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Theorem lnophmlem2 23520
Description: Lemma for lnophmi 23521. (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 23472 . . . . . . . 8  |-  T : ~H
--> ~H
54ffvelrni 5869 . . . . . . 7  |-  ( A  e.  ~H  ->  ( T `  A )  e.  ~H )
62, 5ax-mp 8 . . . . . 6  |-  ( T `
 A )  e. 
~H
74ffvelrni 5869 . . . . . . 7  |-  ( B  e.  ~H  ->  ( T `  B )  e.  ~H )
81, 7ax-mp 8 . . . . . 6  |-  ( T `
 B )  e. 
~H
91, 6, 2, 8polid2i 22659 . . . . 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 22522 . . . . . . . . 9  |-  ( B  +h  A )  =  ( A  +h  B
)
118, 6hvcomi 22522 . . . . . . . . . 10  |-  ( ( T `  B )  +h  ( T `  A ) )  =  ( ( T `  A )  +h  ( T `  B )
)
123lnopaddi 23474 . . . . . . . . . . 11  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  B ) )  =  ( ( T `
 A )  +h  ( T `  B
) ) )
132, 1, 12mp2an 654 . . . . . . . . . 10  |-  ( T `
 ( A  +h  B ) )  =  ( ( T `  A )  +h  ( T `  B )
)
1411, 13eqtr4i 2459 . . . . . . . . 9  |-  ( ( T `  B )  +h  ( T `  A ) )  =  ( T `  ( A  +h  B ) )
1510, 14oveq12i 6093 . . . . . . . 8  |-  ( ( B  +h  A ) 
.ih  ( ( T `
 B )  +h  ( T `  A
) ) )  =  ( ( A  +h  B )  .ih  ( T `  ( A  +h  B ) ) )
161, 2, 8, 6hisubcomi 22606 . . . . . . . . 9  |-  ( ( B  -h  A ) 
.ih  ( ( T `
 B )  -h  ( T `  A
) ) )  =  ( ( A  -h  B )  .ih  (
( T `  A
)  -h  ( T `
 B ) ) )
173lnopsubi 23477 . . . . . . . . . . 11  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  -h  B ) )  =  ( ( T `
 A )  -h  ( T `  B
) ) )
182, 1, 17mp2an 654 . . . . . . . . . 10  |-  ( T `
 ( A  -h  B ) )  =  ( ( T `  A )  -h  ( T `  B )
)
1918oveq2i 6092 . . . . . . . . 9  |-  ( ( A  -h  B ) 
.ih  ( T `  ( A  -h  B
) ) )  =  ( ( A  -h  B )  .ih  (
( T `  A
)  -h  ( T `
 B ) ) )
2016, 19eqtr4i 2459 . . . . . . . 8  |-  ( ( B  -h  A ) 
.ih  ( ( T `
 B )  -h  ( T `  A
) ) )  =  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) )
2115, 20oveq12i 6093 . . . . . . 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 9049 . . . . . . . . . . 11  |-  _i  e.  CC
2322, 1hvmulcli 22517 . . . . . . . . . . . 12  |-  ( _i  .h  B )  e. 
~H
242, 23hvsubcli 22524 . . . . . . . . . . 11  |-  ( A  -h  ( _i  .h  B ) )  e. 
~H
254ffvelrni 5869 . . . . . . . . . . . 12  |-  ( ( A  -h  ( _i  .h  B ) )  e.  ~H  ->  ( T `  ( A  -h  ( _i  .h  B
) ) )  e. 
~H )
2624, 25ax-mp 8 . . . . . . . . . . 11  |-  ( T `
 ( A  -h  ( _i  .h  B
) ) )  e. 
~H
2722, 22, 24, 26his35i 22591 . . . . . . . . . 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 22554 . . . . . . . . . . . 12  |-  ( _i  .h  ( A  -h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  -h  (
_i  .h  ( _i  .h  B ) ) )
2922, 2hvmulcli 22517 . . . . . . . . . . . . . 14  |-  ( _i  .h  A )  e. 
~H
3022, 23hvmulcli 22517 . . . . . . . . . . . . . 14  |-  ( _i  .h  ( _i  .h  B ) )  e. 
~H
3129, 30hvsubvali 22523 . . . . . . . . . . . . 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 22548 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  _i )  .h  B )  =  ( _i  .h  (
_i  .h  B )
)
3332oveq2i 6092 . . . . . . . . . . . . . . 15  |-  ( -u
1  .h  ( ( _i  x.  _i )  .h  B ) )  =  ( -u 1  .h  ( _i  .h  (
_i  .h  B )
) )
34 ixi 9651 . . . . . . . . . . . . . . . . . . 19  |-  ( _i  x.  _i )  = 
-u 1
3534oveq2i 6092 . . . . . . . . . . . . . . . . . 18  |-  ( -u
1  x.  ( _i  x.  _i ) )  =  ( -u 1  x.  -u 1 )
36 ax-1cn 9048 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  CC
3736, 36mul2negi 9481 . . . . . . . . . . . . . . . . . 18  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
38 1t1e1 10126 . . . . . . . . . . . . . . . . . 18  |-  ( 1  x.  1 )  =  1
3935, 37, 383eqtri 2460 . . . . . . . . . . . . . . . . 17  |-  ( -u
1  x.  ( _i  x.  _i ) )  =  1
4039oveq1i 6091 . . . . . . . . . . . . . . . 16  |-  ( (
-u 1  x.  (
_i  x.  _i )
)  .h  B )  =  ( 1  .h  B )
41 neg1cn 10067 . . . . . . . . . . . . . . . . 17  |-  -u 1  e.  CC
4222, 22mulcli 9095 . . . . . . . . . . . . . . . . 17  |-  ( _i  x.  _i )  e.  CC
4341, 42, 1hvmulassi 22548 . . . . . . . . . . . . . . . 16  |-  ( (
-u 1  x.  (
_i  x.  _i )
)  .h  B )  =  ( -u 1  .h  ( ( _i  x.  _i )  .h  B
) )
44 ax-hvmulid 22509 . . . . . . . . . . . . . . . . 17  |-  ( B  e.  ~H  ->  (
1  .h  B )  =  B )
451, 44ax-mp 8 . . . . . . . . . . . . . . . 16  |-  ( 1  .h  B )  =  B
4640, 43, 453eqtr3i 2464 . . . . . . . . . . . . . . 15  |-  ( -u
1  .h  ( ( _i  x.  _i )  .h  B ) )  =  B
4733, 46eqtr3i 2458 . . . . . . . . . . . . . 14  |-  ( -u
1  .h  ( _i  .h  ( _i  .h  B ) ) )  =  B
4847oveq2i 6092 . . . . . . . . . . . . 13  |-  ( ( _i  .h  A )  +h  ( -u 1  .h  ( _i  .h  (
_i  .h  B )
) ) )  =  ( ( _i  .h  A )  +h  B
)
4931, 48eqtri 2456 . . . . . . . . . . . 12  |-  ( ( _i  .h  A )  -h  ( _i  .h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  +h  B
)
5029, 1hvcomi 22522 . . . . . . . . . . . 12  |-  ( ( _i  .h  A )  +h  B )  =  ( B  +h  (
_i  .h  A )
)
5128, 49, 503eqtri 2460 . . . . . . . . . . 11  |-  ( _i  .h  ( A  -h  ( _i  .h  B
) ) )  =  ( B  +h  (
_i  .h  A )
)
5251fveq2i 5731 . . . . . . . . . . . 12  |-  ( T `
 ( _i  .h  ( A  -h  (
_i  .h  B )
) ) )  =  ( T `  ( B  +h  ( _i  .h  A ) ) )
533lnopmuli 23475 . . . . . . . . . . . . 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 654 . . . . . . . . . . . 12  |-  ( T `
 ( _i  .h  ( A  -h  (
_i  .h  B )
) ) )  =  ( _i  .h  ( T `  ( A  -h  ( _i  .h  B
) ) ) )
553lnopaddmuli 23476 . . . . . . . . . . . . 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 1279 . . . . . . . . . . . 12  |-  ( T `
 ( B  +h  ( _i  .h  A
) ) )  =  ( ( T `  B )  +h  (
_i  .h  ( T `  A ) ) )
5752, 54, 563eqtr3i 2464 . . . . . . . . . . 11  |-  ( _i  .h  ( T `  ( A  -h  (
_i  .h  B )
) ) )  =  ( ( T `  B )  +h  (
_i  .h  ( T `  A ) ) )
5851, 57oveq12i 6093 . . . . . . . . . 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 11964 . . . . . . . . . . . . . 14  |-  ( * `
 _i )  = 
-u _i
6059oveq2i 6092 . . . . . . . . . . . . 13  |-  ( _i  x.  ( * `  _i ) )  =  ( _i  x.  -u _i )
6122, 22mulneg2i 9480 . . . . . . . . . . . . 13  |-  ( _i  x.  -u _i )  = 
-u ( _i  x.  _i )
6234negeqi 9299 . . . . . . . . . . . . . 14  |-  -u (
_i  x.  _i )  =  -u -u 1
6336negnegi 9370 . . . . . . . . . . . . . 14  |-  -u -u 1  =  1
6462, 63eqtri 2456 . . . . . . . . . . . . 13  |-  -u (
_i  x.  _i )  =  1
6560, 61, 643eqtri 2460 . . . . . . . . . . . 12  |-  ( _i  x.  ( * `  _i ) )  =  1
6665oveq1i 6091 . . . . . . . . . . 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 23519 . . . . . . . . . . . . 13  |-  ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  e.  RR
6968recni 9102 . . . . . . . . . . . 12  |-  ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  e.  CC
7069mulid2i 9093 . . . . . . . . . . 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 2456 . . . . . . . . . 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 2464 . . . . . . . . 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 22517 . . . . . . . . . . . 12  |-  ( _i  .h  ( T `  A ) )  e. 
~H
741, 29, 8, 73hisubcomi 22606 . . . . . . . . . . 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 6091 . . . . . . . . . . . . . . 15  |-  ( ( _i  x.  _i )  .h  B )  =  ( -u 1  .h  B )
7632, 75eqtr3i 2458 . . . . . . . . . . . . . 14  |-  ( _i  .h  ( _i  .h  B ) )  =  ( -u 1  .h  B )
7776oveq2i 6092 . . . . . . . . . . . . 13  |-  ( ( _i  .h  A )  +h  ( _i  .h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  +h  ( -u 1  .h  B ) )
7822, 2, 23hvdistr1i 22553 . . . . . . . . . . . . 13  |-  ( _i  .h  ( A  +h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  +h  (
_i  .h  ( _i  .h  B ) ) )
7929, 1hvsubvali 22523 . . . . . . . . . . . . 13  |-  ( ( _i  .h  A )  -h  B )  =  ( ( _i  .h  A )  +h  ( -u 1  .h  B ) )
8077, 78, 793eqtr4i 2466 . . . . . . . . . . . 12  |-  ( _i  .h  ( A  +h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  -h  B
)
8180fveq2i 5731 . . . . . . . . . . . . 13  |-  ( T `
 ( _i  .h  ( A  +h  (
_i  .h  B )
) ) )  =  ( T `  (
( _i  .h  A
)  -h  B ) )
822, 23hvaddcli 22521 . . . . . . . . . . . . . 14  |-  ( A  +h  ( _i  .h  B ) )  e. 
~H
833lnopmuli 23475 . . . . . . . . . . . . . 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 654 . . . . . . . . . . . . 13  |-  ( T `
 ( _i  .h  ( A  +h  (
_i  .h  B )
) ) )  =  ( _i  .h  ( T `  ( A  +h  ( _i  .h  B
) ) ) )
853lnopmulsubi 23479 . . . . . . . . . . . . . 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 1279 . . . . . . . . . . . . 13  |-  ( T `
 ( ( _i  .h  A )  -h  B ) )  =  ( ( _i  .h  ( T `  A ) )  -h  ( T `
 B ) )
8781, 84, 863eqtr3i 2464 . . . . . . . . . . . 12  |-  ( _i  .h  ( T `  ( A  +h  (
_i  .h  B )
) ) )  =  ( ( _i  .h  ( T `  A ) )  -h  ( T `
 B ) )
8880, 87oveq12i 6093 . . . . . . . . . . 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 2459 . . . . . . . . . 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 5869 . . . . . . . . . . . 12  |-  ( ( A  +h  ( _i  .h  B ) )  e.  ~H  ->  ( T `  ( A  +h  ( _i  .h  B
) ) )  e. 
~H )
9182, 90ax-mp 8 . . . . . . . . . . 11  |-  ( T `
 ( A  +h  ( _i  .h  B
) ) )  e. 
~H
9222, 22, 82, 91his35i 22591 . . . . . . . . . 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 6091 . . . . . . . . . . 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 23519 . . . . . . . . . . . . 13  |-  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  e.  RR
9594recni 9102 . . . . . . . . . . . 12  |-  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  e.  CC
9695mulid2i 9093 . . . . . . . . . . 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 2456 . . . . . . . . . 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 2460 . . . . . . . . 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 6093 . . . . . . . 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 6092 . . . . . . 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 6093 . . . . . 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 6091 . . . . 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 2456 . . . 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 5731 . . 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 4re 10073 . . . . 5  |-  4  e.  RR
106 4pos 10086 . . . . 5  |-  0  <  4
107105, 106gt0ne0ii 9563 . . . 4  |-  4  =/=  0
1082, 1hvaddcli 22521 . . . . . . . . 9  |-  ( A  +h  B )  e. 
~H
109108, 2, 3, 67lnophmlem1 23519 . . . . . . . 8  |-  ( ( A  +h  B ) 
.ih  ( T `  ( A  +h  B
) ) )  e.  RR
1102, 1hvsubcli 22524 . . . . . . . . 9  |-  ( A  -h  B )  e. 
~H
111110, 2, 3, 67lnophmlem1 23519 . . . . . . . 8  |-  ( ( A  -h  B ) 
.ih  ( T `  ( A  -h  B
) ) )  e.  RR
112109, 111resubcli 9363 . . . . . . 7  |-  ( ( ( A  +h  B
)  .ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  e.  RR
113112recni 9102 . . . . . 6  |-  ( ( ( A  +h  B
)  .ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  e.  CC
11468, 94resubcli 9363 . . . . . . . 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
115114recni 9102 . . . . . . 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
11622, 115mulcli 9095 . . . . . 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
117113, 116addcli 9094 . . . . 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
118105recni 9102 . . . . 5  |-  4  e.  CC
119117, 118cjdivi 11996 . . . 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
) ) )
120107, 119ax-mp 8 . . 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 ) )
121 cjreim 11965 . . . . . . 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
) ) ) ) ) ) ) )
122112, 114, 121mp2an 654 . . . . . 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
) ) ) ) ) ) )
12382, 1, 3, 67lnophmlem1 23519 . . . . . . . . . 10  |-  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  e.  RR
12468, 123resubcli 9363 . . . . . . . . 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
125124recni 9102 . . . . . . . 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
12622, 125mulcli 9095 . . . . . . 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
127113, 126negsubi 9378 . . . . . 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
) ) ) ) ) ) )
128122, 127eqtr4i 2459 . . . . 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
) ) ) ) ) ) )
12922, 115mulneg2i 9480 . . . . . . 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 ) ) ) ) ) )
13069, 95negsubdi2i 9386 . . . . . . . 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 ) ) ) ) )
131130oveq2i 6092 . . . . . . 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
) ) ) ) ) )
132129, 131eqtr3i 2458 . . . . . 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
) ) ) ) ) )
133132oveq2i 6092 . . . . 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
) ) ) ) ) ) )
13413oveq2i 6092 . . . . . . 7  |-  ( ( A  +h  B ) 
.ih  ( T `  ( A  +h  B
) ) )  =  ( ( A  +h  B )  .ih  (
( T `  A
)  +h  ( T `
 B ) ) )
135134, 19oveq12i 6093 . . . . . 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 ) ) ) )
1363lnopaddmuli 23476 . . . . . . . . . 10  |-  ( ( _i  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  ( _i  .h  B
) ) )  =  ( ( T `  A )  +h  (
_i  .h  ( T `  B ) ) ) )
13722, 2, 1, 136mp3an 1279 . . . . . . . . 9  |-  ( T `
 ( A  +h  ( _i  .h  B
) ) )  =  ( ( T `  A )  +h  (
_i  .h  ( T `  B ) ) )
138137oveq2i 6092 . . . . . . . 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 ) ) ) )
1393lnopsubmuli 23478 . . . . . . . . . 10  |-  ( ( _i  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  -h  ( _i  .h  B
) ) )  =  ( ( T `  A )  -h  (
_i  .h  ( T `  B ) ) ) )
14022, 2, 1, 139mp3an 1279 . . . . . . . . 9  |-  ( T `
 ( A  -h  ( _i  .h  B
) ) )  =  ( ( T `  A )  -h  (
_i  .h  ( T `  B ) ) )
141140oveq2i 6092 . . . . . . . 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 ) ) ) )
142138, 141oveq12i 6093 . . . . . . 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 ) ) ) ) )
143142oveq2i 6092 . . . . . 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 ) ) ) ) ) )
144135, 143oveq12i 6093 . . . . 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 ) ) ) ) ) ) )
145128, 133, 1443eqtri 2460 . . . 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 ) ) ) ) ) ) )
146 cjre 11944 . . . . 5  |-  ( 4  e.  RR  ->  (
* `  4 )  =  4 )
147105, 146ax-mp 8 . . . 4  |-  ( * `
 4 )  =  4
148145, 147oveq12i 6093 . . 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 )
149104, 120, 1483eqtrri 2461 . 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 ) ) )
1502, 8, 1, 6polid2i 22659 . 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 )
1516, 1his1i 22602 . 2  |-  ( ( T `  A ) 
.ih  B )  =  ( * `  ( B  .ih  ( T `  A ) ) )
152149, 150, 1513eqtr4i 2466 1  |-  ( A 
.ih  ( T `  B ) )  =  ( ( T `  A )  .ih  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991   _ici 8992    + caddc 8993    x. cmul 8995    - cmin 9291   -ucneg 9292    / cdiv 9677   4c4 10051   *ccj 11901   ~Hchil 22422    +h cva 22423    .h csm 22424    .ih csp 22425    -h cmv 22428   LinOpclo 22450
This theorem is referenced by:  lnophmi  23521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-hilex 22502  ax-hfvadd 22503  ax-hvcom 22504  ax-hvass 22505  ax-hv0cl 22506  ax-hvaddid 22507  ax-hfvmul 22508  ax-hvmulid 22509  ax-hvmulass 22510  ax-hvdistr1 22511  ax-hvdistr2 22512  ax-hvmul0 22513  ax-hfi 22581  ax-his1 22584  ax-his2 22585  ax-his3 22586
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-2 10058  df-3 10059  df-4 10060  df-cj 11904  df-re 11905  df-im 11906  df-hvsub 22474  df-lnop 23344
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