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Theorem lnophmlem2 22597
Description: Lemma for lnophmi 22598. (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 . . . . . . 7  |-  B  e. 
~H
2 lnophmlem.1 . . . . . . . 8  |-  A  e. 
~H
3 lnophmlem.3 . . . . . . . . . 10  |-  T  e. 
LinOp
43lnopfi 22549 . . . . . . . . 9  |-  T : ~H
--> ~H
54ffvelrni 5664 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( T `  A )  e.  ~H )
62, 5ax-mp 8 . . . . . . 7  |-  ( T `
 A )  e. 
~H
74ffvelrni 5664 . . . . . . . 8  |-  ( B  e.  ~H  ->  ( T `  B )  e.  ~H )
81, 7ax-mp 8 . . . . . . 7  |-  ( T `
 B )  e. 
~H
91, 6, 2, 8polid2i 21736 . . . . . 6  |-  ( 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 21599 . . . . . . . . . 10  |-  ( B  +h  A )  =  ( A  +h  B
)
118, 6hvcomi 21599 . . . . . . . . . . 11  |-  ( ( T `  B )  +h  ( T `  A ) )  =  ( ( T `  A )  +h  ( T `  B )
)
123lnopaddi 22551 . . . . . . . . . . . 12  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  B ) )  =  ( ( T `
 A )  +h  ( T `  B
) ) )
132, 1, 12mp2an 653 . . . . . . . . . . 11  |-  ( T `
 ( A  +h  B ) )  =  ( ( T `  A )  +h  ( T `  B )
)
1411, 13eqtr4i 2306 . . . . . . . . . 10  |-  ( ( T `  B )  +h  ( T `  A ) )  =  ( T `  ( A  +h  B ) )
1510, 14oveq12i 5870 . . . . . . . . 9  |-  ( ( B  +h  A ) 
.ih  ( ( T `
 B )  +h  ( T `  A
) ) )  =  ( ( A  +h  B )  .ih  ( T `  ( A  +h  B ) ) )
161, 2, 8, 6hisubcomi 21683 . . . . . . . . . 10  |-  ( ( B  -h  A ) 
.ih  ( ( T `
 B )  -h  ( T `  A
) ) )  =  ( ( A  -h  B )  .ih  (
( T `  A
)  -h  ( T `
 B ) ) )
173lnopsubi 22554 . . . . . . . . . . . 12  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  -h  B ) )  =  ( ( T `
 A )  -h  ( T `  B
) ) )
182, 1, 17mp2an 653 . . . . . . . . . . 11  |-  ( T `
 ( A  -h  B ) )  =  ( ( T `  A )  -h  ( T `  B )
)
1918oveq2i 5869 . . . . . . . . . 10  |-  ( ( A  -h  B ) 
.ih  ( T `  ( A  -h  B
) ) )  =  ( ( A  -h  B )  .ih  (
( T `  A
)  -h  ( T `
 B ) ) )
2016, 19eqtr4i 2306 . . . . . . . . 9  |-  ( ( B  -h  A ) 
.ih  ( ( T `
 B )  -h  ( T `  A
) ) )  =  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) )
2115, 20oveq12i 5870 . . . . . . . 8  |-  ( ( ( 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 8796 . . . . . . . . . . . 12  |-  _i  e.  CC
2322, 1hvmulcli 21594 . . . . . . . . . . . . 13  |-  ( _i  .h  B )  e. 
~H
242, 23hvsubcli 21601 . . . . . . . . . . . 12  |-  ( A  -h  ( _i  .h  B ) )  e. 
~H
254ffvelrni 5664 . . . . . . . . . . . . 13  |-  ( ( A  -h  ( _i  .h  B ) )  e.  ~H  ->  ( T `  ( A  -h  ( _i  .h  B
) ) )  e. 
~H )
2624, 25ax-mp 8 . . . . . . . . . . . 12  |-  ( T `
 ( A  -h  ( _i  .h  B
) ) )  e. 
~H
2722, 22, 24, 26his35i 21668 . . . . . . . . . . 11  |-  ( ( _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 21631 . . . . . . . . . . . . 13  |-  ( _i  .h  ( A  -h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  -h  (
_i  .h  ( _i  .h  B ) ) )
2922, 2hvmulcli 21594 . . . . . . . . . . . . . . 15  |-  ( _i  .h  A )  e. 
~H
3022, 23hvmulcli 21594 . . . . . . . . . . . . . . 15  |-  ( _i  .h  ( _i  .h  B ) )  e. 
~H
3129, 30hvsubvali 21600 . . . . . . . . . . . . . 14  |-  ( ( _i  .h  A )  -h  ( _i  .h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  +h  ( -u 1  .h  ( _i  .h  ( _i  .h  B ) ) ) )
3222, 22, 1hvmulassi 21625 . . . . . . . . . . . . . . . . 17  |-  ( ( _i  x.  _i )  .h  B )  =  ( _i  .h  (
_i  .h  B )
)
3332oveq2i 5869 . . . . . . . . . . . . . . . 16  |-  ( -u
1  .h  ( ( _i  x.  _i )  .h  B ) )  =  ( -u 1  .h  ( _i  .h  (
_i  .h  B )
) )
34 ixi 9397 . . . . . . . . . . . . . . . . . . . 20  |-  ( _i  x.  _i )  = 
-u 1
3534oveq2i 5869 . . . . . . . . . . . . . . . . . . 19  |-  ( -u
1  x.  ( _i  x.  _i ) )  =  ( -u 1  x.  -u 1 )
36 ax-1cn 8795 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  CC
3736, 36mul2negi 9227 . . . . . . . . . . . . . . . . . . 19  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
38 1t1e1 9870 . . . . . . . . . . . . . . . . . . 19  |-  ( 1  x.  1 )  =  1
3935, 37, 383eqtri 2307 . . . . . . . . . . . . . . . . . 18  |-  ( -u
1  x.  ( _i  x.  _i ) )  =  1
4039oveq1i 5868 . . . . . . . . . . . . . . . . 17  |-  ( (
-u 1  x.  (
_i  x.  _i )
)  .h  B )  =  ( 1  .h  B )
41 neg1cn 9813 . . . . . . . . . . . . . . . . . 18  |-  -u 1  e.  CC
4222, 22mulcli 8842 . . . . . . . . . . . . . . . . . 18  |-  ( _i  x.  _i )  e.  CC
4341, 42, 1hvmulassi 21625 . . . . . . . . . . . . . . . . 17  |-  ( (
-u 1  x.  (
_i  x.  _i )
)  .h  B )  =  ( -u 1  .h  ( ( _i  x.  _i )  .h  B
) )
44 ax-hvmulid 21586 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  ~H  ->  (
1  .h  B )  =  B )
451, 44ax-mp 8 . . . . . . . . . . . . . . . . 17  |-  ( 1  .h  B )  =  B
4640, 43, 453eqtr3i 2311 . . . . . . . . . . . . . . . 16  |-  ( -u
1  .h  ( ( _i  x.  _i )  .h  B ) )  =  B
4733, 46eqtr3i 2305 . . . . . . . . . . . . . . 15  |-  ( -u
1  .h  ( _i  .h  ( _i  .h  B ) ) )  =  B
4847oveq2i 5869 . . . . . . . . . . . . . 14  |-  ( ( _i  .h  A )  +h  ( -u 1  .h  ( _i  .h  (
_i  .h  B )
) ) )  =  ( ( _i  .h  A )  +h  B
)
4931, 48eqtri 2303 . . . . . . . . . . . . 13  |-  ( ( _i  .h  A )  -h  ( _i  .h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  +h  B
)
5029, 1hvcomi 21599 . . . . . . . . . . . . 13  |-  ( ( _i  .h  A )  +h  B )  =  ( B  +h  (
_i  .h  A )
)
5128, 49, 503eqtri 2307 . . . . . . . . . . . 12  |-  ( _i  .h  ( A  -h  ( _i  .h  B
) ) )  =  ( B  +h  (
_i  .h  A )
)
5251fveq2i 5528 . . . . . . . . . . . . 13  |-  ( T `
 ( _i  .h  ( A  -h  (
_i  .h  B )
) ) )  =  ( T `  ( B  +h  ( _i  .h  A ) ) )
533lnopmuli 22552 . . . . . . . . . . . . . 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 ) ) ) ) )
5422, 24, 53mp2an 653 . . . . . . . . . . . . 13  |-  ( T `
 ( _i  .h  ( A  -h  (
_i  .h  B )
) ) )  =  ( _i  .h  ( T `  ( A  -h  ( _i  .h  B
) ) ) )
553lnopaddmuli 22553 . . . . . . . . . . . . . 14  |-  ( ( _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 1277 . . . . . . . . . . . . 13  |-  ( T `
 ( B  +h  ( _i  .h  A
) ) )  =  ( ( T `  B )  +h  (
_i  .h  ( T `  A ) ) )
5752, 54, 563eqtr3i 2311 . . . . . . . . . . . 12  |-  ( _i  .h  ( T `  ( A  -h  (
_i  .h  B )
) ) )  =  ( ( T `  B )  +h  (
_i  .h  ( T `  A ) ) )
5851, 57oveq12i 5870 . . . . . . . . . . 11  |-  ( ( _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 11644 . . . . . . . . . . . . . . 15  |-  ( * `
 _i )  = 
-u _i
6059oveq2i 5869 . . . . . . . . . . . . . 14  |-  ( _i  x.  ( * `  _i ) )  =  ( _i  x.  -u _i )
6122, 22mulneg2i 9226 . . . . . . . . . . . . . 14  |-  ( _i  x.  -u _i )  = 
-u ( _i  x.  _i )
6234negeqi 9045 . . . . . . . . . . . . . . 15  |-  -u (
_i  x.  _i )  =  -u -u 1
6336negnegi 9116 . . . . . . . . . . . . . . 15  |-  -u -u 1  =  1
6462, 63eqtri 2303 . . . . . . . . . . . . . 14  |-  -u (
_i  x.  _i )  =  1
6560, 61, 643eqtri 2307 . . . . . . . . . . . . 13  |-  ( _i  x.  ( * `  _i ) )  =  1
6665oveq1i 5868 . . . . . . . . . . . 12  |-  ( ( _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 . . . . . . . . . . . . . . 15  |-  A. x  e.  ~H  ( x  .ih  ( T `  x ) )  e.  RR
6824, 2, 3, 67lnophmlem1 22596 . . . . . . . . . . . . . 14  |-  ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  e.  RR
6968recni 8849 . . . . . . . . . . . . 13  |-  ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  e.  CC
7069mulid2i 8840 . . . . . . . . . . . 12  |-  ( 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 2303 . . . . . . . . . . 11  |-  ( ( _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 2311 . . . . . . . . . 10  |-  ( ( 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 21594 . . . . . . . . . . . . 13  |-  ( _i  .h  ( T `  A ) )  e. 
~H
741, 29, 8, 73hisubcomi 21683 . . . . . . . . . . . 12  |-  ( ( 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 5868 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  _i )  .h  B )  =  ( -u 1  .h  B )
7632, 75eqtr3i 2305 . . . . . . . . . . . . . . 15  |-  ( _i  .h  ( _i  .h  B ) )  =  ( -u 1  .h  B )
7776oveq2i 5869 . . . . . . . . . . . . . 14  |-  ( ( _i  .h  A )  +h  ( _i  .h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  +h  ( -u 1  .h  B ) )
7822, 2, 23hvdistr1i 21630 . . . . . . . . . . . . . 14  |-  ( _i  .h  ( A  +h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  +h  (
_i  .h  ( _i  .h  B ) ) )
7929, 1hvsubvali 21600 . . . . . . . . . . . . . 14  |-  ( ( _i  .h  A )  -h  B )  =  ( ( _i  .h  A )  +h  ( -u 1  .h  B ) )
8077, 78, 793eqtr4i 2313 . . . . . . . . . . . . 13  |-  ( _i  .h  ( A  +h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  -h  B
)
8180fveq2i 5528 . . . . . . . . . . . . . 14  |-  ( T `
 ( _i  .h  ( A  +h  (
_i  .h  B )
) ) )  =  ( T `  (
( _i  .h  A
)  -h  B ) )
822, 23hvaddcli 21598 . . . . . . . . . . . . . . 15  |-  ( A  +h  ( _i  .h  B ) )  e. 
~H
833lnopmuli 22552 . . . . . . . . . . . . . . 15  |-  ( ( _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 653 . . . . . . . . . . . . . 14  |-  ( T `
 ( _i  .h  ( A  +h  (
_i  .h  B )
) ) )  =  ( _i  .h  ( T `  ( A  +h  ( _i  .h  B
) ) ) )
853lnopmulsubi 22556 . . . . . . . . . . . . . . 15  |-  ( ( _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 1277 . . . . . . . . . . . . . 14  |-  ( T `
 ( ( _i  .h  A )  -h  B ) )  =  ( ( _i  .h  ( T `  A ) )  -h  ( T `
 B ) )
8781, 84, 863eqtr3i 2311 . . . . . . . . . . . . 13  |-  ( _i  .h  ( T `  ( A  +h  (
_i  .h  B )
) ) )  =  ( ( _i  .h  ( T `  A ) )  -h  ( T `
 B ) )
8880, 87oveq12i 5870 . . . . . . . . . . . 12  |-  ( ( _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 2306 . . . . . . . . . . 11  |-  ( ( 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 5664 . . . . . . . . . . . . 13  |-  ( ( A  +h  ( _i  .h  B ) )  e.  ~H  ->  ( T `  ( A  +h  ( _i  .h  B
) ) )  e. 
~H )
9182, 90ax-mp 8 . . . . . . . . . . . 12  |-  ( T `
 ( A  +h  ( _i  .h  B
) ) )  e. 
~H
9222, 22, 82, 91his35i 21668 . . . . . . . . . . 11  |-  ( ( _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 5868 . . . . . . . . . . . 12  |-  ( ( _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 22596 . . . . . . . . . . . . . 14  |-  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  e.  RR
9594recni 8849 . . . . . . . . . . . . 13  |-  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  e.  CC
9695mulid2i 8840 . . . . . . . . . . . 12  |-  ( 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 2303 . . . . . . . . . . 11  |-  ( ( _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 2307 . . . . . . . . . 10  |-  ( ( 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 5870 . . . . . . . . 9  |-  ( ( ( 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 5869 . . . . . . . 8  |-  ( _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 5870 . . . . . . 7  |-  ( ( ( ( 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 5868 . . . . . 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 )
) ) ) ) ) )  /  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 2303 . . . . 5  |-  ( 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 5528 . . . 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
) )
105 4re 9819 . . . . . 6  |-  4  e.  RR
106 4pos 9832 . . . . . 6  |-  0  <  4
107105, 106gt0ne0ii 9309 . . . . 5  |-  4  =/=  0
1082, 1hvaddcli 21598 . . . . . . . . . 10  |-  ( A  +h  B )  e. 
~H
109108, 2, 3, 67lnophmlem1 22596 . . . . . . . . 9  |-  ( ( A  +h  B ) 
.ih  ( T `  ( A  +h  B
) ) )  e.  RR
1102, 1hvsubcli 21601 . . . . . . . . . 10  |-  ( A  -h  B )  e. 
~H
111110, 2, 3, 67lnophmlem1 22596 . . . . . . . . 9  |-  ( ( A  -h  B ) 
.ih  ( T `  ( A  -h  B
) ) )  e.  RR
112109, 111resubcli 9109 . . . . . . . 8  |-  ( ( ( A  +h  B
)  .ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  e.  RR
113112recni 8849 . . . . . . 7  |-  ( ( ( A  +h  B
)  .ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  e.  CC
11468, 94resubcli 9109 . . . . . . . . 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
115114recni 8849 . . . . . . . 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
11622, 115mulcli 8842 . . . . . . 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
117113, 116addcli 8841 . . . . . 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
) ) ) ) ) ) )  e.  CC
118105recni 8849 . . . . . 6  |-  4  e.  CC
119117, 118cjdivi 11676 . . . . 5  |-  ( 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 . . . 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 ) )  =  ( ( * `  ( ( ( ( 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 ) )
121104, 120eqtri 2303 . . 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
) )
122 cjreim 11645 . . . . . . 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
) ) ) ) ) ) ) )
123112, 114, 122mp2an 653 . . . . . 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
) ) ) ) ) ) )
12482, 1, 3, 67lnophmlem1 22596 . . . . . . . . . 10  |-  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  e.  RR
12568, 124resubcli 9109 . . . . . . . . 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
126125recni 8849 . . . . . . . 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
12722, 126mulcli 8842 . . . . . . 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
128113, 127negsubi 9124 . . . . . 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
) ) ) ) ) ) )
129123, 128eqtr4i 2306 . . . . 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
) ) ) ) ) ) )
13022, 115mulneg2i 9226 . . . . . . 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 ) ) ) ) ) )
13169, 95negsubdi2i 9132 . . . . . . . 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 ) ) ) ) )
132131oveq2i 5869 . . . . . . 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
) ) ) ) ) )
133130, 132eqtr3i 2305 . . . . . 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
) ) ) ) ) )
134133oveq2i 5869 . . . . 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
) ) ) ) ) ) )
13513oveq2i 5869 . . . . . . 7  |-  ( ( A  +h  B ) 
.ih  ( T `  ( A  +h  B
) ) )  =  ( ( A  +h  B )  .ih  (
( T `  A
)  +h  ( T `
 B ) ) )
136135, 19oveq12i 5870 . . . . . 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 ) ) ) )
1373lnopaddmuli 22553 . . . . . . . . . 10  |-  ( ( _i  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  ( _i  .h  B
) ) )  =  ( ( T `  A )  +h  (
_i  .h  ( T `  B ) ) ) )
13822, 2, 1, 137mp3an 1277 . . . . . . . . 9  |-  ( T `
 ( A  +h  ( _i  .h  B
) ) )  =  ( ( T `  A )  +h  (
_i  .h  ( T `  B ) ) )
139138oveq2i 5869 . . . . . . . 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 ) ) ) )
1403lnopsubmuli 22555 . . . . . . . . . 10  |-  ( ( _i  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  -h  ( _i  .h  B
) ) )  =  ( ( T `  A )  -h  (
_i  .h  ( T `  B ) ) ) )
14122, 2, 1, 140mp3an 1277 . . . . . . . . 9  |-  ( T `
 ( A  -h  ( _i  .h  B
) ) )  =  ( ( T `  A )  -h  (
_i  .h  ( T `  B ) ) )
142141oveq2i 5869 . . . . . . . 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 ) ) ) )
143139, 142oveq12i 5870 . . . . . . 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 ) ) ) ) )
144143oveq2i 5869 . . . . . 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 ) ) ) ) ) )
145136, 144oveq12i 5870 . . . . 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 ) ) ) ) ) ) )
146129, 134, 1453eqtri 2307 . . . 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 ) ) ) ) ) ) )
147 cjre 11624 . . . . 5  |-  ( 4  e.  RR  ->  (
* `  4 )  =  4 )
148105, 147ax-mp 8 . . . 4  |-  ( * `
 4 )  =  4
149146, 148oveq12i 5870 . . 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 )
150121, 149eqtr2i 2304 . 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 ) ) )
1512, 8, 1, 6polid2i 21736 . 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 )
1526, 1his1i 21679 . 2  |-  ( ( T `  A ) 
.ih  B )  =  ( * `  ( B  .ih  ( T `  A ) ) )
153150, 151, 1523eqtr4i 2313 1  |-  ( A 
.ih  ( T `  B ) )  =  ( ( T `  A )  .ih  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   4c4 9797   *ccj 11581   ~Hchil 21499    +h cva 21500    .h csm 21501    .ih csp 21502    -h cmv 21505   LinOpclo 21527
This theorem is referenced by:  lnophmi  22598
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvdistr1 21588  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-3 9805  df-4 9806  df-cj 11584  df-re 11585  df-im 11586  df-hvsub 21551  df-lnop 22421
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