Theorem cntrsubgnsg 18935

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 766	. . . . . . . . 9 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑋 ⊆ 𝑍)
3		simprr 770	. . . . . . . . 9 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
4	2, 3	sseldd 3922	. . . . . . . 8 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑍)
5		eqid 2738	. . . . . . . . . 10 ⊢ (Base‘𝑀) = (Base‘𝑀)
6		eqid 2738	. . . . . . . . . 10 ⊢ (Cntz‘𝑀) = (Cntz‘𝑀)
7	5, 6	cntrval 18913	. . . . . . . . 9 ⊢ ((Cntz‘𝑀)‘(Base‘𝑀)) = (Cntr‘𝑀)
8		cntrnsg.z	. . . . . . . . 9 ⊢ 𝑍 = (Cntr‘𝑀)
9	7, 8	eqtr4i 2769	. . . . . . . 8 ⊢ ((Cntz‘𝑀)‘(Base‘𝑀)) = 𝑍
10	4, 9	eleqtrrdi 2850	. . . . . . 7 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ ((Cntz‘𝑀)‘(Base‘𝑀)))
11		simprl 768	. . . . . . 7 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ (Base‘𝑀))
12		eqid 2738	. . . . . . . 8 ⊢ (+g‘𝑀) = (+g‘𝑀)
13	12, 6	cntzi 18923	. . . . . . 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 7283	. . . . 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 18748	. . . . . . 7 ⊢ (𝑋 ∈ (SubGrp‘𝑀) → 𝑀 ∈ Grp)
17	16	ad2antrr 723	. . . . . 6 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑀 ∈ Grp)
18	5	subgss 18744	. . . . . . . 8 ⊢ (𝑋 ∈ (SubGrp‘𝑀) → 𝑋 ⊆ (Base‘𝑀))
19	18	ad2antrr 723	. . . . . . 7 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑋 ⊆ (Base‘𝑀))
20	19, 3	sseldd 3922	. . . . . 6 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ (Base‘𝑀))
21		eqid 2738	. . . . . . 7 ⊢ (-g‘𝑀) = (-g‘𝑀)
22	5, 12, 21	grppncan 18654	. . . . . 6 ⊢ ((𝑀 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → ((𝑦(+g‘𝑀)𝑥)(-g‘𝑀)𝑥) = 𝑦)
23	17, 20, 11, 22	syl3anc 1370	. . . . 5 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → ((𝑦(+g‘𝑀)𝑥)(-g‘𝑀)𝑥) = 𝑦)
24	15, 23	eqtr3d 2780	. . . 4 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → ((𝑥(+g‘𝑀)𝑦)(-g‘𝑀)𝑥) = 𝑦)
25	24, 3	eqeltrd 2839	. . 3 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → ((𝑥(+g‘𝑀)𝑦)(-g‘𝑀)𝑥) ∈ 𝑋)
26	25	ralrimivva 3120	. 2 ⊢ ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ 𝑋 ((𝑥(+g‘𝑀)𝑦)(-g‘𝑀)𝑥) ∈ 𝑋)
27	5, 12, 21	isnsg3 18776	. 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 1539   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3887  ‘cfv 6427  (class class class)co 7268  Basecbs 16900  +_gcplusg 16950  Grpcgrp 18565  -_gcsg 18567  SubGrpcsubg 18737  NrmSGrpcnsg 18738  Cntzccntz 18909  Cntrccntr 18910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5222  ax-nul 5229  ax-pow 5287  ax-pr 5351  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3432  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-iun 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5485  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-iota 6385  df-fun 6429  df-fn 6430  df-f 6431  df-f1 6432  df-fo 6433  df-f1o 6434  df-fv 6435  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7821  df-2nd 7822  df-0g 17140  df-mgm 18314  df-sgrp 18363  df-mnd 18374  df-grp 18568  df-minusg 18569  df-sbg 18570  df-subg 18740  df-nsg 18741  df-cntz 18911  df-cntr 18912

This theorem is referenced by:  cntrnsg  18936