Theorem nlmngp 24996

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 2760	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2760	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
3		eqid 2760	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		eqid 2760	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2760	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2760	. . . 4 ⊢ (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊))
7	1, 2, 3, 4, 5, 6	isnlm 24994	. . 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 502	. 2 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing))
9	8	simp1d 1160	1 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ‘cfv 6538  (class class class)co 7420   · cmul 11205  Basecbs 17387  Scalarcsca 17431   ·_𝑠 cvsca 17432  LModclmod 21135  normcnm 24895  NrmGrpcngp 24896  NrmRingcnrg 24898  NrmModcnlm 24899

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-ov 7423  df-nlm 24905

This theorem is used by:  nlmdsdi  25000  nlmdsdir  25001  nlmmul0or  25002  nlmvscnlem2  25004  nlmvscnlem1  25005  nlmvscn  25006  nlmtlm  25013  lssnlm  25020  ngpocelbl  25023  isnmhm2  25071  idnmhm  25073  0nmhm  25074  nmoleub2lem  25435  nmoleub2lem3  25436  nmoleub2lem2  25437  nmoleub3  25440  nmhmcn  25441  ncvsm1  25475  ncvsdif  25476  ncvspi  25477  ncvs1  25478  ncvspds  25482  cphngp  25494  ipcnlem2  25565  ipcnlem1  25566  csscld  25570  bnngp  25663  cssbn  25696