Theorem phllvec 19974

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 2622	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2622	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
3		eqid 2622	. . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
4		eqid 2622	. . 3 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		eqid 2622	. . 3 ⊢ (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
6		eqid 2622	. . 3 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
7	1, 2, 3, 4, 5, 6	isphl 19973	. 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 1076	1 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec)

Syntax hints:   → wi 4   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  *_𝑟cstv 15943  Scalarcsca 15944  ·_𝑖cip 15946  0_gc0g 16100  *-Ringcsr 18844   LMHom clmhm 19019  LVecclvec 19102  ringLModcrglmod 19169  PreHilcphl 19969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-iota 5851  df-fv 5896  df-ov 6653  df-phl 19971

This theorem is referenced by:  phllmod  19975  obsne0  20069  obslbs  20074  cphlvec  22975  tchclm  23031  ipcau2  23033  tchcph  23036