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Metamath Proof Explorer

Theorem ip0l

Description: Inner product with a zero first argument. Part of proof of Theorem 6.44 of Ponnusamy p. 361. (Contributed by NM, 5-Feb-2007) (Revised by Mario Carneiro, 7-Oct-2015)

Ref Expression
Hypotheses phlsrng.f 
|- F = ( Scalar ` W )
phllmhm.h 
|- ., = ( .i ` W )
phllmhm.v 
|- V = ( Base ` W )
ip0l.z 
|- Z = ( 0g ` F )
ip0l.o 
|- .0. = ( 0g ` W )
Assertion ip0l 
|- ( ( W e. PreHil /\ A e. V ) -> ( .0. ., A ) = Z )

Proof

Step Hyp Ref Expression
1 phlsrng.f 
 |-  F = ( Scalar ` W )
2 phllmhm.h 
 |-  ., = ( .i ` W )
3 phllmhm.v 
 |-  V = ( Base ` W )
4 ip0l.z 
 |-  Z = ( 0g ` F )
5 ip0l.o 
 |-  .0. = ( 0g ` W )
6 phllmod 
 |-  ( W e. PreHil -> W e. LMod )
7 lmodgrp 
 |-  ( W e. LMod -> W e. Grp )
8 3 5 grpidcl 
 |-  ( W e. Grp -> .0. e. V )
9 6 7 8 3syl 
 |-  ( W e. PreHil -> .0. e. V )
10 9 adantr 
 |-  ( ( W e. PreHil /\ A e. V ) -> .0. e. V )
11 oveq1 
 |-  ( x = .0. -> ( x ., A ) = ( .0. ., A ) )
12 eqid 
 |-  ( x e. V |-> ( x ., A ) ) = ( x e. V |-> ( x ., A ) )
13 ovex 
 |-  ( .0. ., A ) e. _V
14 11 12 13 fvmpt 
 |-  ( .0. e. V -> ( ( x e. V |-> ( x ., A ) ) ` .0. ) = ( .0. ., A ) )
15 10 14 syl 
 |-  ( ( W e. PreHil /\ A e. V ) -> ( ( x e. V |-> ( x ., A ) ) ` .0. ) = ( .0. ., A ) )
16 1 2 3 12 phllmhm 
 |-  ( ( W e. PreHil /\ A e. V ) -> ( x e. V |-> ( x ., A ) ) e. ( W LMHom ( ringLMod ` F ) ) )
17 lmghm 
 |-  ( ( x e. V |-> ( x ., A ) ) e. ( W LMHom ( ringLMod ` F ) ) -> ( x e. V |-> ( x ., A ) ) e. ( W GrpHom ( ringLMod ` F ) ) )
18 rlm0 
 |-  ( 0g ` F ) = ( 0g ` ( ringLMod ` F ) )
19 4 18 eqtri 
 |-  Z = ( 0g ` ( ringLMod ` F ) )
20 5 19 ghmid 
 |-  ( ( x e. V |-> ( x ., A ) ) e. ( W GrpHom ( ringLMod ` F ) ) -> ( ( x e. V |-> ( x ., A ) ) ` .0. ) = Z )
21 16 17 20 3syl 
 |-  ( ( W e. PreHil /\ A e. V ) -> ( ( x e. V |-> ( x ., A ) ) ` .0. ) = Z )
22 15 21 eqtr3d 
 |-  ( ( W e. PreHil /\ A e. V ) -> ( .0. ., A ) = Z )