Theorem cntrsubgnsg 19557

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 488	. 2 ⊢ ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) → 𝑋 ∈ (SubGrp‘𝑀))
2		simplr 781	. . . . . . . . 9 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑋 ⊆ 𝑍)
3		simprr 785	. . . . . . . . 9 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
4	2, 3	sseldd 3909	. . . . . . . 8 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑍)
5		eqid 2760	. . . . . . . . . 10 ⊢ (Base‘𝑀) = (Base‘𝑀)
6		eqid 2760	. . . . . . . . . 10 ⊢ (Cntz‘𝑀) = (Cntz‘𝑀)
7	5, 6	cntrval 19533	. . . . . . . . 9 ⊢ ((Cntz‘𝑀)‘(Base‘𝑀)) = (Cntr‘𝑀)
8		cntrnsg.z	. . . . . . . . 9 ⊢ 𝑍 = (Cntr‘𝑀)
9	7, 8	eqtr4i 2786	. . . . . . . 8 ⊢ ((Cntz‘𝑀)‘(Base‘𝑀)) = 𝑍
10	4, 9	eleqtrrdi 2871	. . . . . . 7 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ ((Cntz‘𝑀)‘(Base‘𝑀)))
11		simprl 783	. . . . . . 7 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ (Base‘𝑀))
12		eqid 2760	. . . . . . . 8 ⊢ (+g‘𝑀) = (+g‘𝑀)
13	12, 6	cntzi 19543	. . . . . . 7 ⊢ ((𝑦 ∈ ((Cntz‘𝑀)‘(Base‘𝑀)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
14	10, 11, 13	syl2anc 596	. . . . . 6 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
15	14	oveq1d 7435	. . . . 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 19341	. . . . . . 7 ⊢ (𝑋 ∈ (SubGrp‘𝑀) → 𝑀 ∈ Grp)
17	16	ad2antrr 739	. . . . . 6 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑀 ∈ Grp)
18	5	subgss 19337	. . . . . . . 8 ⊢ (𝑋 ∈ (SubGrp‘𝑀) → 𝑋 ⊆ (Base‘𝑀))
19	18	ad2antrr 739	. . . . . . 7 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑋 ⊆ (Base‘𝑀))
20	19, 3	sseldd 3909	. . . . . 6 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ (Base‘𝑀))
21		eqid 2760	. . . . . . 7 ⊢ (-g‘𝑀) = (-g‘𝑀)
22	5, 12, 21	grppncan 19241	. . . . . 6 ⊢ ((𝑀 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → ((𝑦(+g‘𝑀)𝑥)(-g‘𝑀)𝑥) = 𝑦)
23	17, 20, 11, 22	syl3anc 1398	. . . . 5 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → ((𝑦(+g‘𝑀)𝑥)(-g‘𝑀)𝑥) = 𝑦)
24	15, 23	eqtr3d 2797	. . . 4 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → ((𝑥(+g‘𝑀)𝑦)(-g‘𝑀)𝑥) = 𝑦)
25	24, 3	eqeltrd 2860	. . 3 ⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → ((𝑥(+g‘𝑀)𝑦)(-g‘𝑀)𝑥) ∈ 𝑋)
26	25	ralrimivva 3205	. 2 ⊢ ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ 𝑋 ((𝑥(+g‘𝑀)𝑦)(-g‘𝑀)𝑥) ∈ 𝑋)
27	5, 12, 21	isnsg3 19370	. 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 595	1 ⊢ ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) → 𝑋 ∈ (NrmSGrp‘𝑀))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  +_gcplusg 17428  Grpcgrp 19144  -_gcsg 19146  SubGrpcsubg 19330  NrmSGrpcnsg 19331  Cntzccntz 19529  Cntrccntr 19530

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-0g 17612  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-grp 19147  df-minusg 19148  df-sbg 19149  df-subg 19333  df-nsg 19334  df-cntz 19531  df-cntr 19532

This theorem is used by:  cntrnsg  19558