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Theorem lnophmlem2 26609
Description: Lemma for lnophmi 26610. (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 26561 . . . . . . . 8  |-  T : ~H
--> ~H
54ffvelrni 6018 . . . . . . 7  |-  ( A  e.  ~H  ->  ( T `  A )  e.  ~H )
62, 5ax-mp 5 . . . . . 6  |-  ( T `
 A )  e. 
~H
74ffvelrni 6018 . . . . . . 7  |-  ( B  e.  ~H  ->  ( T `  B )  e.  ~H )
81, 7ax-mp 5 . . . . . 6  |-  ( T `
 B )  e. 
~H
91, 6, 2, 8polid2i 25747 . . . . 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 25609 . . . . . . . . 9  |-  ( B  +h  A )  =  ( A  +h  B
)
118, 6hvcomi 25609 . . . . . . . . . 10  |-  ( ( T `  B )  +h  ( T `  A ) )  =  ( ( T `  A )  +h  ( T `  B )
)
123lnopaddi 26563 . . . . . . . . . . 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 2499 . . . . . . . . 9  |-  ( ( T `  B )  +h  ( T `  A ) )  =  ( T `  ( A  +h  B ) )
1510, 14oveq12i 6294 . . . . . . . 8  |-  ( ( B  +h  A ) 
.ih  ( ( T `
 B )  +h  ( T `  A
) ) )  =  ( ( A  +h  B )  .ih  ( T `  ( A  +h  B ) ) )
161, 2, 8, 6hisubcomi 25694 . . . . . . . . 9  |-  ( ( B  -h  A ) 
.ih  ( ( T `
 B )  -h  ( T `  A
) ) )  =  ( ( A  -h  B )  .ih  (
( T `  A
)  -h  ( T `
 B ) ) )
173lnopsubi 26566 . . . . . . . . . . 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 6293 . . . . . . . . 9  |-  ( ( A  -h  B ) 
.ih  ( T `  ( A  -h  B
) ) )  =  ( ( A  -h  B )  .ih  (
( T `  A
)  -h  ( T `
 B ) ) )
2016, 19eqtr4i 2499 . . . . . . . 8  |-  ( ( B  -h  A ) 
.ih  ( ( T `
 B )  -h  ( T `  A
) ) )  =  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) )
2115, 20oveq12i 6294 . . . . . . 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 9547 . . . . . . . . . . 11  |-  _i  e.  CC
2322, 1hvmulcli 25604 . . . . . . . . . . . 12  |-  ( _i  .h  B )  e. 
~H
242, 23hvsubcli 25611 . . . . . . . . . . 11  |-  ( A  -h  ( _i  .h  B ) )  e. 
~H
254ffvelrni 6018 . . . . . . . . . . . 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 25679 . . . . . . . . . 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 25642 . . . . . . . . . . . 12  |-  ( _i  .h  ( A  -h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  -h  (
_i  .h  ( _i  .h  B ) ) )
2922, 2hvmulcli 25604 . . . . . . . . . . . . . 14  |-  ( _i  .h  A )  e. 
~H
3022, 23hvmulcli 25604 . . . . . . . . . . . . . 14  |-  ( _i  .h  ( _i  .h  B ) )  e. 
~H
3129, 30hvsubvali 25610 . . . . . . . . . . . . 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 25636 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  _i )  .h  B )  =  ( _i  .h  (
_i  .h  B )
)
3332oveq2i 6293 . . . . . . . . . . . . . . 15  |-  ( -u
1  .h  ( ( _i  x.  _i )  .h  B ) )  =  ( -u 1  .h  ( _i  .h  (
_i  .h  B )
) )
34 ixi 10174 . . . . . . . . . . . . . . . . . . 19  |-  ( _i  x.  _i )  = 
-u 1
3534oveq2i 6293 . . . . . . . . . . . . . . . . . 18  |-  ( -u
1  x.  ( _i  x.  _i ) )  =  ( -u 1  x.  -u 1 )
36 ax-1cn 9546 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  CC
3736, 36mul2negi 10000 . . . . . . . . . . . . . . . . . 18  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
38 1t1e1 10679 . . . . . . . . . . . . . . . . . 18  |-  ( 1  x.  1 )  =  1
3935, 37, 383eqtri 2500 . . . . . . . . . . . . . . . . 17  |-  ( -u
1  x.  ( _i  x.  _i ) )  =  1
4039oveq1i 6292 . . . . . . . . . . . . . . . 16  |-  ( (
-u 1  x.  (
_i  x.  _i )
)  .h  B )  =  ( 1  .h  B )
41 neg1cn 10635 . . . . . . . . . . . . . . . . 17  |-  -u 1  e.  CC
4222, 22mulcli 9597 . . . . . . . . . . . . . . . . 17  |-  ( _i  x.  _i )  e.  CC
4341, 42, 1hvmulassi 25636 . . . . . . . . . . . . . . . 16  |-  ( (
-u 1  x.  (
_i  x.  _i )
)  .h  B )  =  ( -u 1  .h  ( ( _i  x.  _i )  .h  B
) )
44 ax-hvmulid 25596 . . . . . . . . . . . . . . . . 17  |-  ( B  e.  ~H  ->  (
1  .h  B )  =  B )
451, 44ax-mp 5 . . . . . . . . . . . . . . . 16  |-  ( 1  .h  B )  =  B
4640, 43, 453eqtr3i 2504 . . . . . . . . . . . . . . 15  |-  ( -u
1  .h  ( ( _i  x.  _i )  .h  B ) )  =  B
4733, 46eqtr3i 2498 . . . . . . . . . . . . . 14  |-  ( -u
1  .h  ( _i  .h  ( _i  .h  B ) ) )  =  B
4847oveq2i 6293 . . . . . . . . . . . . 13  |-  ( ( _i  .h  A )  +h  ( -u 1  .h  ( _i  .h  (
_i  .h  B )
) ) )  =  ( ( _i  .h  A )  +h  B
)
4931, 48eqtri 2496 . . . . . . . . . . . 12  |-  ( ( _i  .h  A )  -h  ( _i  .h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  +h  B
)
5029, 1hvcomi 25609 . . . . . . . . . . . 12  |-  ( ( _i  .h  A )  +h  B )  =  ( B  +h  (
_i  .h  A )
)
5128, 49, 503eqtri 2500 . . . . . . . . . . 11  |-  ( _i  .h  ( A  -h  ( _i  .h  B
) ) )  =  ( B  +h  (
_i  .h  A )
)
5251fveq2i 5867 . . . . . . . . . . . 12  |-  ( T `
 ( _i  .h  ( A  -h  (
_i  .h  B )
) ) )  =  ( T `  ( B  +h  ( _i  .h  A ) ) )
533lnopmuli 26564 . . . . . . . . . . . . 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 26565 . . . . . . . . . . . . 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 1324 . . . . . . . . . . . 12  |-  ( T `
 ( B  +h  ( _i  .h  A
) ) )  =  ( ( T `  B )  +h  (
_i  .h  ( T `  A ) ) )
5752, 54, 563eqtr3i 2504 . . . . . . . . . . 11  |-  ( _i  .h  ( T `  ( A  -h  (
_i  .h  B )
) ) )  =  ( ( T `  B )  +h  (
_i  .h  ( T `  A ) ) )
5851, 57oveq12i 6294 . . . . . . . . . 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 12949 . . . . . . . . . . . . . 14  |-  ( * `
 _i )  = 
-u _i
6059oveq2i 6293 . . . . . . . . . . . . 13  |-  ( _i  x.  ( * `  _i ) )  =  ( _i  x.  -u _i )
6122, 22mulneg2i 9999 . . . . . . . . . . . . 13  |-  ( _i  x.  -u _i )  = 
-u ( _i  x.  _i )
6234negeqi 9809 . . . . . . . . . . . . . 14  |-  -u (
_i  x.  _i )  =  -u -u 1
63 negneg1e1 10639 . . . . . . . . . . . . . 14  |-  -u -u 1  =  1
6462, 63eqtri 2496 . . . . . . . . . . . . 13  |-  -u (
_i  x.  _i )  =  1
6560, 61, 643eqtri 2500 . . . . . . . . . . . 12  |-  ( _i  x.  ( * `  _i ) )  =  1
6665oveq1i 6292 . . . . . . . . . . 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 26608 . . . . . . . . . . . . 13  |-  ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  e.  RR
6968recni 9604 . . . . . . . . . . . 12  |-  ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  e.  CC
7069mulid2i 9595 . . . . . . . . . . 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 2496 . . . . . . . . . 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 2504 . . . . . . . . 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 25604 . . . . . . . . . . . 12  |-  ( _i  .h  ( T `  A ) )  e. 
~H
741, 29, 8, 73hisubcomi 25694 . . . . . . . . . . 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 6292 . . . . . . . . . . . . . . 15  |-  ( ( _i  x.  _i )  .h  B )  =  ( -u 1  .h  B )
7632, 75eqtr3i 2498 . . . . . . . . . . . . . 14  |-  ( _i  .h  ( _i  .h  B ) )  =  ( -u 1  .h  B )
7776oveq2i 6293 . . . . . . . . . . . . 13  |-  ( ( _i  .h  A )  +h  ( _i  .h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  +h  ( -u 1  .h  B ) )
7822, 2, 23hvdistr1i 25641 . . . . . . . . . . . . 13  |-  ( _i  .h  ( A  +h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  +h  (
_i  .h  ( _i  .h  B ) ) )
7929, 1hvsubvali 25610 . . . . . . . . . . . . 13  |-  ( ( _i  .h  A )  -h  B )  =  ( ( _i  .h  A )  +h  ( -u 1  .h  B ) )
8077, 78, 793eqtr4i 2506 . . . . . . . . . . . 12  |-  ( _i  .h  ( A  +h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  -h  B
)
8180fveq2i 5867 . . . . . . . . . . . . 13  |-  ( T `
 ( _i  .h  ( A  +h  (
_i  .h  B )
) ) )  =  ( T `  (
( _i  .h  A
)  -h  B ) )
822, 23hvaddcli 25608 . . . . . . . . . . . . . 14  |-  ( A  +h  ( _i  .h  B ) )  e. 
~H
833lnopmuli 26564 . . . . . . . . . . . . . 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 26568 . . . . . . . . . . . . . 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 1324 . . . . . . . . . . . . 13  |-  ( T `
 ( ( _i  .h  A )  -h  B ) )  =  ( ( _i  .h  ( T `  A ) )  -h  ( T `
 B ) )
8781, 84, 863eqtr3i 2504 . . . . . . . . . . . 12  |-  ( _i  .h  ( T `  ( A  +h  (
_i  .h  B )
) ) )  =  ( ( _i  .h  ( T `  A ) )  -h  ( T `
 B ) )
8880, 87oveq12i 6294 . . . . . . . . . . 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 2499 . . . . . . . . . 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 6018 . . . . . . . . . . . 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 25679 . . . . . . . . . 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 6292 . . . . . . . . . . 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 26608 . . . . . . . . . . . . 13  |-  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  e.  RR
9594recni 9604 . . . . . . . . . . . 12  |-  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  e.  CC
9695mulid2i 9595 . . . . . . . . . . 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 2496 . . . . . . . . . 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 2500 . . . . . . . . 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 6294 . . . . . . . 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 6293 . . . . . . 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 6294 . . . . . 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 6292 . . . . 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 2496 . . . 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 5867 . . 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 10628 . . . 4  |-  4  =/=  0
1062, 1hvaddcli 25608 . . . . . . . . 9  |-  ( A  +h  B )  e. 
~H
107106, 2, 3, 67lnophmlem1 26608 . . . . . . . 8  |-  ( ( A  +h  B ) 
.ih  ( T `  ( A  +h  B
) ) )  e.  RR
1082, 1hvsubcli 25611 . . . . . . . . 9  |-  ( A  -h  B )  e. 
~H
109108, 2, 3, 67lnophmlem1 26608 . . . . . . . 8  |-  ( ( A  -h  B ) 
.ih  ( T `  ( A  -h  B
) ) )  e.  RR
110107, 109resubcli 9877 . . . . . . 7  |-  ( ( ( A  +h  B
)  .ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  e.  RR
111110recni 9604 . . . . . 6  |-  ( ( ( A  +h  B
)  .ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  e.  CC
11268, 94resubcli 9877 . . . . . . . 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 9604 . . . . . . 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 9597 . . . . . 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 9596 . . . . 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 10608 . . . . . 6  |-  4  e.  RR
117116recni 9604 . . . . 5  |-  4  e.  CC
118115, 117cjdivi 12981 . . . 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 12950 . . . . . . 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 26608 . . . . . . . . . 10  |-  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  e.  RR
12368, 122resubcli 9877 . . . . . . . . 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 9604 . . . . . . . 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 9597 . . . . . . 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 9893 . . . . . 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 2499 . . . . 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 9999 . . . . . . 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 9901 . . . . . . . 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 6293 . . . . . . 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 2498 . . . . . 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 6293 . . . . 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 6293 . . . . . . 7  |-  ( ( A  +h  B ) 
.ih  ( T `  ( A  +h  B
) ) )  =  ( ( A  +h  B )  .ih  (
( T `  A
)  +h  ( T `
 B ) ) )
134133, 19oveq12i 6294 . . . . . 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 26565 . . . . . . . . . 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 1324 . . . . . . . . 9  |-  ( T `
 ( A  +h  ( _i  .h  B
) ) )  =  ( ( T `  A )  +h  (
_i  .h  ( T `  B ) ) )
137136oveq2i 6293 . . . . . . . 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 26567 . . . . . . . . . 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 1324 . . . . . . . . 9  |-  ( T `
 ( A  -h  ( _i  .h  B
) ) )  =  ( ( T `  A )  -h  (
_i  .h  ( T `  B ) ) )
140139oveq2i 6293 . . . . . . . 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 6294 . . . . . . 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 6293 . . . . . 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 6294 . . . . 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 2500 . . . 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 12929 . . . . 5  |-  ( 4  e.  RR  ->  (
* `  4 )  =  4 )
146116, 145ax-mp 5 . . . 4  |-  ( * `
 4 )  =  4
147144, 146oveq12i 6294 . . 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 2501 . 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 25747 . 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 25690 . 2  |-  ( ( T `  A ) 
.ih  B )  =  ( * `  ( B  .ih  ( T `  A ) ) )
151148, 149, 1503eqtr4i 2506 1  |-  ( A 
.ih  ( T `  B ) )  =  ( ( T `  A )  .ih  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489   _ici 9490    + caddc 9491    x. cmul 9493    - cmin 9801   -ucneg 9802    / cdiv 10202   4c4 10583   *ccj 12886   ~Hchil 25509    +h cva 25510    .h csm 25511    .ih csp 25512    -h cmv 25515   LinOpclo 25537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-hilex 25589  ax-hfvadd 25590  ax-hvcom 25591  ax-hvass 25592  ax-hv0cl 25593  ax-hvaddid 25594  ax-hfvmul 25595  ax-hvmulid 25596  ax-hvmulass 25597  ax-hvdistr1 25598  ax-hvdistr2 25599  ax-hvmul0 25600  ax-hfi 25669  ax-his1 25672  ax-his2 25673  ax-his3 25674
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-2 10590  df-3 10591  df-4 10592  df-cj 12889  df-re 12890  df-im 12891  df-hvsub 25561  df-lnop 26433
This theorem is referenced by:  lnophmi  26610
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