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Theorem phllvec 21056
Description: A pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
Assertion
Ref Expression
phllvec (π‘Š ∈ PreHil β†’ π‘Š ∈ LVec)

Proof of Theorem phllvec
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . 3 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
2 eqid 2733 . . 3 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
3 eqid 2733 . . 3 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
4 eqid 2733 . . 3 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
5 eqid 2733 . . 3 (*π‘Ÿβ€˜(Scalarβ€˜π‘Š)) = (*π‘Ÿβ€˜(Scalarβ€˜π‘Š))
6 eqid 2733 . . 3 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
71, 2, 3, 4, 5, 6isphl 21055 . 2 (π‘Š ∈ PreHil ↔ (π‘Š ∈ LVec ∧ (Scalarβ€˜π‘Š) ∈ *-Ring ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘Š)((𝑦 ∈ (Baseβ€˜π‘Š) ↦ (𝑦(Β·π‘–β€˜π‘Š)π‘₯)) ∈ (π‘Š LMHom (ringLModβ€˜(Scalarβ€˜π‘Š))) ∧ ((π‘₯(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)) β†’ π‘₯ = (0gβ€˜π‘Š)) ∧ βˆ€π‘¦ ∈ (Baseβ€˜π‘Š)((*π‘Ÿβ€˜(Scalarβ€˜π‘Š))β€˜(π‘₯(Β·π‘–β€˜π‘Š)𝑦)) = (𝑦(Β·π‘–β€˜π‘Š)π‘₯))))
87simp1bi 1146 1 (π‘Š ∈ PreHil β†’ π‘Š ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   ↦ cmpt 5192  β€˜cfv 6500  (class class class)co 7361  Basecbs 17091  *π‘Ÿcstv 17143  Scalarcsca 17144  Β·π‘–cip 17146  0gc0g 17329  *-Ringcsr 20346   LMHom clmhm 20524  LVecclvec 20607  ringLModcrglmod 20675  PreHilcphl 21051
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 5267
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 2941  df-ral 3062  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-iota 6452  df-fv 6508  df-ov 7364  df-phl 21053
This theorem is referenced by:  phllmod  21057  phlssphl  21086  obsne0  21154  obslbs  21159  cphlvec  24562  phclm  24619  ipcau2  24621  tcphcph  24624
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