Theorem phllmhm 19977

Description: The inner product of a pre-Hilbert space is linear in its left argument. (Contributed by Mario Carneiro, 7-Oct-2015.)

Hypotheses

Ref	Expression
phlsrng.f	|- F = (Scalar ` W )
phllmhm.h	|- ., = ( .i ` W )
phllmhm.v	|- V = ( Base ` W )
phllmhm.g	|- G = ( x e. V |-> ( x ., A ) )

Assertion

Ref	Expression
phllmhm	|- ( ( W e. PreHil /\ A e. V ) -> G e. ( W LMHom (ringLMod ` F ) ) )

Distinct variable groups:   x, A   x, .,   x, V   x, W

Allowed substitution hints:   F( x)   G( x)

Proof of Theorem phllmhm

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		phllmhm.v	. . . . 5 |- V = ( Base ` W )
2		phlsrng.f	. . . . 5 |- F = (Scalar ` W )
3		phllmhm.h	. . . . 5 |- ., = ( .i ` W )
4		eqid 2622	. . . . 5 |- ( 0g ` W ) = ( 0g ` W )
5		eqid 2622	. . . . 5 |- ( *r ` F ) = ( *r ` F )
6		eqid 2622	. . . . 5 |- ( 0g ` F ) = ( 0g ` F )
7	1, 2, 3, 4, 5, 6	isphl 19973	. . . 4 |- ( W e. PreHil <-> ( W e. LVec /\ F e. *Ring /\ A. y e. V ( ( x e. V |-> ( x ., y ) ) e. ( W LMHom (ringLMod ` F ) ) /\ ( ( y ., y ) = ( 0g ` F ) -> y = ( 0g ` W ) ) /\ A. x e. V ( ( *r ` F ) ` ( y ., x ) ) = ( x ., y ) ) ) )
8	7	simp3bi 1078	. . 3 |- ( W e. PreHil -> A. y e. V ( ( x e. V |-> ( x ., y ) ) e. ( W LMHom (ringLMod ` F ) ) /\ ( ( y ., y ) = ( 0g ` F ) -> y = ( 0g ` W ) ) /\ A. x e. V ( ( *r ` F ) ` ( y ., x ) ) = ( x ., y ) ) )
9		simp1 1061	. . . 4 |- ( ( ( x e. V |-> ( x ., y ) ) e. ( W LMHom (ringLMod ` F ) ) /\ ( ( y ., y ) = ( 0g ` F ) -> y = ( 0g ` W ) ) /\ A. x e. V ( ( *r ` F ) ` ( y ., x ) ) = ( x ., y ) ) -> ( x e. V |-> ( x ., y ) ) e. ( W LMHom (ringLMod ` F ) ) )
10	9	ralimi 2952	. . 3 |- ( A. y e. V ( ( x e. V |-> ( x ., y ) ) e. ( W LMHom (ringLMod ` F ) ) /\ ( ( y ., y ) = ( 0g ` F ) -> y = ( 0g ` W ) ) /\ A. x e. V ( ( *r ` F ) ` ( y ., x ) ) = ( x ., y ) ) -> A. y e. V ( x e. V |-> ( x ., y ) ) e. ( W LMHom (ringLMod ` F ) ) )
11	8, 10	syl 17	. 2 |- ( W e. PreHil -> A. y e. V ( x e. V |-> ( x ., y ) ) e. ( W LMHom (ringLMod ` F ) ) )
12		oveq2 6658	. . . . . 6 |- ( y = A -> ( x ., y ) = ( x ., A ) )
13	12	mpteq2dv 4745	. . . . 5 |- ( y = A -> ( x e. V |-> ( x ., y ) ) = ( x e. V |-> ( x ., A ) ) )
14		phllmhm.g	. . . . 5 |- G = ( x e. V |-> ( x ., A ) )
15	13, 14	syl6eqr 2674	. . . 4 |- ( y = A -> ( x e. V |-> ( x ., y ) ) = G )
16	15	eleq1d 2686	. . 3 |- ( y = A -> ( ( x e. V |-> ( x ., y ) ) e. ( W LMHom (ringLMod ` F ) ) <-> G e. ( W LMHom (ringLMod ` F ) ) ) )
17	16	rspccva 3308	. 2 |- ( ( A. y e. V ( x e. V |-> ( x ., y ) ) e. ( W LMHom (ringLMod ` F ) ) /\ A e. V ) -> G e. ( W LMHom (ringLMod ` F ) ) )
18	11, 17	sylan 488	1 |- ( ( W e. PreHil /\ A e. V ) -> G e. ( W LMHom (ringLMod ` F ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ 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:  ipcl  19978  ip0l  19981  ipdir  19984  ipass  19990