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Theorem nlmlmod 22852
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 2825 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2825 . . . 4 (norm‘𝑊) = (norm‘𝑊)
3 eqid 2825 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 eqid 2825 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2825 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6 eqid 2825 . . . 4 (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊))
71, 2, 3, 4, 5, 6isnlm 22849 . . 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 493 . 2 (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing))
98simp2d 1179 1 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1113   = wceq 1658  wcel 2166  wral 3117  cfv 6123  (class class class)co 6905   · cmul 10257  Basecbs 16222  Scalarcsca 16308   ·𝑠 cvsca 16309  LModclmod 19219  normcnm 22751  NrmGrpcngp 22752  NrmRingcnrg 22754  NrmModcnlm 22755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-nul 5013
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131  df-ov 6908  df-nlm 22761
This theorem is referenced by:  nlmdsdi  22855  nlmdsdir  22856  nlmmul0or  22857  nlmvscnlem2  22859  nlmvscn  22861  nlmtlm  22868  nvclmod  22872  isnvc2  22873  lssnlm  22875  ngpocelbl  22878  idnmhm  22928  0nmhm  22929  nmhmplusg  22931  nmhmcn  23289  cphlmod  23343  bnlmod  23511  nmmulg  30557
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