Theorem nlmngp 24739

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 2764	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2764	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
3		eqid 2764	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		eqid 2764	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2764	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2764	. . . 4 ⊢ (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊))
7	1, 2, 3, 4, 5, 6	isnlm 24737	. . 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 500	. 2 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing))
9	8	simp1d 1156	1 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)

Syntax hints:   → wi 4   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ‘cfv 6523  (class class class)co 7398   · cmul 11080  Basecbs 17247  Scalarcsca 17291   ·_𝑠 cvsca 17292  LModclmod 20929  normcnm 24638  NrmGrpcngp 24639  NrmRingcnrg 24641  NrmModcnlm 24642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-nul 5258

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-iota 6479  df-fv 6531  df-ov 7401  df-nlm 24648

This theorem is referenced by:  nlmdsdi  24743  nlmdsdir  24744  nlmmul0or  24745  nlmvscnlem2  24747  nlmvscnlem1  24748  nlmvscn  24749  nlmtlm  24756  lssnlm  24763  ngpocelbl  24766  isnmhm2  24814  idnmhm  24816  0nmhm  24817  nmoleub2lem  25178  nmoleub2lem3  25179  nmoleub2lem2  25180  nmoleub3  25183  nmhmcn  25184  ncvsm1  25218  ncvsdif  25219  ncvspi  25220  ncvs1  25221  ncvspds  25225  cphngp  25237  ipcnlem2  25308  ipcnlem1  25309  csscld  25313  bnngp  25406  cssbn  25439