Theorem nlmngp 24656

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 2737	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2737	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
3		eqid 2737	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		eqid 2737	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2737	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2737	. . . 4 ⊢ (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊))
7	1, 2, 3, 4, 5, 6	isnlm 24654	. . 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 496	. 2 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing))
9	8	simp1d 1143	1 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ‘cfv 6494  (class class class)co 7362   · cmul 11038  Basecbs 17174  Scalarcsca 17218   ·_𝑠 cvsca 17219  LModclmod 20850  normcnm 24555  NrmGrpcngp 24556  NrmRingcnrg 24558  NrmModcnlm 24559

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6450  df-fv 6502  df-ov 7365  df-nlm 24565

This theorem is referenced by:  nlmdsdi  24660  nlmdsdir  24661  nlmmul0or  24662  nlmvscnlem2  24664  nlmvscnlem1  24665  nlmvscn  24666  nlmtlm  24673  lssnlm  24680  ngpocelbl  24683  isnmhm2  24731  idnmhm  24733  0nmhm  24734  nmoleub2lem  25095  nmoleub2lem3  25096  nmoleub2lem2  25097  nmoleub3  25100  nmhmcn  25101  ncvsm1  25135  ncvsdif  25136  ncvspi  25137  ncvs1  25138  ncvspds  25142  cphngp  25154  ipcnlem2  25225  ipcnlem1  25226  csscld  25230  bnngp  25323  cssbn  25356