Theorem phllvec 21647

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 2737	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2737	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
3		eqid 2737	. . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
4		eqid 2737	. . 3 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		eqid 2737	. . 3 ⊢ (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
6		eqid 2737	. . 3 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
7	1, 2, 3, 4, 5, 6	isphl 21646	. 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 1146	1 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061   ↦ cmpt 5225  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  *_𝑟cstv 17299  Scalarcsca 17300  ·_𝑖cip 17302  0_gc0g 17484  *-Ringcsr 20839   LMHom clmhm 21018  LVecclvec 21101  ringLModcrglmod 21171  PreHilcphl 21642

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-iota 6514  df-fv 6569  df-ov 7434  df-phl 21644

This theorem is referenced by:  phllmod  21648  phlssphl  21677  obsne0  21745  obslbs  21750  cphlvec  25209  phclm  25266  ipcau2  25268  tcphcph  25271