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Theorem nlmngp 22421
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.
StepHypRef Expression
1 eqid 2621 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2621 . . . 4 (norm‘𝑊) = (norm‘𝑊)
3 eqid 2621 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 eqid 2621 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2621 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6 eqid 2621 . . . 4 (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊))
71, 2, 3, 4, 5, 6isnlm 22419 . . 3 (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ (Base‘𝑊)((norm‘𝑊)‘(𝑥( ·𝑠𝑊)𝑦)) = (((norm‘(Scalar‘𝑊))‘𝑥) · ((norm‘𝑊)‘𝑦))))
87simplbi 476 . 2 (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing))
98simp1d 1071 1 (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987  wral 2908  cfv 5857  (class class class)co 6615   · cmul 9901  Basecbs 15800  Scalarcsca 15884   ·𝑠 cvsca 15885  LModclmod 18803  normcnm 22321  NrmGrpcngp 22322  NrmRingcnrg 22324  NrmModcnlm 22325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4759
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-ov 6618  df-nlm 22331
This theorem is referenced by:  nlmdsdi  22425  nlmdsdir  22426  nlmmul0or  22427  nlmvscnlem2  22429  nlmvscnlem1  22430  nlmvscn  22431  nlmtlm  22438  lssnlm  22445  ngpocelbl  22448  isnmhm2  22496  idnmhm  22498  0nmhm  22499  nmoleub2lem  22854  nmoleub2lem3  22855  nmoleub2lem2  22856  nmoleub3  22859  nmhmcn  22860  ncvsm1  22894  ncvsdif  22895  ncvspi  22896  ncvs1  22897  ncvspds  22901  cphngp  22913  ipcnlem2  22983  ipcnlem1  22984  csscld  22988  bnngp  23079
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