Theorem phllvec 20343

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 2825	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2825	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
3		eqid 2825	. . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
4		eqid 2825	. . 3 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		eqid 2825	. . 3 ⊢ (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
6		eqid 2825	. . 3 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
7	1, 2, 3, 4, 5, 6	isphl 20342	. 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 1179	1 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec)

Syntax hints:   → wi 4   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  ∀wral 3117   ↦ cmpt 4954  ‘cfv 6127  (class class class)co 6910  Basecbs 16229  *_𝑟cstv 16314  Scalarcsca 16315  ·_𝑖cip 16317  0_gc0g 16460  *-Ringcsr 19207   LMHom clmhm 19385  LVecclvec 19468  ringLModcrglmod 19537  PreHilcphl 20338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5015

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-iota 6090  df-fv 6135  df-ov 6913  df-phl 20340

This theorem is referenced by:  phllmod  20344  phlssphl  20373  obsne0  20439  obslbs  20444  cphlvec  23351  phclm  23407  ipcau2  23409  tcphcph  23412