Metamath Proof Explorer


Theorem cphsubdi

Description: Distributive law for inner product subtraction. Complex version of ipsubdi . (Contributed by Mario Carneiro, 16-Oct-2015)

Ref Expression
Hypotheses cphipcj.h
|- ., = ( .i ` W )
cphipcj.v
|- V = ( Base ` W )
cphsubdir.m
|- .- = ( -g ` W )
Assertion cphsubdi
|- ( ( W e. CPreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( A ., ( B .- C ) ) = ( ( A ., B ) - ( A ., C ) ) )

Proof

Step Hyp Ref Expression
1 cphipcj.h
 |-  ., = ( .i ` W )
2 cphipcj.v
 |-  V = ( Base ` W )
3 cphsubdir.m
 |-  .- = ( -g ` W )
4 cphphl
 |-  ( W e. CPreHil -> W e. PreHil )
5 eqid
 |-  ( Scalar ` W ) = ( Scalar ` W )
6 eqid
 |-  ( -g ` ( Scalar ` W ) ) = ( -g ` ( Scalar ` W ) )
7 5 1 2 3 6 ipsubdi
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( A ., ( B .- C ) ) = ( ( A ., B ) ( -g ` ( Scalar ` W ) ) ( A ., C ) ) )
8 4 7 sylan
 |-  ( ( W e. CPreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( A ., ( B .- C ) ) = ( ( A ., B ) ( -g ` ( Scalar ` W ) ) ( A ., C ) ) )
9 cphclm
 |-  ( W e. CPreHil -> W e. CMod )
10 9 adantr
 |-  ( ( W e. CPreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> W e. CMod )
11 4 adantr
 |-  ( ( W e. CPreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> W e. PreHil )
12 simpr1
 |-  ( ( W e. CPreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> A e. V )
13 simpr2
 |-  ( ( W e. CPreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> B e. V )
14 eqid
 |-  ( Base ` ( Scalar ` W ) ) = ( Base ` ( Scalar ` W ) )
15 5 1 2 14 ipcl
 |-  ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( A ., B ) e. ( Base ` ( Scalar ` W ) ) )
16 11 12 13 15 syl3anc
 |-  ( ( W e. CPreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( A ., B ) e. ( Base ` ( Scalar ` W ) ) )
17 simpr3
 |-  ( ( W e. CPreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> C e. V )
18 5 1 2 14 ipcl
 |-  ( ( W e. PreHil /\ A e. V /\ C e. V ) -> ( A ., C ) e. ( Base ` ( Scalar ` W ) ) )
19 11 12 17 18 syl3anc
 |-  ( ( W e. CPreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( A ., C ) e. ( Base ` ( Scalar ` W ) ) )
20 5 14 clmsub
 |-  ( ( W e. CMod /\ ( A ., B ) e. ( Base ` ( Scalar ` W ) ) /\ ( A ., C ) e. ( Base ` ( Scalar ` W ) ) ) -> ( ( A ., B ) - ( A ., C ) ) = ( ( A ., B ) ( -g ` ( Scalar ` W ) ) ( A ., C ) ) )
21 10 16 19 20 syl3anc
 |-  ( ( W e. CPreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A ., B ) - ( A ., C ) ) = ( ( A ., B ) ( -g ` ( Scalar ` W ) ) ( A ., C ) ) )
22 8 21 eqtr4d
 |-  ( ( W e. CPreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( A ., ( B .- C ) ) = ( ( A ., B ) - ( A ., C ) ) )