Theorem nlmngp 24614

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 2735	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2735	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
3		eqid 2735	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		eqid 2735	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2735	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2735	. . . 4 ⊢ (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊))
7	1, 2, 3, 4, 5, 6	isnlm 24612	. . 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 1540   ∈ wcel 2108  ∀wral 3051  ‘cfv 6530  (class class class)co 7403   · cmul 11132  Basecbs 17226  Scalarcsca 17272   ·_𝑠 cvsca 17273  LModclmod 20815  normcnm 24513  NrmGrpcngp 24514  NrmRingcnrg 24516  NrmModcnlm 24517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-nul 5276

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6483  df-fv 6538  df-ov 7406  df-nlm 24523

This theorem is referenced by:  nlmdsdi  24618  nlmdsdir  24619  nlmmul0or  24620  nlmvscnlem2  24622  nlmvscnlem1  24623  nlmvscn  24624  nlmtlm  24631  lssnlm  24638  ngpocelbl  24641  isnmhm2  24689  idnmhm  24691  0nmhm  24692  nmoleub2lem  25063  nmoleub2lem3  25064  nmoleub2lem2  25065  nmoleub3  25068  nmhmcn  25069  ncvsm1  25104  ncvsdif  25105  ncvspi  25106  ncvs1  25107  ncvspds  25111  cphngp  25123  ipcnlem2  25194  ipcnlem1  25195  csscld  25199  bnngp  25292  cssbn  25325