Theorem cntrsubgnsg 17773

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 e. (SubGrp ` M ) /\ X C_ Z ) -> X e. (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 473	. 2 |- ( ( X e. (SubGrp ` M ) /\ X C_ Z ) -> X e. (SubGrp ` M ) )
2		simplr 792	. . . . . . . . 9 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> X C_ Z )
3		simprr 796	. . . . . . . . 9 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> y e. X )
4	2, 3	sseldd 3604	. . . . . . . 8 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> y e. Z )
5		eqid 2622	. . . . . . . . . 10 |- ( Base ` M ) = ( Base ` M )
6		eqid 2622	. . . . . . . . . 10 |- (Cntz ` M ) = (Cntz ` M )
7	5, 6	cntrval 17752	. . . . . . . . 9 |- ( (Cntz ` M ) ` ( Base ` M ) ) = (Cntr ` M )
8		cntrnsg.z	. . . . . . . . 9 |- Z = (Cntr ` M )
9	7, 8	eqtr4i 2647	. . . . . . . 8 |- ( (Cntz ` M ) ` ( Base ` M ) ) = Z
10	4, 9	syl6eleqr 2712	. . . . . . 7 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> y e. ( (Cntz ` M ) ` ( Base ` M ) ) )
11		simprl 794	. . . . . . 7 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> x e. ( Base ` M ) )
12		eqid 2622	. . . . . . . 8 |- ( +g ` M ) = ( +g ` M )
13	12, 6	cntzi 17762	. . . . . . 7 |- ( ( y e. ( (Cntz ` M ) ` ( Base ` M ) ) /\ x e. ( Base ` M ) ) -> ( y ( +g ` M ) x ) = ( x ( +g ` M ) y ) )
14	10, 11, 13	syl2anc 693	. . . . . 6 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> ( y ( +g ` M ) x ) = ( x ( +g ` M ) y ) )
15	14	oveq1d 6665	. . . . 5 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> ( ( y ( +g ` M ) x ) ( -g ` M ) x ) = ( ( x ( +g ` M ) y ) ( -g ` M ) x ) )
16		subgrcl 17599	. . . . . . 7 |- ( X e. (SubGrp ` M ) -> M e. Grp )
17	16	ad2antrr 762	. . . . . 6 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> M e. Grp )
18	5	subgss 17595	. . . . . . . 8 |- ( X e. (SubGrp ` M ) -> X C_ ( Base ` M ) )
19	18	ad2antrr 762	. . . . . . 7 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> X C_ ( Base ` M ) )
20	19, 3	sseldd 3604	. . . . . 6 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> y e. ( Base ` M ) )
21		eqid 2622	. . . . . . 7 |- ( -g ` M ) = ( -g ` M )
22	5, 12, 21	grppncan 17506	. . . . . 6 |- ( ( M e. Grp /\ y e. ( Base ` M ) /\ x e. ( Base ` M ) ) -> ( ( y ( +g ` M ) x ) ( -g ` M ) x ) = y )
23	17, 20, 11, 22	syl3anc 1326	. . . . 5 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> ( ( y ( +g ` M ) x ) ( -g ` M ) x ) = y )
24	15, 23	eqtr3d 2658	. . . 4 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> ( ( x ( +g ` M ) y ) ( -g ` M ) x ) = y )
25	24, 3	eqeltrd 2701	. . 3 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> ( ( x ( +g ` M ) y ) ( -g ` M ) x ) e. X )
26	25	ralrimivva 2971	. 2 |- ( ( X e. (SubGrp ` M ) /\ X C_ Z ) -> A. x e. ( Base ` M ) A. y e. X ( ( x ( +g ` M ) y ) ( -g ` M ) x ) e. X )
27	5, 12, 21	isnsg3 17628	. 2 |- ( X e. (NrmSGrp ` M ) <-> ( X e. (SubGrp ` M ) /\ A. x e. ( Base ` M ) A. y e. X ( ( x ( +g ` M ) y ) ( -g ` M ) x ) e. X ) )
28	1, 26, 27	sylanbrc 698	1 |- ( ( X e. (SubGrp ` M ) /\ X C_ Z ) -> X e. (NrmSGrp ` M ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   Grpcgrp 17422   -gcsg 17424  SubGrpcsubg 17588  NrmSGrpcnsg 17589  Cntzccntz 17748  Cntrccntr 17749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-nsg 17592  df-cntz 17750  df-cntr 17751

This theorem is referenced by:  cntrnsg  17774