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Theorem nlmlmod 24415
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 2730 . . . 4 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
2 eqid 2730 . . . 4 (normβ€˜π‘Š) = (normβ€˜π‘Š)
3 eqid 2730 . . . 4 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
4 eqid 2730 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
5 eqid 2730 . . . 4 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
6 eqid 2730 . . . 4 (normβ€˜(Scalarβ€˜π‘Š)) = (normβ€˜(Scalarβ€˜π‘Š))
71, 2, 3, 4, 5, 6isnlm 24412 . . 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 496 . 2 (π‘Š ∈ NrmMod β†’ (π‘Š ∈ NrmGrp ∧ π‘Š ∈ LMod ∧ (Scalarβ€˜π‘Š) ∈ NrmRing))
98simp2d 1141 1 (π‘Š ∈ NrmMod β†’ π‘Š ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  β€˜cfv 6542  (class class class)co 7411   Β· cmul 11117  Basecbs 17148  Scalarcsca 17204   ·𝑠 cvsca 17205  LModclmod 20614  normcnm 24305  NrmGrpcngp 24306  NrmRingcnrg 24308  NrmModcnlm 24309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-ov 7414  df-nlm 24315
This theorem is referenced by:  nlmdsdi  24418  nlmdsdir  24419  nlmmul0or  24420  nlmvscnlem2  24422  nlmvscn  24424  nlmtlm  24431  nvclmod  24435  isnvc2  24436  lssnlm  24438  ngpocelbl  24441  idnmhm  24491  0nmhm  24492  nmhmplusg  24494  nmhmcn  24867  cphlmod  24922  bnlmod  25091  nmmulg  33246
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