HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  lnophmlem2 Structured version   Unicode version

Theorem lnophmlem2 26914
Description: Lemma for lnophmi 26915. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnophmlem.1  |-  A  e. 
~H
lnophmlem.2  |-  B  e. 
~H
lnophmlem.3  |-  T  e. 
LinOp
lnophmlem.4  |-  A. x  e.  ~H  ( x  .ih  ( T `  x ) )  e.  RR
Assertion
Ref Expression
lnophmlem2  |-  ( A 
.ih  ( T `  B ) )  =  ( ( T `  A )  .ih  B
)
Distinct variable groups:    x, A    x, B    x, T

Proof of Theorem lnophmlem2
StepHypRef Expression
1 lnophmlem.2 . . . . . 6  |-  B  e. 
~H
2 lnophmlem.1 . . . . . . 7  |-  A  e. 
~H
3 lnophmlem.3 . . . . . . . . 9  |-  T  e. 
LinOp
43lnopfi 26866 . . . . . . . 8  |-  T : ~H
--> ~H
54ffvelrni 6015 . . . . . . 7  |-  ( A  e.  ~H  ->  ( T `  A )  e.  ~H )
62, 5ax-mp 5 . . . . . 6  |-  ( T `
 A )  e. 
~H
74ffvelrni 6015 . . . . . . 7  |-  ( B  e.  ~H  ->  ( T `  B )  e.  ~H )
81, 7ax-mp 5 . . . . . 6  |-  ( T `
 B )  e. 
~H
91, 6, 2, 8polid2i 26052 . . . . 5  |-  ( B 
.ih  ( T `  A ) )  =  ( ( ( ( ( B  +h  A
)  .ih  ( ( T `  B )  +h  ( T `  A
) ) )  -  ( ( B  -h  A )  .ih  (
( T `  B
)  -h  ( T `
 A ) ) ) )  +  ( _i  x.  ( ( ( B  +h  (
_i  .h  A )
)  .ih  ( ( T `  B )  +h  ( _i  .h  ( T `  A )
) ) )  -  ( ( B  -h  ( _i  .h  A
) )  .ih  (
( T `  B
)  -h  ( _i  .h  ( T `  A ) ) ) ) ) ) )  /  4 )
101, 2hvcomi 25914 . . . . . . . . 9  |-  ( B  +h  A )  =  ( A  +h  B
)
118, 6hvcomi 25914 . . . . . . . . . 10  |-  ( ( T `  B )  +h  ( T `  A ) )  =  ( ( T `  A )  +h  ( T `  B )
)
123lnopaddi 26868 . . . . . . . . . . 11  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  B ) )  =  ( ( T `
 A )  +h  ( T `  B
) ) )
132, 1, 12mp2an 672 . . . . . . . . . 10  |-  ( T `
 ( A  +h  B ) )  =  ( ( T `  A )  +h  ( T `  B )
)
1411, 13eqtr4i 2475 . . . . . . . . 9  |-  ( ( T `  B )  +h  ( T `  A ) )  =  ( T `  ( A  +h  B ) )
1510, 14oveq12i 6293 . . . . . . . 8  |-  ( ( B  +h  A ) 
.ih  ( ( T `
 B )  +h  ( T `  A
) ) )  =  ( ( A  +h  B )  .ih  ( T `  ( A  +h  B ) ) )
161, 2, 8, 6hisubcomi 25999 . . . . . . . . 9  |-  ( ( B  -h  A ) 
.ih  ( ( T `
 B )  -h  ( T `  A
) ) )  =  ( ( A  -h  B )  .ih  (
( T `  A
)  -h  ( T `
 B ) ) )
173lnopsubi 26871 . . . . . . . . . . 11  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  -h  B ) )  =  ( ( T `
 A )  -h  ( T `  B
) ) )
182, 1, 17mp2an 672 . . . . . . . . . 10  |-  ( T `
 ( A  -h  B ) )  =  ( ( T `  A )  -h  ( T `  B )
)
1918oveq2i 6292 . . . . . . . . 9  |-  ( ( A  -h  B ) 
.ih  ( T `  ( A  -h  B
) ) )  =  ( ( A  -h  B )  .ih  (
( T `  A
)  -h  ( T `
 B ) ) )
2016, 19eqtr4i 2475 . . . . . . . 8  |-  ( ( B  -h  A ) 
.ih  ( ( T `
 B )  -h  ( T `  A
) ) )  =  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) )
2115, 20oveq12i 6293 . . . . . . 7  |-  ( ( ( B  +h  A
)  .ih  ( ( T `  B )  +h  ( T `  A
) ) )  -  ( ( B  -h  A )  .ih  (
( T `  B
)  -h  ( T `
 A ) ) ) )  =  ( ( ( A  +h  B )  .ih  ( T `  ( A  +h  B ) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )
22 ax-icn 9554 . . . . . . . . . . 11  |-  _i  e.  CC
2322, 1hvmulcli 25909 . . . . . . . . . . . 12  |-  ( _i  .h  B )  e. 
~H
242, 23hvsubcli 25916 . . . . . . . . . . 11  |-  ( A  -h  ( _i  .h  B ) )  e. 
~H
254ffvelrni 6015 . . . . . . . . . . . 12  |-  ( ( A  -h  ( _i  .h  B ) )  e.  ~H  ->  ( T `  ( A  -h  ( _i  .h  B
) ) )  e. 
~H )
2624, 25ax-mp 5 . . . . . . . . . . 11  |-  ( T `
 ( A  -h  ( _i  .h  B
) ) )  e. 
~H
2722, 22, 24, 26his35i 25984 . . . . . . . . . 10  |-  ( ( _i  .h  ( A  -h  ( _i  .h  B ) ) ) 
.ih  ( _i  .h  ( T `  ( A  -h  ( _i  .h  B ) ) ) ) )  =  ( ( _i  x.  (
* `  _i )
)  x.  ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) ) )
2822, 2, 23hvsubdistr1i 25947 . . . . . . . . . . . 12  |-  ( _i  .h  ( A  -h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  -h  (
_i  .h  ( _i  .h  B ) ) )
2922, 2hvmulcli 25909 . . . . . . . . . . . . . 14  |-  ( _i  .h  A )  e. 
~H
3022, 23hvmulcli 25909 . . . . . . . . . . . . . 14  |-  ( _i  .h  ( _i  .h  B ) )  e. 
~H
3129, 30hvsubvali 25915 . . . . . . . . . . . . 13  |-  ( ( _i  .h  A )  -h  ( _i  .h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  +h  ( -u 1  .h  ( _i  .h  ( _i  .h  B ) ) ) )
3222, 22, 1hvmulassi 25941 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  _i )  .h  B )  =  ( _i  .h  (
_i  .h  B )
)
3332oveq2i 6292 . . . . . . . . . . . . . . 15  |-  ( -u
1  .h  ( ( _i  x.  _i )  .h  B ) )  =  ( -u 1  .h  ( _i  .h  (
_i  .h  B )
) )
34 ixi 10185 . . . . . . . . . . . . . . . . . . 19  |-  ( _i  x.  _i )  = 
-u 1
3534oveq2i 6292 . . . . . . . . . . . . . . . . . 18  |-  ( -u
1  x.  ( _i  x.  _i ) )  =  ( -u 1  x.  -u 1 )
36 ax-1cn 9553 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  CC
3736, 36mul2negi 10011 . . . . . . . . . . . . . . . . . 18  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
38 1t1e1 10690 . . . . . . . . . . . . . . . . . 18  |-  ( 1  x.  1 )  =  1
3935, 37, 383eqtri 2476 . . . . . . . . . . . . . . . . 17  |-  ( -u
1  x.  ( _i  x.  _i ) )  =  1
4039oveq1i 6291 . . . . . . . . . . . . . . . 16  |-  ( (
-u 1  x.  (
_i  x.  _i )
)  .h  B )  =  ( 1  .h  B )
41 neg1cn 10646 . . . . . . . . . . . . . . . . 17  |-  -u 1  e.  CC
4222, 22mulcli 9604 . . . . . . . . . . . . . . . . 17  |-  ( _i  x.  _i )  e.  CC
4341, 42, 1hvmulassi 25941 . . . . . . . . . . . . . . . 16  |-  ( (
-u 1  x.  (
_i  x.  _i )
)  .h  B )  =  ( -u 1  .h  ( ( _i  x.  _i )  .h  B
) )
44 ax-hvmulid 25901 . . . . . . . . . . . . . . . . 17  |-  ( B  e.  ~H  ->  (
1  .h  B )  =  B )
451, 44ax-mp 5 . . . . . . . . . . . . . . . 16  |-  ( 1  .h  B )  =  B
4640, 43, 453eqtr3i 2480 . . . . . . . . . . . . . . 15  |-  ( -u
1  .h  ( ( _i  x.  _i )  .h  B ) )  =  B
4733, 46eqtr3i 2474 . . . . . . . . . . . . . 14  |-  ( -u
1  .h  ( _i  .h  ( _i  .h  B ) ) )  =  B
4847oveq2i 6292 . . . . . . . . . . . . 13  |-  ( ( _i  .h  A )  +h  ( -u 1  .h  ( _i  .h  (
_i  .h  B )
) ) )  =  ( ( _i  .h  A )  +h  B
)
4931, 48eqtri 2472 . . . . . . . . . . . 12  |-  ( ( _i  .h  A )  -h  ( _i  .h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  +h  B
)
5029, 1hvcomi 25914 . . . . . . . . . . . 12  |-  ( ( _i  .h  A )  +h  B )  =  ( B  +h  (
_i  .h  A )
)
5128, 49, 503eqtri 2476 . . . . . . . . . . 11  |-  ( _i  .h  ( A  -h  ( _i  .h  B
) ) )  =  ( B  +h  (
_i  .h  A )
)
5251fveq2i 5859 . . . . . . . . . . . 12  |-  ( T `
 ( _i  .h  ( A  -h  (
_i  .h  B )
) ) )  =  ( T `  ( B  +h  ( _i  .h  A ) ) )
533lnopmuli 26869 . . . . . . . . . . . . 13  |-  ( ( _i  e.  CC  /\  ( A  -h  (
_i  .h  B )
)  e.  ~H )  ->  ( T `  (
_i  .h  ( A  -h  ( _i  .h  B
) ) ) )  =  ( _i  .h  ( T `  ( A  -h  ( _i  .h  B ) ) ) ) )
5422, 24, 53mp2an 672 . . . . . . . . . . . 12  |-  ( T `
 ( _i  .h  ( A  -h  (
_i  .h  B )
) ) )  =  ( _i  .h  ( T `  ( A  -h  ( _i  .h  B
) ) ) )
553lnopaddmuli 26870 . . . . . . . . . . . . 13  |-  ( ( _i  e.  CC  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( T `  ( B  +h  ( _i  .h  A
) ) )  =  ( ( T `  B )  +h  (
_i  .h  ( T `  A ) ) ) )
5622, 1, 2, 55mp3an 1325 . . . . . . . . . . . 12  |-  ( T `
 ( B  +h  ( _i  .h  A
) ) )  =  ( ( T `  B )  +h  (
_i  .h  ( T `  A ) ) )
5752, 54, 563eqtr3i 2480 . . . . . . . . . . 11  |-  ( _i  .h  ( T `  ( A  -h  (
_i  .h  B )
) ) )  =  ( ( T `  B )  +h  (
_i  .h  ( T `  A ) ) )
5851, 57oveq12i 6293 . . . . . . . . . 10  |-  ( ( _i  .h  ( A  -h  ( _i  .h  B ) ) ) 
.ih  ( _i  .h  ( T `  ( A  -h  ( _i  .h  B ) ) ) ) )  =  ( ( B  +h  (
_i  .h  A )
)  .ih  ( ( T `  B )  +h  ( _i  .h  ( T `  A )
) ) )
59 cji 12974 . . . . . . . . . . . . . 14  |-  ( * `
 _i )  = 
-u _i
6059oveq2i 6292 . . . . . . . . . . . . 13  |-  ( _i  x.  ( * `  _i ) )  =  ( _i  x.  -u _i )
6122, 22mulneg2i 10010 . . . . . . . . . . . . 13  |-  ( _i  x.  -u _i )  = 
-u ( _i  x.  _i )
6234negeqi 9818 . . . . . . . . . . . . . 14  |-  -u (
_i  x.  _i )  =  -u -u 1
63 negneg1e1 10650 . . . . . . . . . . . . . 14  |-  -u -u 1  =  1
6462, 63eqtri 2472 . . . . . . . . . . . . 13  |-  -u (
_i  x.  _i )  =  1
6560, 61, 643eqtri 2476 . . . . . . . . . . . 12  |-  ( _i  x.  ( * `  _i ) )  =  1
6665oveq1i 6291 . . . . . . . . . . 11  |-  ( ( _i  x.  ( * `
 _i ) )  x.  ( ( A  -h  ( _i  .h  B ) )  .ih  ( T `  ( A  -h  ( _i  .h  B ) ) ) ) )  =  ( 1  x.  ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) ) )
67 lnophmlem.4 . . . . . . . . . . . . . 14  |-  A. x  e.  ~H  ( x  .ih  ( T `  x ) )  e.  RR
6824, 2, 3, 67lnophmlem1 26913 . . . . . . . . . . . . 13  |-  ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  e.  RR
6968recni 9611 . . . . . . . . . . . 12  |-  ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  e.  CC
7069mulid2i 9602 . . . . . . . . . . 11  |-  ( 1  x.  ( ( A  -h  ( _i  .h  B ) )  .ih  ( T `  ( A  -h  ( _i  .h  B ) ) ) ) )  =  ( ( A  -h  (
_i  .h  B )
)  .ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )
7166, 70eqtri 2472 . . . . . . . . . 10  |-  ( ( _i  x.  ( * `
 _i ) )  x.  ( ( A  -h  ( _i  .h  B ) )  .ih  ( T `  ( A  -h  ( _i  .h  B ) ) ) ) )  =  ( ( A  -h  (
_i  .h  B )
)  .ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )
7227, 58, 713eqtr3i 2480 . . . . . . . . 9  |-  ( ( B  +h  ( _i  .h  A ) ) 
.ih  ( ( T `
 B )  +h  ( _i  .h  ( T `  A )
) ) )  =  ( ( A  -h  ( _i  .h  B
) )  .ih  ( T `  ( A  -h  ( _i  .h  B
) ) ) )
7322, 6hvmulcli 25909 . . . . . . . . . . . 12  |-  ( _i  .h  ( T `  A ) )  e. 
~H
741, 29, 8, 73hisubcomi 25999 . . . . . . . . . . 11  |-  ( ( B  -h  ( _i  .h  A ) ) 
.ih  ( ( T `
 B )  -h  ( _i  .h  ( T `  A )
) ) )  =  ( ( ( _i  .h  A )  -h  B )  .ih  (
( _i  .h  ( T `  A )
)  -h  ( T `
 B ) ) )
7534oveq1i 6291 . . . . . . . . . . . . . . 15  |-  ( ( _i  x.  _i )  .h  B )  =  ( -u 1  .h  B )
7632, 75eqtr3i 2474 . . . . . . . . . . . . . 14  |-  ( _i  .h  ( _i  .h  B ) )  =  ( -u 1  .h  B )
7776oveq2i 6292 . . . . . . . . . . . . 13  |-  ( ( _i  .h  A )  +h  ( _i  .h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  +h  ( -u 1  .h  B ) )
7822, 2, 23hvdistr1i 25946 . . . . . . . . . . . . 13  |-  ( _i  .h  ( A  +h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  +h  (
_i  .h  ( _i  .h  B ) ) )
7929, 1hvsubvali 25915 . . . . . . . . . . . . 13  |-  ( ( _i  .h  A )  -h  B )  =  ( ( _i  .h  A )  +h  ( -u 1  .h  B ) )
8077, 78, 793eqtr4i 2482 . . . . . . . . . . . 12  |-  ( _i  .h  ( A  +h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  -h  B
)
8180fveq2i 5859 . . . . . . . . . . . . 13  |-  ( T `
 ( _i  .h  ( A  +h  (
_i  .h  B )
) ) )  =  ( T `  (
( _i  .h  A
)  -h  B ) )
822, 23hvaddcli 25913 . . . . . . . . . . . . . 14  |-  ( A  +h  ( _i  .h  B ) )  e. 
~H
833lnopmuli 26869 . . . . . . . . . . . . . 14  |-  ( ( _i  e.  CC  /\  ( A  +h  (
_i  .h  B )
)  e.  ~H )  ->  ( T `  (
_i  .h  ( A  +h  ( _i  .h  B
) ) ) )  =  ( _i  .h  ( T `  ( A  +h  ( _i  .h  B ) ) ) ) )
8422, 82, 83mp2an 672 . . . . . . . . . . . . 13  |-  ( T `
 ( _i  .h  ( A  +h  (
_i  .h  B )
) ) )  =  ( _i  .h  ( T `  ( A  +h  ( _i  .h  B
) ) ) )
853lnopmulsubi 26873 . . . . . . . . . . . . . 14  |-  ( ( _i  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( (
_i  .h  A )  -h  B ) )  =  ( ( _i  .h  ( T `  A ) )  -h  ( T `
 B ) ) )
8622, 2, 1, 85mp3an 1325 . . . . . . . . . . . . 13  |-  ( T `
 ( ( _i  .h  A )  -h  B ) )  =  ( ( _i  .h  ( T `  A ) )  -h  ( T `
 B ) )
8781, 84, 863eqtr3i 2480 . . . . . . . . . . . 12  |-  ( _i  .h  ( T `  ( A  +h  (
_i  .h  B )
) ) )  =  ( ( _i  .h  ( T `  A ) )  -h  ( T `
 B ) )
8880, 87oveq12i 6293 . . . . . . . . . . 11  |-  ( ( _i  .h  ( A  +h  ( _i  .h  B ) ) ) 
.ih  ( _i  .h  ( T `  ( A  +h  ( _i  .h  B ) ) ) ) )  =  ( ( ( _i  .h  A )  -h  B
)  .ih  ( (
_i  .h  ( T `  A ) )  -h  ( T `  B
) ) )
8974, 88eqtr4i 2475 . . . . . . . . . 10  |-  ( ( B  -h  ( _i  .h  A ) ) 
.ih  ( ( T `
 B )  -h  ( _i  .h  ( T `  A )
) ) )  =  ( ( _i  .h  ( A  +h  (
_i  .h  B )
) )  .ih  (
_i  .h  ( T `  ( A  +h  (
_i  .h  B )
) ) ) )
904ffvelrni 6015 . . . . . . . . . . . 12  |-  ( ( A  +h  ( _i  .h  B ) )  e.  ~H  ->  ( T `  ( A  +h  ( _i  .h  B
) ) )  e. 
~H )
9182, 90ax-mp 5 . . . . . . . . . . 11  |-  ( T `
 ( A  +h  ( _i  .h  B
) ) )  e. 
~H
9222, 22, 82, 91his35i 25984 . . . . . . . . . 10  |-  ( ( _i  .h  ( A  +h  ( _i  .h  B ) ) ) 
.ih  ( _i  .h  ( T `  ( A  +h  ( _i  .h  B ) ) ) ) )  =  ( ( _i  x.  (
* `  _i )
)  x.  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) ) )
9365oveq1i 6291 . . . . . . . . . . 11  |-  ( ( _i  x.  ( * `
 _i ) )  x.  ( ( A  +h  ( _i  .h  B ) )  .ih  ( T `  ( A  +h  ( _i  .h  B ) ) ) ) )  =  ( 1  x.  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) ) )
9482, 2, 3, 67lnophmlem1 26913 . . . . . . . . . . . . 13  |-  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  e.  RR
9594recni 9611 . . . . . . . . . . . 12  |-  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  e.  CC
9695mulid2i 9602 . . . . . . . . . . 11  |-  ( 1  x.  ( ( A  +h  ( _i  .h  B ) )  .ih  ( T `  ( A  +h  ( _i  .h  B ) ) ) ) )  =  ( ( A  +h  (
_i  .h  B )
)  .ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )
9793, 96eqtri 2472 . . . . . . . . . 10  |-  ( ( _i  x.  ( * `
 _i ) )  x.  ( ( A  +h  ( _i  .h  B ) )  .ih  ( T `  ( A  +h  ( _i  .h  B ) ) ) ) )  =  ( ( A  +h  (
_i  .h  B )
)  .ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )
9889, 92, 973eqtri 2476 . . . . . . . . 9  |-  ( ( B  -h  ( _i  .h  A ) ) 
.ih  ( ( T `
 B )  -h  ( _i  .h  ( T `  A )
) ) )  =  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) )
9972, 98oveq12i 6293 . . . . . . . 8  |-  ( ( ( B  +h  (
_i  .h  A )
)  .ih  ( ( T `  B )  +h  ( _i  .h  ( T `  A )
) ) )  -  ( ( B  -h  ( _i  .h  A
) )  .ih  (
( T `  B
)  -h  ( _i  .h  ( T `  A ) ) ) ) )  =  ( ( ( A  -h  ( _i  .h  B
) )  .ih  ( T `  ( A  -h  ( _i  .h  B
) ) ) )  -  ( ( A  +h  ( _i  .h  B ) )  .ih  ( T `  ( A  +h  ( _i  .h  B ) ) ) ) )
10099oveq2i 6292 . . . . . . 7  |-  ( _i  x.  ( ( ( B  +h  ( _i  .h  A ) ) 
.ih  ( ( T `
 B )  +h  ( _i  .h  ( T `  A )
) ) )  -  ( ( B  -h  ( _i  .h  A
) )  .ih  (
( T `  B
)  -h  ( _i  .h  ( T `  A ) ) ) ) ) )  =  ( _i  x.  (
( ( A  -h  ( _i  .h  B
) )  .ih  ( T `  ( A  -h  ( _i  .h  B
) ) ) )  -  ( ( A  +h  ( _i  .h  B ) )  .ih  ( T `  ( A  +h  ( _i  .h  B ) ) ) ) ) )
10121, 100oveq12i 6293 . . . . . 6  |-  ( ( ( ( B  +h  A )  .ih  (
( T `  B
)  +h  ( T `
 A ) ) )  -  ( ( B  -h  A ) 
.ih  ( ( T `
 B )  -h  ( T `  A
) ) ) )  +  ( _i  x.  ( ( ( B  +h  ( _i  .h  A ) )  .ih  ( ( T `  B )  +h  (
_i  .h  ( T `  A ) ) ) )  -  ( ( B  -h  ( _i  .h  A ) ) 
.ih  ( ( T `
 B )  -h  ( _i  .h  ( T `  A )
) ) ) ) ) )  =  ( ( ( ( A  +h  B )  .ih  ( T `  ( A  +h  B ) ) )  -  ( ( A  -h  B ) 
.ih  ( T `  ( A  -h  B
) ) ) )  +  ( _i  x.  ( ( ( A  -h  ( _i  .h  B ) )  .ih  ( T `  ( A  -h  ( _i  .h  B ) ) ) )  -  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) ) ) ) )
102101oveq1i 6291 . . . . 5  |-  ( ( ( ( ( B  +h  A )  .ih  ( ( T `  B )  +h  ( T `  A )
) )  -  (
( B  -h  A
)  .ih  ( ( T `  B )  -h  ( T `  A
) ) ) )  +  ( _i  x.  ( ( ( B  +h  ( _i  .h  A ) )  .ih  ( ( T `  B )  +h  (
_i  .h  ( T `  A ) ) ) )  -  ( ( B  -h  ( _i  .h  A ) ) 
.ih  ( ( T `
 B )  -h  ( _i  .h  ( T `  A )
) ) ) ) ) )  /  4
)  =  ( ( ( ( ( A  +h  B )  .ih  ( T `  ( A  +h  B ) ) )  -  ( ( A  -h  B ) 
.ih  ( T `  ( A  -h  B
) ) ) )  +  ( _i  x.  ( ( ( A  -h  ( _i  .h  B ) )  .ih  ( T `  ( A  -h  ( _i  .h  B ) ) ) )  -  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) ) ) ) )  /  4
)
1039, 102eqtri 2472 . . . 4  |-  ( B 
.ih  ( T `  A ) )  =  ( ( ( ( ( A  +h  B
)  .ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  +  ( _i  x.  ( ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) ) ) )  / 
4 )
104103fveq2i 5859 . . 3  |-  ( * `
 ( B  .ih  ( T `  A ) ) )  =  ( * `  ( ( ( ( ( A  +h  B )  .ih  ( T `  ( A  +h  B ) ) )  -  ( ( A  -h  B ) 
.ih  ( T `  ( A  -h  B
) ) ) )  +  ( _i  x.  ( ( ( A  -h  ( _i  .h  B ) )  .ih  ( T `  ( A  -h  ( _i  .h  B ) ) ) )  -  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) ) ) ) )  /  4
) )
105 4ne0 10639 . . . 4  |-  4  =/=  0
1062, 1hvaddcli 25913 . . . . . . . . 9  |-  ( A  +h  B )  e. 
~H
107106, 2, 3, 67lnophmlem1 26913 . . . . . . . 8  |-  ( ( A  +h  B ) 
.ih  ( T `  ( A  +h  B
) ) )  e.  RR
1082, 1hvsubcli 25916 . . . . . . . . 9  |-  ( A  -h  B )  e. 
~H
109108, 2, 3, 67lnophmlem1 26913 . . . . . . . 8  |-  ( ( A  -h  B ) 
.ih  ( T `  ( A  -h  B
) ) )  e.  RR
110107, 109resubcli 9886 . . . . . . 7  |-  ( ( ( A  +h  B
)  .ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  e.  RR
111110recni 9611 . . . . . 6  |-  ( ( ( A  +h  B
)  .ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  e.  CC
11268, 94resubcli 9886 . . . . . . . 8  |-  ( ( ( A  -h  (
_i  .h  B )
)  .ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) )  e.  RR
113112recni 9611 . . . . . . 7  |-  ( ( ( A  -h  (
_i  .h  B )
)  .ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) )  e.  CC
11422, 113mulcli 9604 . . . . . 6  |-  ( _i  x.  ( ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) ) )  e.  CC
115111, 114addcli 9603 . . . . 5  |-  ( ( ( ( A  +h  B )  .ih  ( T `  ( A  +h  B ) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  +  ( _i  x.  ( ( ( A  -h  (
_i  .h  B )
)  .ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) ) ) )  e.  CC
116 4re 10619 . . . . . 6  |-  4  e.  RR
117116recni 9611 . . . . 5  |-  4  e.  CC
118115, 117cjdivi 13006 . . . 4  |-  ( 4  =/=  0  ->  (
* `  ( (
( ( ( A  +h  B )  .ih  ( T `  ( A  +h  B ) ) )  -  ( ( A  -h  B ) 
.ih  ( T `  ( A  -h  B
) ) ) )  +  ( _i  x.  ( ( ( A  -h  ( _i  .h  B ) )  .ih  ( T `  ( A  -h  ( _i  .h  B ) ) ) )  -  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) ) ) ) )  /  4
) )  =  ( ( * `  (
( ( ( A  +h  B )  .ih  ( T `  ( A  +h  B ) ) )  -  ( ( A  -h  B ) 
.ih  ( T `  ( A  -h  B
) ) ) )  +  ( _i  x.  ( ( ( A  -h  ( _i  .h  B ) )  .ih  ( T `  ( A  -h  ( _i  .h  B ) ) ) )  -  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) ) ) ) ) )  / 
( * `  4
) ) )
119105, 118ax-mp 5 . . 3  |-  ( * `
 ( ( ( ( ( A  +h  B )  .ih  ( T `  ( A  +h  B ) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  +  ( _i  x.  ( ( ( A  -h  (
_i  .h  B )
)  .ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) ) ) )  / 
4 ) )  =  ( ( * `  ( ( ( ( A  +h  B ) 
.ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  +  ( _i  x.  ( ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) ) ) ) )  /  ( * ` 
4 ) )
120 cjreim 12975 . . . . . . 7  |-  ( ( ( ( ( A  +h  B )  .ih  ( T `  ( A  +h  B ) ) )  -  ( ( A  -h  B ) 
.ih  ( T `  ( A  -h  B
) ) ) )  e.  RR  /\  (
( ( A  -h  ( _i  .h  B
) )  .ih  ( T `  ( A  -h  ( _i  .h  B
) ) ) )  -  ( ( A  +h  ( _i  .h  B ) )  .ih  ( T `  ( A  +h  ( _i  .h  B ) ) ) ) )  e.  RR )  ->  ( * `  ( ( ( ( A  +h  B ) 
.ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  +  ( _i  x.  ( ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) ) ) ) )  =  ( ( ( ( A  +h  B
)  .ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  -  ( _i  x.  ( ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) ) ) ) )
121110, 112, 120mp2an 672 . . . . . 6  |-  ( * `
 ( ( ( ( A  +h  B
)  .ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  +  ( _i  x.  ( ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) ) ) ) )  =  ( ( ( ( A  +h  B
)  .ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  -  ( _i  x.  ( ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) ) ) )
12282, 1, 3, 67lnophmlem1 26913 . . . . . . . . . 10  |-  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  e.  RR
12368, 122resubcli 9886 . . . . . . . . 9  |-  ( ( ( A  -h  (
_i  .h  B )
)  .ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) )  e.  RR
124123recni 9611 . . . . . . . 8  |-  ( ( ( A  -h  (
_i  .h  B )
)  .ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) )  e.  CC
12522, 124mulcli 9604 . . . . . . 7  |-  ( _i  x.  ( ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) ) )  e.  CC
126111, 125negsubi 9902 . . . . . 6  |-  ( ( ( ( A  +h  B )  .ih  ( T `  ( A  +h  B ) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  +  -u ( _i  x.  (
( ( A  -h  ( _i  .h  B
) )  .ih  ( T `  ( A  -h  ( _i  .h  B
) ) ) )  -  ( ( A  +h  ( _i  .h  B ) )  .ih  ( T `  ( A  +h  ( _i  .h  B ) ) ) ) ) ) )  =  ( ( ( ( A  +h  B
)  .ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  -  ( _i  x.  ( ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) ) ) )
127121, 126eqtr4i 2475 . . . . 5  |-  ( * `
 ( ( ( ( A  +h  B
)  .ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  +  ( _i  x.  ( ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) ) ) ) )  =  ( ( ( ( A  +h  B
)  .ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  +  -u (
_i  x.  ( (
( A  -h  (
_i  .h  B )
)  .ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) ) ) )
12822, 113mulneg2i 10010 . . . . . . 7  |-  ( _i  x.  -u ( ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) ) )  =  -u ( _i  x.  (
( ( A  -h  ( _i  .h  B
) )  .ih  ( T `  ( A  -h  ( _i  .h  B
) ) ) )  -  ( ( A  +h  ( _i  .h  B ) )  .ih  ( T `  ( A  +h  ( _i  .h  B ) ) ) ) ) )
12969, 95negsubdi2i 9911 . . . . . . . 8  |-  -u (
( ( A  -h  ( _i  .h  B
) )  .ih  ( T `  ( A  -h  ( _i  .h  B
) ) ) )  -  ( ( A  +h  ( _i  .h  B ) )  .ih  ( T `  ( A  +h  ( _i  .h  B ) ) ) ) )  =  ( ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) )  -  ( ( A  -h  ( _i  .h  B ) )  .ih  ( T `  ( A  -h  ( _i  .h  B ) ) ) ) )
130129oveq2i 6292 . . . . . . 7  |-  ( _i  x.  -u ( ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) ) )  =  ( _i  x.  ( ( ( A  +h  (
_i  .h  B )
)  .ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  -  ( ( A  -h  ( _i  .h  B
) )  .ih  ( T `  ( A  -h  ( _i  .h  B
) ) ) ) ) )
131128, 130eqtr3i 2474 . . . . . 6  |-  -u (
_i  x.  ( (
( A  -h  (
_i  .h  B )
)  .ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) ) )  =  ( _i  x.  ( ( ( A  +h  (
_i  .h  B )
)  .ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  -  ( ( A  -h  ( _i  .h  B
) )  .ih  ( T `  ( A  -h  ( _i  .h  B
) ) ) ) ) )
132131oveq2i 6292 . . . . 5  |-  ( ( ( ( A  +h  B )  .ih  ( T `  ( A  +h  B ) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  +  -u ( _i  x.  (
( ( A  -h  ( _i  .h  B
) )  .ih  ( T `  ( A  -h  ( _i  .h  B
) ) ) )  -  ( ( A  +h  ( _i  .h  B ) )  .ih  ( T `  ( A  +h  ( _i  .h  B ) ) ) ) ) ) )  =  ( ( ( ( A  +h  B
)  .ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  +  ( _i  x.  ( ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  -  ( ( A  -h  ( _i  .h  B
) )  .ih  ( T `  ( A  -h  ( _i  .h  B
) ) ) ) ) ) )
13313oveq2i 6292 . . . . . . 7  |-  ( ( A  +h  B ) 
.ih  ( T `  ( A  +h  B
) ) )  =  ( ( A  +h  B )  .ih  (
( T `  A
)  +h  ( T `
 B ) ) )
134133, 19oveq12i 6293 . . . . . 6  |-  ( ( ( A  +h  B
)  .ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  =  ( ( ( A  +h  B
)  .ih  ( ( T `  A )  +h  ( T `  B
) ) )  -  ( ( A  -h  B )  .ih  (
( T `  A
)  -h  ( T `
 B ) ) ) )
1353lnopaddmuli 26870 . . . . . . . . . 10  |-  ( ( _i  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  ( _i  .h  B
) ) )  =  ( ( T `  A )  +h  (
_i  .h  ( T `  B ) ) ) )
13622, 2, 1, 135mp3an 1325 . . . . . . . . 9  |-  ( T `
 ( A  +h  ( _i  .h  B
) ) )  =  ( ( T `  A )  +h  (
_i  .h  ( T `  B ) ) )
137136oveq2i 6292 . . . . . . . 8  |-  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  =  ( ( A  +h  ( _i  .h  B
) )  .ih  (
( T `  A
)  +h  ( _i  .h  ( T `  B ) ) ) )
1383lnopsubmuli 26872 . . . . . . . . . 10  |-  ( ( _i  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  -h  ( _i  .h  B
) ) )  =  ( ( T `  A )  -h  (
_i  .h  ( T `  B ) ) ) )
13922, 2, 1, 138mp3an 1325 . . . . . . . . 9  |-  ( T `
 ( A  -h  ( _i  .h  B
) ) )  =  ( ( T `  A )  -h  (
_i  .h  ( T `  B ) ) )
140139oveq2i 6292 . . . . . . . 8  |-  ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  =  ( ( A  -h  ( _i  .h  B
) )  .ih  (
( T `  A
)  -h  ( _i  .h  ( T `  B ) ) ) )
141137, 140oveq12i 6293 . . . . . . 7  |-  ( ( ( A  +h  (
_i  .h  B )
)  .ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  -  ( ( A  -h  ( _i  .h  B
) )  .ih  ( T `  ( A  -h  ( _i  .h  B
) ) ) ) )  =  ( ( ( A  +h  (
_i  .h  B )
)  .ih  ( ( T `  A )  +h  ( _i  .h  ( T `  B )
) ) )  -  ( ( A  -h  ( _i  .h  B
) )  .ih  (
( T `  A
)  -h  ( _i  .h  ( T `  B ) ) ) ) )
142141oveq2i 6292 . . . . . 6  |-  ( _i  x.  ( ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  -  ( ( A  -h  ( _i  .h  B
) )  .ih  ( T `  ( A  -h  ( _i  .h  B
) ) ) ) ) )  =  ( _i  x.  ( ( ( A  +h  (
_i  .h  B )
)  .ih  ( ( T `  A )  +h  ( _i  .h  ( T `  B )
) ) )  -  ( ( A  -h  ( _i  .h  B
) )  .ih  (
( T `  A
)  -h  ( _i  .h  ( T `  B ) ) ) ) ) )
143134, 142oveq12i 6293 . . . . 5  |-  ( ( ( ( A  +h  B )  .ih  ( T `  ( A  +h  B ) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  +  ( _i  x.  ( ( ( A  +h  (
_i  .h  B )
)  .ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  -  ( ( A  -h  ( _i  .h  B
) )  .ih  ( T `  ( A  -h  ( _i  .h  B
) ) ) ) ) ) )  =  ( ( ( ( A  +h  B ) 
.ih  ( ( T `
 A )  +h  ( T `  B
) ) )  -  ( ( A  -h  B )  .ih  (
( T `  A
)  -h  ( T `
 B ) ) ) )  +  ( _i  x.  ( ( ( A  +h  (
_i  .h  B )
)  .ih  ( ( T `  A )  +h  ( _i  .h  ( T `  B )
) ) )  -  ( ( A  -h  ( _i  .h  B
) )  .ih  (
( T `  A
)  -h  ( _i  .h  ( T `  B ) ) ) ) ) ) )
144127, 132, 1433eqtri 2476 . . . 4  |-  ( * `
 ( ( ( ( A  +h  B
)  .ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  +  ( _i  x.  ( ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) ) ) ) )  =  ( ( ( ( A  +h  B
)  .ih  ( ( T `  A )  +h  ( T `  B
) ) )  -  ( ( A  -h  B )  .ih  (
( T `  A
)  -h  ( T `
 B ) ) ) )  +  ( _i  x.  ( ( ( A  +h  (
_i  .h  B )
)  .ih  ( ( T `  A )  +h  ( _i  .h  ( T `  B )
) ) )  -  ( ( A  -h  ( _i  .h  B
) )  .ih  (
( T `  A
)  -h  ( _i  .h  ( T `  B ) ) ) ) ) ) )
145 cjre 12954 . . . . 5  |-  ( 4  e.  RR  ->  (
* `  4 )  =  4 )
146116, 145ax-mp 5 . . . 4  |-  ( * `
 4 )  =  4
147144, 146oveq12i 6293 . . 3  |-  ( ( * `  ( ( ( ( A  +h  B )  .ih  ( T `  ( A  +h  B ) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  +  ( _i  x.  ( ( ( A  -h  (
_i  .h  B )
)  .ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  -  ( ( A  +h  ( _i  .h  B
) )  .ih  ( T `  ( A  +h  ( _i  .h  B
) ) ) ) ) ) ) )  /  ( * ` 
4 ) )  =  ( ( ( ( ( A  +h  B
)  .ih  ( ( T `  A )  +h  ( T `  B
) ) )  -  ( ( A  -h  B )  .ih  (
( T `  A
)  -h  ( T `
 B ) ) ) )  +  ( _i  x.  ( ( ( A  +h  (
_i  .h  B )
)  .ih  ( ( T `  A )  +h  ( _i  .h  ( T `  B )
) ) )  -  ( ( A  -h  ( _i  .h  B
) )  .ih  (
( T `  A
)  -h  ( _i  .h  ( T `  B ) ) ) ) ) ) )  /  4 )
148104, 119, 1473eqtrri 2477 . 2  |-  ( ( ( ( ( A  +h  B )  .ih  ( ( T `  A )  +h  ( T `  B )
) )  -  (
( A  -h  B
)  .ih  ( ( T `  A )  -h  ( T `  B
) ) ) )  +  ( _i  x.  ( ( ( A  +h  ( _i  .h  B ) )  .ih  ( ( T `  A )  +h  (
_i  .h  ( T `  B ) ) ) )  -  ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( ( T `
 A )  -h  ( _i  .h  ( T `  B )
) ) ) ) ) )  /  4
)  =  ( * `
 ( B  .ih  ( T `  A ) ) )
1492, 8, 1, 6polid2i 26052 . 2  |-  ( A 
.ih  ( T `  B ) )  =  ( ( ( ( ( A  +h  B
)  .ih  ( ( T `  A )  +h  ( T `  B
) ) )  -  ( ( A  -h  B )  .ih  (
( T `  A
)  -h  ( T `
 B ) ) ) )  +  ( _i  x.  ( ( ( A  +h  (
_i  .h  B )
)  .ih  ( ( T `  A )  +h  ( _i  .h  ( T `  B )
) ) )  -  ( ( A  -h  ( _i  .h  B
) )  .ih  (
( T `  A
)  -h  ( _i  .h  ( T `  B ) ) ) ) ) ) )  /  4 )
1506, 1his1i 25995 . 2  |-  ( ( T `  A ) 
.ih  B )  =  ( * `  ( B  .ih  ( T `  A ) ) )
151148, 149, 1503eqtr4i 2482 1  |-  ( A 
.ih  ( T `  B ) )  =  ( ( T `  A )  .ih  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496   _ici 9497    + caddc 9498    x. cmul 9500    - cmin 9810   -ucneg 9811    / cdiv 10213   4c4 10594   *ccj 12911   ~Hchil 25814    +h cva 25815    .h csm 25816    .ih csp 25817    -h cmv 25820   LinOpclo 25842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-hilex 25894  ax-hfvadd 25895  ax-hvcom 25896  ax-hvass 25897  ax-hv0cl 25898  ax-hvaddid 25899  ax-hfvmul 25900  ax-hvmulid 25901  ax-hvmulass 25902  ax-hvdistr1 25903  ax-hvdistr2 25904  ax-hvmul0 25905  ax-hfi 25974  ax-his1 25977  ax-his2 25978  ax-his3 25979
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-2 10601  df-3 10602  df-4 10603  df-cj 12914  df-re 12915  df-im 12916  df-hvsub 25866  df-lnop 26738
This theorem is referenced by:  lnophmi  26915
  Copyright terms: Public domain W3C validator