Metamath Proof Explorer

Theorem ipassr2

Description: "Associative" law for inner product. Conjugate version of ipassr . (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.f 
|- K = ( Base ` F )
ipass.s 
|- .x. = ( .s ` W )
ipass.p 
|- .X. = ( .r ` F )
ipassr.i 
|- .* = ( *r ` F )
Assertion ipassr2 
|- ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. K ) ) -> ( ( A ., B ) .X. C ) = ( A ., ( ( .* ` C ) .x. B ) ) )

Proof

Step Hyp Ref Expression
1 phlsrng.f 
 |-  F = ( Scalar ` W )
2 phllmhm.h 
 |-  ., = ( .i ` W )
3 phllmhm.v 
 |-  V = ( Base ` W )
4 ipdir.f 
 |-  K = ( Base ` F )
5 ipass.s 
 |-  .x. = ( .s ` W )
6 ipass.p 
 |-  .X. = ( .r ` F )
7 ipassr.i 
 |-  .* = ( *r ` F )
8 simpl 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. K ) ) -> W e. PreHil )
9 simpr1 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. K ) ) -> A e. V )
10 simpr2 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. K ) ) -> B e. V )
11 1 phlsrng 
 |-  ( W e. PreHil -> F e. *Ring )
12 simpr3 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. K ) ) -> C e. K )
13 7 4 srngcl 
 |-  ( ( F e. *Ring /\ C e. K ) -> ( .* ` C ) e. K )
14 11 12 13 syl2an2r 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. K ) ) -> ( .* ` C ) e. K )
15 1 2 3 4 5 6 7 ipassr 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ ( .* ` C ) e. K ) ) -> ( A ., ( ( .* ` C ) .x. B ) ) = ( ( A ., B ) .X. ( .* ` ( .* ` C ) ) ) )
16 8 9 10 14 15 syl13anc 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. K ) ) -> ( A ., ( ( .* ` C ) .x. B ) ) = ( ( A ., B ) .X. ( .* ` ( .* ` C ) ) ) )
17 7 4 srngnvl 
 |-  ( ( F e. *Ring /\ C e. K ) -> ( .* ` ( .* ` C ) ) = C )
18 11 12 17 syl2an2r 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. K ) ) -> ( .* ` ( .* ` C ) ) = C )
19 18 oveq2d 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. K ) ) -> ( ( A ., B ) .X. ( .* ` ( .* ` C ) ) ) = ( ( A ., B ) .X. C ) )
20 16 19 eqtr2d 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. K ) ) -> ( ( A ., B ) .X. C ) = ( A ., ( ( .* ` C ) .x. B ) ) )