Theorem nlmngp 24719

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 2740	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2740	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
3		eqid 2740	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		eqid 2740	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2740	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2740	. . . 4 ⊢ (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊))
7	1, 2, 3, 4, 5, 6	isnlm 24717	. . 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 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ‘cfv 6573  (class class class)co 7448   · cmul 11189  Basecbs 17258  Scalarcsca 17314   ·_𝑠 cvsca 17315  LModclmod 20880  normcnm 24610  NrmGrpcngp 24611  NrmRingcnrg 24613  NrmModcnlm 24614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451  df-nlm 24620

This theorem is referenced by:  nlmdsdi  24723  nlmdsdir  24724  nlmmul0or  24725  nlmvscnlem2  24727  nlmvscnlem1  24728  nlmvscn  24729  nlmtlm  24736  lssnlm  24743  ngpocelbl  24746  isnmhm2  24794  idnmhm  24796  0nmhm  24797  nmoleub2lem  25166  nmoleub2lem3  25167  nmoleub2lem2  25168  nmoleub3  25171  nmhmcn  25172  ncvsm1  25207  ncvsdif  25208  ncvspi  25209  ncvs1  25210  ncvspds  25214  cphngp  25226  ipcnlem2  25297  ipcnlem1  25298  csscld  25302  bnngp  25395  cssbn  25428