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 e. PreHil -> W e. 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 e. PreHil <-> ( W e. LVec /\ (Scalar ` W ) e. *Ring /\ A. x e. ( Base ` W ) ( ( y e. ( Base ` W ) |-> ( y ( .i ` W ) x ) ) e. ( W LMHom (ringLMod ` (Scalar ` W ) ) ) /\ ( ( x ( .i ` W ) x ) = ( 0g ` (Scalar ` W ) ) -> x = ( 0g ` W ) ) /\ A. y e. ( Base ` W ) ( ( *r ` (Scalar ` W ) ) ` ( x ( .i ` W ) y ) ) = ( y ( .i ` W ) x ) ) ) )
8	7	simp1bi 1076	1 |- ( W e. PreHil -> W e. LVec )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Basecbs 15857   *rcstv 15943  Scalarcsca 15944   .icip 15946   0gc0g 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