MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nlmlmod Structured version   Visualization version   GIF version

Theorem nlmlmod 23291
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 2824 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2824 . . . 4 (norm‘𝑊) = (norm‘𝑊)
3 eqid 2824 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 eqid 2824 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2824 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6 eqid 2824 . . . 4 (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊))
71, 2, 3, 4, 5, 6isnlm 23288 . . 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 501 . 2 (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing))
98simp2d 1140 1 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  wral 3133  cfv 6343  (class class class)co 7149   · cmul 10540  Basecbs 16483  Scalarcsca 16568   ·𝑠 cvsca 16569  LModclmod 19634  normcnm 23190  NrmGrpcngp 23191  NrmRingcnrg 23193  NrmModcnlm 23194
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5196
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7152  df-nlm 23200
This theorem is referenced by:  nlmdsdi  23294  nlmdsdir  23295  nlmmul0or  23296  nlmvscnlem2  23298  nlmvscn  23300  nlmtlm  23307  nvclmod  23311  isnvc2  23312  lssnlm  23314  ngpocelbl  23317  idnmhm  23367  0nmhm  23368  nmhmplusg  23370  nmhmcn  23732  cphlmod  23786  bnlmod  23954  nmmulg  31270
  Copyright terms: Public domain W3C validator