊆ (Base‘𝑀))
1918ad2antrr 705 . . . . . . 7 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑋 ⊆ (Base‘𝑀))
2019, 3sseldd 3753 . . . . . 6 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → 𝑦 ∈ (Base‘𝑀))
21 eqid 2771 . . . . . . 7 (-g𝑀) = (-g𝑀)
225, 12, 21grppncan 17714 . . . . . 6 ((𝑀 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → ((𝑦(+g𝑀)𝑥)(-g𝑀)𝑥) = 𝑦)
2317, 20, 11, 22syl3anc 1476 . . . . 5 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → ((𝑦(+g𝑀)𝑥)(-g𝑀)𝑥) = 𝑦)
2415, 23eqtr3d 2807 . . . 4 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → ((𝑥(+g𝑀)𝑦)(-g𝑀)𝑥) = 𝑦)
2524, 3eqeltrd 2850 . . 3 (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦𝑋)) → ((𝑥(+g𝑀)𝑦)(-g𝑀)𝑥) ∈ 𝑋)
2625ralrimivva 3120 . 2 ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) → ∀𝑥 ∈ (Base‘𝑀)∀𝑦𝑋 ((𝑥(+g𝑀)𝑦)(-g𝑀)𝑥) ∈ 𝑋)
275, 12, 21isnsg3 17836 . 2 (𝑋 ∈ (NrmSGrp‘𝑀) ↔ (𝑋 ∈ (SubGrp‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝑀)∀𝑦𝑋 ((𝑥(+g𝑀)𝑦)(-g𝑀)𝑥) ∈ 𝑋))
281, 26, 27sylanbrc 572 1 ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) → 𝑋 ∈ (NrmSGrp‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  wss 3723  cfv 6030  (class class class)co 6796  Basecbs 16064  +gcplusg 16149  Grpcgrp 17630  -gcsg 17632  SubGrpcsubg 17796  NrmSGrpcnsg 17797  Cntzccntz 17955  Cntrccntr 17956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-grp 17633  df-minusg 17634  df-sbg 17635  df-subg 17799  df-nsg 17800  df-cntz 17957  df-cntr 17958
This theorem is referenced by:  cntrnsg  17981
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