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Theorem nlmlmod 22529
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 2651 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2651 . . . 4 (norm‘𝑊) = (norm‘𝑊)
3 eqid 2651 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 eqid 2651 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2651 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6 eqid 2651 . . . 4 (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊))
71, 2, 3, 4, 5, 6isnlm 22526 . . 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 475 . 2 (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing))
98simp2d 1094 1 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1054   = wceq 1523  wcel 2030  wral 2941  cfv 5926  (class class class)co 6690   · cmul 9979  Basecbs 15904  Scalarcsca 15991   ·𝑠 cvsca 15992  LModclmod 18911  normcnm 22428  NrmGrpcngp 22429  NrmRingcnrg 22431  NrmModcnlm 22432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-nlm 22438
This theorem is referenced by:  nlmdsdi  22532  nlmdsdir  22533  nlmmul0or  22534  nlmvscnlem2  22536  nlmvscn  22538  nlmtlm  22545  nvclmod  22549  isnvc2  22550  lssnlm  22552  ngpocelbl  22555  idnmhm  22605  0nmhm  22606  nmhmplusg  22608  nmhmcn  22966  cphlmod  23020  bnlmod  23186  nmmulg  30140
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