Theorem nlmngp 24619

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 2734	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2734	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
3		eqid 2734	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		eqid 2734	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2734	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2734	. . . 4 ⊢ (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊))
7	1, 2, 3, 4, 5, 6	isnlm 24617	. . 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 497	. 2 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing))
9	8	simp1d 1142	1 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ‘cfv 6490  (class class class)co 7356   · cmul 11029  Basecbs 17134  Scalarcsca 17178   ·_𝑠 cvsca 17179  LModclmod 20809  normcnm 24518  NrmGrpcngp 24519  NrmRingcnrg 24521  NrmModcnlm 24522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-nul 5249

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-iota 6446  df-fv 6498  df-ov 7359  df-nlm 24528

This theorem is referenced by:  nlmdsdi  24623  nlmdsdir  24624  nlmmul0or  24625  nlmvscnlem2  24627  nlmvscnlem1  24628  nlmvscn  24629  nlmtlm  24636  lssnlm  24643  ngpocelbl  24646  isnmhm2  24694  idnmhm  24696  0nmhm  24697  nmoleub2lem  25068  nmoleub2lem3  25069  nmoleub2lem2  25070  nmoleub3  25073  nmhmcn  25074  ncvsm1  25108  ncvsdif  25109  ncvspi  25110  ncvs1  25111  ncvspds  25115  cphngp  25127  ipcnlem2  25198  ipcnlem1  25199  csscld  25203  bnngp  25296  cssbn  25329