Theorem nlmngp 24713

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 2734	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2734	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
3		eqid 2734	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		eqid 2734	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2734	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2734	. . . 4 ⊢ (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊))
7	1, 2, 3, 4, 5, 6	isnlm 24711	. . 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 1141	1 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105  ∀wral 3058  ‘cfv 6562  (class class class)co 7430   · cmul 11157  Basecbs 17244  Scalarcsca 17300   ·_𝑠 cvsca 17301  LModclmod 20874  normcnm 24604  NrmGrpcngp 24605  NrmRingcnrg 24607  NrmModcnlm 24608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-nul 5311

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-ov 7433  df-nlm 24614

This theorem is referenced by:  nlmdsdi  24717  nlmdsdir  24718  nlmmul0or  24719  nlmvscnlem2  24721  nlmvscnlem1  24722  nlmvscn  24723  nlmtlm  24730  lssnlm  24737  ngpocelbl  24740  isnmhm2  24788  idnmhm  24790  0nmhm  24791  nmoleub2lem  25160  nmoleub2lem3  25161  nmoleub2lem2  25162  nmoleub3  25165  nmhmcn  25166  ncvsm1  25201  ncvsdif  25202  ncvspi  25203  ncvs1  25204  ncvspds  25208  cphngp  25220  ipcnlem2  25291  ipcnlem1  25292  csscld  25296  bnngp  25389  cssbn  25422