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Theorem cphlmod 22727
Description: A complex pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)
Assertion
Ref Expression
cphlmod (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod)

Proof of Theorem cphlmod
StepHypRef Expression
1 cphnlm 22725 . 2 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
2 nlmlmod 22240 . 2 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
31, 2syl 17 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1976  LModclmod 18635  NrmModcnlm 22143  ℂPreHilccph 22719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-nul 4712
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-xp 5034  df-cnv 5036  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fv 5798  df-ov 6530  df-nlm 22149  df-cph 22721
This theorem is referenced by:  cphclm  22742  cph2ass  22766  cphtchnm  22782  nmparlem  22791  cphipval2  22793  4cphipval2  22794  cphipval  22795  minveclem1  22948  minveclem2  22950  minveclem4  22956  minveclem6  22958  pjthlem1  22961  pjthlem2  22962
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