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Theorem nlmlmod 22387
Description: A normed module is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
nlmlmod (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)

Proof of Theorem nlmlmod
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2626 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2626 . . . 4 (norm‘𝑊) = (norm‘𝑊)
3 eqid 2626 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 eqid 2626 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2626 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6 eqid 2626 . . . 4 (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊))
71, 2, 3, 4, 5, 6isnlm 22384 . . 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))
98simp2d 1072 1 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1992  wral 2912  cfv 5850  (class class class)co 6605   · cmul 9886  Basecbs 15776  Scalarcsca 15860   ·𝑠 cvsca 15861  LModclmod 18779  normcnm 22286  NrmGrpcngp 22287  NrmRingcnrg 22289  NrmModcnlm 22290
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-nul 4754
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 1883  df-eu 2478  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5813  df-fv 5858  df-ov 6608  df-nlm 22296
This theorem is referenced by:  nlmdsdi  22390  nlmdsdir  22391  nlmmul0or  22392  nlmvscnlem2  22394  nlmvscn  22396  nlmtlm  22403  nvclmod  22407  isnvc2  22408  lssnlm  22410  ngpocelbl  22413  idnmhm  22463  0nmhm  22464  nmhmplusg  22466  nmhmcn  22823  cphlmod  22877  bnlmod  23043  nmmulg  29786
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