Theorem phllvec 21538

Description: A pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)

Assertion

Ref	Expression
phllvec	⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec)

Proof of Theorem phllvec

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2729	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
3		eqid 2729	. . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
4		eqid 2729	. . 3 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		eqid 2729	. . 3 ⊢ (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
6		eqid 2729	. . 3 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
7	1, 2, 3, 4, 5, 6	isphl 21537	. 2 ⊢ (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ (Scalar‘𝑊) ∈ *-Ring ∧ ∀𝑥 ∈ (Base‘𝑊)((𝑦 ∈ (Base‘𝑊) ↦ (𝑦(·𝑖‘𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))) ∧ ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g‘𝑊)) ∧ ∀𝑦 ∈ (Base‘𝑊)((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥))))
8	7	simp1bi 1145	1 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ↦ cmpt 5188  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  *_𝑟cstv 17222  Scalarcsca 17223  ·_𝑖cip 17225  0_gc0g 17402  *-Ringcsr 20747   LMHom clmhm 20926  LVecclvec 21009  ringLModcrglmod 21079  PreHilcphl 21533

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-iota 6464  df-fv 6519  df-ov 7390  df-phl 21535

This theorem is referenced by:  phllmod  21539  phlssphl  21568  obsne0  21634  obslbs  21639  cphlvec  25075  phclm  25132  ipcau2  25134  tcphcph  25137