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Theorem nlmlmod 24195
Description: A normed module is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
nlmlmod (π‘Š ∈ NrmMod β†’ π‘Š ∈ LMod)

Proof of Theorem nlmlmod
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . 4 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
2 eqid 2733 . . . 4 (normβ€˜π‘Š) = (normβ€˜π‘Š)
3 eqid 2733 . . . 4 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
4 eqid 2733 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
5 eqid 2733 . . . 4 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
6 eqid 2733 . . . 4 (normβ€˜(Scalarβ€˜π‘Š)) = (normβ€˜(Scalarβ€˜π‘Š))
71, 2, 3, 4, 5, 6isnlm 24192 . . 3 (π‘Š ∈ NrmMod ↔ ((π‘Š ∈ NrmGrp ∧ π‘Š ∈ LMod ∧ (Scalarβ€˜π‘Š) ∈ NrmRing) ∧ βˆ€π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘¦ ∈ (Baseβ€˜π‘Š)((normβ€˜π‘Š)β€˜(π‘₯( ·𝑠 β€˜π‘Š)𝑦)) = (((normβ€˜(Scalarβ€˜π‘Š))β€˜π‘₯) Β· ((normβ€˜π‘Š)β€˜π‘¦))))
87simplbi 499 . 2 (π‘Š ∈ NrmMod β†’ (π‘Š ∈ NrmGrp ∧ π‘Š ∈ LMod ∧ (Scalarβ€˜π‘Š) ∈ NrmRing))
98simp2d 1144 1 (π‘Š ∈ NrmMod β†’ π‘Š ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  β€˜cfv 6544  (class class class)co 7409   Β· cmul 11115  Basecbs 17144  Scalarcsca 17200   ·𝑠 cvsca 17201  LModclmod 20471  normcnm 24085  NrmGrpcngp 24086  NrmRingcnrg 24088  NrmModcnlm 24089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-nlm 24095
This theorem is referenced by:  nlmdsdi  24198  nlmdsdir  24199  nlmmul0or  24200  nlmvscnlem2  24202  nlmvscn  24204  nlmtlm  24211  nvclmod  24215  isnvc2  24216  lssnlm  24218  ngpocelbl  24221  idnmhm  24271  0nmhm  24272  nmhmplusg  24274  nmhmcn  24636  cphlmod  24691  bnlmod  24860  nmmulg  32948
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