Theorem nlmngp 22700

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 2770	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2770	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
3		eqid 2770	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		eqid 2770	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2770	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2770	. . . 4 ⊢ (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊))
7	1, 2, 3, 4, 5, 6	isnlm 22698	. . 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 479	. 2 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing))
9	8	simp1d 1135	1 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)

Syntax hints:   → wi 4   ∧ w3a 1070   = wceq 1630   ∈ wcel 2144  ∀wral 3060  ‘cfv 6031  (class class class)co 6792   · cmul 10142  Basecbs 16063  Scalarcsca 16151   ·_𝑠 cvsca 16152  LModclmod 19072  normcnm 22600  NrmGrpcngp 22601  NrmRingcnrg 22603  NrmModcnlm 22604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-nul 4920

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039  df-ov 6795  df-nlm 22610

This theorem is referenced by:  nlmdsdi  22704  nlmdsdir  22705  nlmmul0or  22706  nlmvscnlem2  22708  nlmvscnlem1  22709  nlmvscn  22710  nlmtlm  22717  lssnlm  22724  ngpocelbl  22727  isnmhm2  22775  idnmhm  22777  0nmhm  22778  nmoleub2lem  23132  nmoleub2lem3  23133  nmoleub2lem2  23134  nmoleub3  23137  nmhmcn  23138  ncvsm1  23172  ncvsdif  23173  ncvspi  23174  ncvs1  23175  ncvspds  23179  cphngp  23191  ipcnlem2  23261  ipcnlem1  23262  csscld  23266  bnngp  23357