Theorem nlmngp 23280

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 2821	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2821	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
3		eqid 2821	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		eqid 2821	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2821	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2821	. . . 4 ⊢ (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊))
7	1, 2, 3, 4, 5, 6	isnlm 23278	. . 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 1138	1 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ‘cfv 6350  (class class class)co 7150   · cmul 10536  Basecbs 16477  Scalarcsca 16562   ·_𝑠 cvsca 16563  LModclmod 19628  normcnm 23180  NrmGrpcngp 23181  NrmRingcnrg 23183  NrmModcnlm 23184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-nul 5203

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-iota 6309  df-fv 6358  df-ov 7153  df-nlm 23190

This theorem is referenced by:  nlmdsdi  23284  nlmdsdir  23285  nlmmul0or  23286  nlmvscnlem2  23288  nlmvscnlem1  23289  nlmvscn  23290  nlmtlm  23297  lssnlm  23304  ngpocelbl  23307  isnmhm2  23355  idnmhm  23357  0nmhm  23358  nmoleub2lem  23712  nmoleub2lem3  23713  nmoleub2lem2  23714  nmoleub3  23717  nmhmcn  23718  ncvsm1  23752  ncvsdif  23753  ncvspi  23754  ncvs1  23755  ncvspds  23759  cphngp  23771  ipcnlem2  23841  ipcnlem1  23842  csscld  23846  bnngp  23939  cssbn  23972