Theorem cntrsubgnsg 19209

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) ∧ 𝑋 ⊆ 𝑍) → 𝑋 ∈ (NrmSGrp‘𝑀))

Proof of Theorem cntrsubgnsg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 483	. 2 ⊢ ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) → 𝑋 ∈ (SubGrp‘𝑀))
2		simplr 767	. . . . . . . . 9 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑋 ⊆ 𝑍)
3		simprr 771	. . . . . . . . 9 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
4	2, 3	sseldd 3983	. . . . . . . 8 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑍)
5		eqid 2732	. . . . . . . . . 10 ⊢ (Base‘𝑀) = (Base‘𝑀)
6		eqid 2732	. . . . . . . . . 10 ⊢ (Cntz‘𝑀) = (Cntz‘𝑀)
7	5, 6	cntrval 19185	. . . . . . . . 9 ⊢ ((Cntz‘𝑀)‘(Base‘𝑀)) = (Cntr‘𝑀)
8		cntrnsg.z	. . . . . . . . 9 ⊢ 𝑍 = (Cntr‘𝑀)
9	7, 8	eqtr4i 2763	. . . . . . . 8 ⊢ ((Cntz‘𝑀)‘(Base‘𝑀)) = 𝑍
10	4, 9	eleqtrrdi 2844	. . . . . . 7 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ ((Cntz‘𝑀)‘(Base‘𝑀)))
11		simprl 769	. . . . . . 7 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ (Base‘𝑀))
12		eqid 2732	. . . . . . . 8 ⊢ (+g‘𝑀) = (+g‘𝑀)
13	12, 6	cntzi 19195	. . . . . . 7 ⊢ ((𝑦 ∈ ((Cntz‘𝑀)‘(Base‘𝑀)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
14	10, 11, 13	syl2anc 584	. . . . . 6 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
15	14	oveq1d 7426	. . . . 5 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → ((𝑦(+g‘𝑀)𝑥)(-g‘𝑀)𝑥) = ((𝑥(+g‘𝑀)𝑦)(-g‘𝑀)𝑥))
16		subgrcl 19013	. . . . . . 7 ⊢ (𝑋 ∈ (SubGrp‘𝑀) → 𝑀 ∈ Grp)
17	16	ad2antrr 724	. . . . . 6 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑀 ∈ Grp)
18	5	subgss 19009	. . . . . . . 8 ⊢ (𝑋 ∈ (SubGrp‘𝑀) → 𝑋 ⊆ (Base‘𝑀))
19	18	ad2antrr 724	. . . . . . 7 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑋 ⊆ (Base‘𝑀))
20	19, 3	sseldd 3983	. . . . . 6 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ (Base‘𝑀))
21		eqid 2732	. . . . . . 7 ⊢ (-g‘𝑀) = (-g‘𝑀)
22	5, 12, 21	grppncan 18916	. . . . . 6 ⊢ ((𝑀 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → ((𝑦(+g‘𝑀)𝑥)(-g‘𝑀)𝑥) = 𝑦)
23	17, 20, 11, 22	syl3anc 1371	. . . . 5 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → ((𝑦(+g‘𝑀)𝑥)(-g‘𝑀)𝑥) = 𝑦)
24	15, 23	eqtr3d 2774	. . . 4 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → ((𝑥(+g‘𝑀)𝑦)(-g‘𝑀)𝑥) = 𝑦)
25	24, 3	eqeltrd 2833	. . 3 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → ((𝑥(+g‘𝑀)𝑦)(-g‘𝑀)𝑥) ∈ 𝑋)
26	25	ralrimivva 3200	. 2 ⊢ ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ 𝑋 ((𝑥(+g‘𝑀)𝑦)(-g‘𝑀)𝑥) ∈ 𝑋)
27	5, 12, 21	isnsg3 19042	. 2 ⊢ (𝑋 ∈ (NrmSGrp‘𝑀) ↔ (𝑋 ∈ (SubGrp‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ 𝑋 ((𝑥(+g‘𝑀)𝑦)(-g‘𝑀)𝑥) ∈ 𝑋))
28	1, 26, 27	sylanbrc 583	1 ⊢ ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) → 𝑋 ∈ (NrmSGrp‘𝑀))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ⊆ wss 3948  ‘cfv 6543  (class class class)co 7411  Basecbs 17146  +_gcplusg 17199  Grpcgrp 18821  -_gcsg 18823  SubGrpcsubg 19002  NrmSGrpcnsg 19003  Cntzccntz 19181  Cntrccntr 19182

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-0g 17389  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-grp 18824  df-minusg 18825  df-sbg 18826  df-subg 19005  df-nsg 19006  df-cntz 19183  df-cntr 19184

This theorem is referenced by:  cntrnsg  19210