Theorem nlmngp 24193

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 2732	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2732	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
3		eqid 2732	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		eqid 2732	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2732	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2732	. . . 4 ⊢ (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊))
7	1, 2, 3, 4, 5, 6	isnlm 24191	. . 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 498	. 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 1541   ∈ wcel 2106  ∀wral 3061  ‘cfv 6543  (class class class)co 7408   · cmul 11114  Basecbs 17143  Scalarcsca 17199   ·_𝑠 cvsca 17200  LModclmod 20470  normcnm 24084  NrmGrpcngp 24085  NrmRingcnrg 24087  NrmModcnlm 24088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411  df-nlm 24094

This theorem is referenced by:  nlmdsdi  24197  nlmdsdir  24198  nlmmul0or  24199  nlmvscnlem2  24201  nlmvscnlem1  24202  nlmvscn  24203  nlmtlm  24210  lssnlm  24217  ngpocelbl  24220  isnmhm2  24268  idnmhm  24270  0nmhm  24271  nmoleub2lem  24629  nmoleub2lem3  24630  nmoleub2lem2  24631  nmoleub3  24634  nmhmcn  24635  ncvsm1  24670  ncvsdif  24671  ncvspi  24672  ncvs1  24673  ncvspds  24677  cphngp  24689  ipcnlem2  24760  ipcnlem1  24761  csscld  24765  bnngp  24858  cssbn  24891