Metamath Proof Explorer

Theorem ipdir

Description: Distributive law for inner product (right-distributivity). Equation I3 of Ponnusamy p. 362. (Contributed by NM, 25-Aug-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 )
ipdir.g 
|- .+ = ( +g ` W )
ipdir.p 
|- .+^ = ( +g ` F )
Assertion ipdir 
|- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A .+ B ) ., C ) = ( ( A ., C ) .+^ ( B ., C ) ) )

Proof

Step Hyp Ref Expression
1 phlsrng.f 
 |-  F = ( Scalar ` W )
2 phllmhm.h 
 |-  ., = ( .i ` W )
3 phllmhm.v 
 |-  V = ( Base ` W )
4 ipdir.g 
 |-  .+ = ( +g ` W )
5 ipdir.p 
 |-  .+^ = ( +g ` F )
6 eqid 
 |-  ( x e. V |-> ( x ., C ) ) = ( x e. V |-> ( x ., C ) )
7 1 2 3 6 phllmhm 
 |-  ( ( W e. PreHil /\ C e. V ) -> ( x e. V |-> ( x ., C ) ) e. ( W LMHom ( ringLMod ` F ) ) )
8 7 3ad2antr3 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( x e. V |-> ( x ., C ) ) e. ( W LMHom ( ringLMod ` F ) ) )
9 lmghm 
 |-  ( ( x e. V |-> ( x ., C ) ) e. ( W LMHom ( ringLMod ` F ) ) -> ( x e. V |-> ( x ., C ) ) e. ( W GrpHom ( ringLMod ` F ) ) )
10 8 9 syl 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( x e. V |-> ( x ., C ) ) e. ( W GrpHom ( ringLMod ` F ) ) )
11 simpr1 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> A e. V )
12 simpr2 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> B e. V )
13 rlmplusg 
 |-  ( +g ` F ) = ( +g ` ( ringLMod ` F ) )
14 5 13 eqtri 
 |-  .+^ = ( +g ` ( ringLMod ` F ) )
15 3 4 14 ghmlin 
 |-  ( ( ( x e. V |-> ( x ., C ) ) e. ( W GrpHom ( ringLMod ` F ) ) /\ A e. V /\ B e. V ) -> ( ( x e. V |-> ( x ., C ) ) ` ( A .+ B ) ) = ( ( ( x e. V |-> ( x ., C ) ) ` A ) .+^ ( ( x e. V |-> ( x ., C ) ) ` B ) ) )
16 10 11 12 15 syl3anc 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( x e. V |-> ( x ., C ) ) ` ( A .+ B ) ) = ( ( ( x e. V |-> ( x ., C ) ) ` A ) .+^ ( ( x e. V |-> ( x ., C ) ) ` B ) ) )
17 phllmod 
 |-  ( W e. PreHil -> W e. LMod )
18 3 4 lmodvacl 
 |-  ( ( W e. LMod /\ A e. V /\ B e. V ) -> ( A .+ B ) e. V )
19 17 18 syl3an1 
 |-  ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( A .+ B ) e. V )
20 19 3adant3r3 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( A .+ B ) e. V )
21 oveq1 
 |-  ( x = ( A .+ B ) -> ( x ., C ) = ( ( A .+ B ) ., C ) )
22 ovex 
 |-  ( x ., C ) e. _V
23 21 6 22 fvmpt3i 
 |-  ( ( A .+ B ) e. V -> ( ( x e. V |-> ( x ., C ) ) ` ( A .+ B ) ) = ( ( A .+ B ) ., C ) )
24 20 23 syl 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( x e. V |-> ( x ., C ) ) ` ( A .+ B ) ) = ( ( A .+ B ) ., C ) )
25 oveq1 
 |-  ( x = A -> ( x ., C ) = ( A ., C ) )
26 25 6 22 fvmpt3i 
 |-  ( A e. V -> ( ( x e. V |-> ( x ., C ) ) ` A ) = ( A ., C ) )
27 11 26 syl 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( x e. V |-> ( x ., C ) ) ` A ) = ( A ., C ) )
28 oveq1 
 |-  ( x = B -> ( x ., C ) = ( B ., C ) )
29 28 6 22 fvmpt3i 
 |-  ( B e. V -> ( ( x e. V |-> ( x ., C ) ) ` B ) = ( B ., C ) )
30 12 29 syl 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( x e. V |-> ( x ., C ) ) ` B ) = ( B ., C ) )
31 27 30 oveq12d 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( ( x e. V |-> ( x ., C ) ) ` A ) .+^ ( ( x e. V |-> ( x ., C ) ) ` B ) ) = ( ( A ., C ) .+^ ( B ., C ) ) )
32 16 24 31 3eqtr3d 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A .+ B ) ., C ) = ( ( A ., C ) .+^ ( B ., C ) ) )