Theorem cntrsubgnsg 14169

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 109	. 2 |- ( ( X e. (SubGrp ` M ) /\ X C_ Z ) -> X e. (SubGrp ` M ) )
2		simplr 533	. . . . . . . . 9 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> X C_ Z )
3		simprr 537	. . . . . . . . 9 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> y e. X )
4	2, 3	sseldd 3249	. . . . . . . 8 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> y e. Z )
5		eqid 2238	. . . . . . . . . 10 |- ( Base ` M ) = ( Base ` M )
6		eqid 2238	. . . . . . . . . 10 |- (Cntz ` M ) = (Cntz ` M )
7	5, 6	cntrval 14145	. . . . . . . . 9 |- ( (Cntz ` M ) ` ( Base ` M ) ) = (Cntr ` M )
8		cntrnsg.z	. . . . . . . . 9 |- Z = (Cntr ` M )
9	7, 8	eqtr4i 2262	. . . . . . . 8 |- ( (Cntz ` M ) ` ( Base ` M ) ) = Z
10	4, 9	eleqtrrdi 2332	. . . . . . 7 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> y e. ( (Cntz ` M ) ` ( Base ` M ) ) )
11		simprl 535	. . . . . . 7 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> x e. ( Base ` M ) )
12		eqid 2238	. . . . . . . 8 |- ( +g ` M ) = ( +g ` M )
13	12, 6	cntzi 14156	. . . . . . 7 |- ( ( y e. ( (Cntz ` M ) ` ( Base ` M ) ) /\ x e. ( Base ` M ) ) -> ( y ( +g ` M ) x ) = ( x ( +g ` M ) y ) )
14	10, 11, 13	syl2anc 415	. . . . . 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 6100	. . . . 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 14035	. . . . . . 7 |- ( X e. (SubGrp ` M ) -> M e. Grp )
17	16	ad2antrr 492	. . . . . 6 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> M e. Grp )
18	5	subgss 14030	. . . . . . . 8 |- ( X e. (SubGrp ` M ) -> X C_ ( Base ` M ) )
19	18	ad2antrr 492	. . . . . . 7 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> X C_ ( Base ` M ) )
20	19, 3	sseldd 3249	. . . . . 6 |- ( ( ( X e. (SubGrp ` M ) /\ X C_ Z ) /\ ( x e. ( Base ` M ) /\ y e. X ) ) -> y e. ( Base ` M ) )
21		eqid 2238	. . . . . . 7 |- ( -g ` M ) = ( -g ` M )
22	5, 12, 21	grppncan 13949	. . . . . 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 1278	. . . . 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 2273	. . . 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 2315	. . 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 2632	. 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 14063	. 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 421	1 |- ( ( X e. (SubGrp ` M ) /\ X C_ Z ) -> X e. (NrmSGrp ` M ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   ` cfv 5377  (class class class)co 6085   Basecbs 13404   +g cplusg 13484   Grpcgrp 13858   -gcsg 13860  SubGrpcsubg 14023  NrmSGrpcnsg 14024  Cntzccntz 14140  Cntrccntr 14141

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1re 8274  ax-addrcl 8277

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-inn 9308  df-2 9366  df-ndx 13407  df-slot 13408  df-base 13410  df-plusg 13497  df-0g 13665  df-mgm 13729  df-sgrp 13770  df-mnd 13783  df-grp 13861  df-minusg 13862  df-sbg 13863  df-subg 14026  df-nsg 14027  df-cntz 14142  df-cntr 14143

This theorem is used by:  cntrnsg  14170