Theorem nlmngp 22858

Description: A normed module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)

Assertion

Ref	Expression
nlmngp	⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)

Proof of Theorem nlmngp

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2825	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2825	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
3		eqid 2825	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		eqid 2825	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2825	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2825	. . . 4 ⊢ (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊))
7	1, 2, 3, 4, 5, 6	isnlm 22856	. . 3 ⊢ (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ (Base‘𝑊)((norm‘𝑊)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (((norm‘(Scalar‘𝑊))‘𝑥) · ((norm‘𝑊)‘𝑦))))
8	7	simplbi 493	. 2 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing))
9	8	simp1d 1176	1 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)

Syntax hints:   → wi 4   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  ∀wral 3117  ‘cfv 6127  (class class class)co 6910   · cmul 10264  Basecbs 16229  Scalarcsca 16315   ·_𝑠 cvsca 16316  LModclmod 19226  normcnm 22758  NrmGrpcngp 22759  NrmRingcnrg 22761  NrmModcnlm 22762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5015

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-iota 6090  df-fv 6135  df-ov 6913  df-nlm 22768

This theorem is referenced by:  nlmdsdi  22862  nlmdsdir  22863  nlmmul0or  22864  nlmvscnlem2  22866  nlmvscnlem1  22867  nlmvscn  22868  nlmtlm  22875  lssnlm  22882  ngpocelbl  22885  isnmhm2  22933  idnmhm  22935  0nmhm  22936  nmoleub2lem  23290  nmoleub2lem3  23291  nmoleub2lem2  23292  nmoleub3  23295  nmhmcn  23296  ncvsm1  23330  ncvsdif  23331  ncvspi  23332  ncvs1  23333  ncvspds  23337  cphngp  23349  ipcnlem2  23419  ipcnlem1  23420  csscld  23424  bnngp  23517  cssbn  23550