Theorem nlmngp 24805

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 2769	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2769	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
3		eqid 2769	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		eqid 2769	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2769	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2769	. . . 4 ⊢ (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊))
7	1, 2, 3, 4, 5, 6	isnlm 24803	. . 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 501	. 2 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing))
9	8	simp1d 1158	1 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ‘cfv 6539  (class class class)co 7413   · cmul 11107  Basecbs 17271  Scalarcsca 17315   ·_𝑠 cvsca 17316  LModclmod 20961  normcnm 24704  NrmGrpcngp 24705  NrmRingcnrg 24707  NrmModcnlm 24708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5273

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6495  df-fv 6547  df-ov 7416  df-nlm 24714

This theorem is referenced by:  nlmdsdi  24809  nlmdsdir  24810  nlmmul0or  24811  nlmvscnlem2  24813  nlmvscnlem1  24814  nlmvscn  24815  nlmtlm  24822  lssnlm  24829  ngpocelbl  24832  isnmhm2  24880  idnmhm  24882  0nmhm  24883  nmoleub2lem  25244  nmoleub2lem3  25245  nmoleub2lem2  25246  nmoleub3  25249  nmhmcn  25250  ncvsm1  25284  ncvsdif  25285  ncvspi  25286  ncvs1  25287  ncvspds  25291  cphngp  25303  ipcnlem2  25374  ipcnlem1  25375  csscld  25379  bnngp  25472  cssbn  25505