Theorem cntrsubgnsg 19307

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 482	. 2 ⊢ ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) → 𝑋 ∈ (SubGrp‘𝑀))
2		simplr 769	. . . . . . . . 9 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑋 ⊆ 𝑍)
3		simprr 773	. . . . . . . . 9 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
4	2, 3	sseldd 3918	. . . . . . . 8 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑍)
5		eqid 2735	. . . . . . . . . 10 ⊢ (Base‘𝑀) = (Base‘𝑀)
6		eqid 2735	. . . . . . . . . 10 ⊢ (Cntz‘𝑀) = (Cntz‘𝑀)
7	5, 6	cntrval 19283	. . . . . . . . 9 ⊢ ((Cntz‘𝑀)‘(Base‘𝑀)) = (Cntr‘𝑀)
8		cntrnsg.z	. . . . . . . . 9 ⊢ 𝑍 = (Cntr‘𝑀)
9	7, 8	eqtr4i 2761	. . . . . . . 8 ⊢ ((Cntz‘𝑀)‘(Base‘𝑀)) = 𝑍
10	4, 9	eleqtrrdi 2846	. . . . . . 7 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ ((Cntz‘𝑀)‘(Base‘𝑀)))
11		simprl 771	. . . . . . 7 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ (Base‘𝑀))
12		eqid 2735	. . . . . . . 8 ⊢ (+g‘𝑀) = (+g‘𝑀)
13	12, 6	cntzi 19293	. . . . . . 7 ⊢ ((𝑦 ∈ ((Cntz‘𝑀)‘(Base‘𝑀)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
14	10, 11, 13	syl2anc 585	. . . . . 6 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
15	14	oveq1d 7371	. . . . 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 19096	. . . . . . 7 ⊢ (𝑋 ∈ (SubGrp‘𝑀) → 𝑀 ∈ Grp)
17	16	ad2antrr 727	. . . . . 6 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑀 ∈ Grp)
18	5	subgss 19092	. . . . . . . 8 ⊢ (𝑋 ∈ (SubGrp‘𝑀) → 𝑋 ⊆ (Base‘𝑀))
19	18	ad2antrr 727	. . . . . . 7 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑋 ⊆ (Base‘𝑀))
20	19, 3	sseldd 3918	. . . . . 6 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ (Base‘𝑀))
21		eqid 2735	. . . . . . 7 ⊢ (-g‘𝑀) = (-g‘𝑀)
22	5, 12, 21	grppncan 18996	. . . . . 6 ⊢ ((𝑀 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → ((𝑦(+g‘𝑀)𝑥)(-g‘𝑀)𝑥) = 𝑦)
23	17, 20, 11, 22	syl3anc 1374	. . . . 5 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → ((𝑦(+g‘𝑀)𝑥)(-g‘𝑀)𝑥) = 𝑦)
24	15, 23	eqtr3d 2772	. . . 4 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → ((𝑥(+g‘𝑀)𝑦)(-g‘𝑀)𝑥) = 𝑦)
25	24, 3	eqeltrd 2835	. . 3 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → ((𝑥(+g‘𝑀)𝑦)(-g‘𝑀)𝑥) ∈ 𝑋)
26	25	ralrimivva 3178	. 2 ⊢ ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ 𝑋 ((𝑥(+g‘𝑀)𝑦)(-g‘𝑀)𝑥) ∈ 𝑋)
27	5, 12, 21	isnsg3 19124	. 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 584	1 ⊢ ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) → 𝑋 ∈ (NrmSGrp‘𝑀))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3049   ⊆ wss 3885  ‘cfv 6487  (class class class)co 7356  Basecbs 17168  +_gcplusg 17209  Grpcgrp 18898  -_gcsg 18900  SubGrpcsubg 19085  NrmSGrpcnsg 19086  Cntzccntz 19279  Cntrccntr 19280

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rmo 3340  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-0g 17393  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-grp 18901  df-minusg 18902  df-sbg 18903  df-subg 19088  df-nsg 19089  df-cntz 19281  df-cntr 19282

This theorem is referenced by:  cntrnsg  19308