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Theorem lnophmlem2 25553
Description: Lemma for lnophmi 25554. (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 25505 . . . . . . . 8  |-  T : ~H
--> ~H
54ffvelrni 5938 . . . . . . 7  |-  ( A  e.  ~H  ->  ( T `  A )  e.  ~H )
62, 5ax-mp 5 . . . . . 6  |-  ( T `
 A )  e. 
~H
74ffvelrni 5938 . . . . . . 7  |-  ( B  e.  ~H  ->  ( T `  B )  e.  ~H )
81, 7ax-mp 5 . . . . . 6  |-  ( T `
 B )  e. 
~H
91, 6, 2, 8polid2i 24691 . . . . 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 24553 . . . . . . . . 9  |-  ( B  +h  A )  =  ( A  +h  B
)
118, 6hvcomi 24553 . . . . . . . . . 10  |-  ( ( T `  B )  +h  ( T `  A ) )  =  ( ( T `  A )  +h  ( T `  B )
)
123lnopaddi 25507 . . . . . . . . . . 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 2482 . . . . . . . . 9  |-  ( ( T `  B )  +h  ( T `  A ) )  =  ( T `  ( A  +h  B ) )
1510, 14oveq12i 6199 . . . . . . . 8  |-  ( ( B  +h  A ) 
.ih  ( ( T `
 B )  +h  ( T `  A
) ) )  =  ( ( A  +h  B )  .ih  ( T `  ( A  +h  B ) ) )
161, 2, 8, 6hisubcomi 24638 . . . . . . . . 9  |-  ( ( B  -h  A ) 
.ih  ( ( T `
 B )  -h  ( T `  A
) ) )  =  ( ( A  -h  B )  .ih  (
( T `  A
)  -h  ( T `
 B ) ) )
173lnopsubi 25510 . . . . . . . . . . 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 6198 . . . . . . . . 9  |-  ( ( A  -h  B ) 
.ih  ( T `  ( A  -h  B
) ) )  =  ( ( A  -h  B )  .ih  (
( T `  A
)  -h  ( T `
 B ) ) )
2016, 19eqtr4i 2482 . . . . . . . 8  |-  ( ( B  -h  A ) 
.ih  ( ( T `
 B )  -h  ( T `  A
) ) )  =  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) )
2115, 20oveq12i 6199 . . . . . . 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 9439 . . . . . . . . . . 11  |-  _i  e.  CC
2322, 1hvmulcli 24548 . . . . . . . . . . . 12  |-  ( _i  .h  B )  e. 
~H
242, 23hvsubcli 24555 . . . . . . . . . . 11  |-  ( A  -h  ( _i  .h  B ) )  e. 
~H
254ffvelrni 5938 . . . . . . . . . . . 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 24623 . . . . . . . . . 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 24586 . . . . . . . . . . . 12  |-  ( _i  .h  ( A  -h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  -h  (
_i  .h  ( _i  .h  B ) ) )
2922, 2hvmulcli 24548 . . . . . . . . . . . . . 14  |-  ( _i  .h  A )  e. 
~H
3022, 23hvmulcli 24548 . . . . . . . . . . . . . 14  |-  ( _i  .h  ( _i  .h  B ) )  e. 
~H
3129, 30hvsubvali 24554 . . . . . . . . . . . . 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 24580 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  _i )  .h  B )  =  ( _i  .h  (
_i  .h  B )
)
3332oveq2i 6198 . . . . . . . . . . . . . . 15  |-  ( -u
1  .h  ( ( _i  x.  _i )  .h  B ) )  =  ( -u 1  .h  ( _i  .h  (
_i  .h  B )
) )
34 ixi 10063 . . . . . . . . . . . . . . . . . . 19  |-  ( _i  x.  _i )  = 
-u 1
3534oveq2i 6198 . . . . . . . . . . . . . . . . . 18  |-  ( -u
1  x.  ( _i  x.  _i ) )  =  ( -u 1  x.  -u 1 )
36 ax-1cn 9438 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  CC
3736, 36mul2negi 9890 . . . . . . . . . . . . . . . . . 18  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
38 1t1e1 10567 . . . . . . . . . . . . . . . . . 18  |-  ( 1  x.  1 )  =  1
3935, 37, 383eqtri 2483 . . . . . . . . . . . . . . . . 17  |-  ( -u
1  x.  ( _i  x.  _i ) )  =  1
4039oveq1i 6197 . . . . . . . . . . . . . . . 16  |-  ( (
-u 1  x.  (
_i  x.  _i )
)  .h  B )  =  ( 1  .h  B )
41 neg1cn 10523 . . . . . . . . . . . . . . . . 17  |-  -u 1  e.  CC
4222, 22mulcli 9489 . . . . . . . . . . . . . . . . 17  |-  ( _i  x.  _i )  e.  CC
4341, 42, 1hvmulassi 24580 . . . . . . . . . . . . . . . 16  |-  ( (
-u 1  x.  (
_i  x.  _i )
)  .h  B )  =  ( -u 1  .h  ( ( _i  x.  _i )  .h  B
) )
44 ax-hvmulid 24540 . . . . . . . . . . . . . . . . 17  |-  ( B  e.  ~H  ->  (
1  .h  B )  =  B )
451, 44ax-mp 5 . . . . . . . . . . . . . . . 16  |-  ( 1  .h  B )  =  B
4640, 43, 453eqtr3i 2487 . . . . . . . . . . . . . . 15  |-  ( -u
1  .h  ( ( _i  x.  _i )  .h  B ) )  =  B
4733, 46eqtr3i 2481 . . . . . . . . . . . . . 14  |-  ( -u
1  .h  ( _i  .h  ( _i  .h  B ) ) )  =  B
4847oveq2i 6198 . . . . . . . . . . . . 13  |-  ( ( _i  .h  A )  +h  ( -u 1  .h  ( _i  .h  (
_i  .h  B )
) ) )  =  ( ( _i  .h  A )  +h  B
)
4931, 48eqtri 2479 . . . . . . . . . . . 12  |-  ( ( _i  .h  A )  -h  ( _i  .h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  +h  B
)
5029, 1hvcomi 24553 . . . . . . . . . . . 12  |-  ( ( _i  .h  A )  +h  B )  =  ( B  +h  (
_i  .h  A )
)
5128, 49, 503eqtri 2483 . . . . . . . . . . 11  |-  ( _i  .h  ( A  -h  ( _i  .h  B
) ) )  =  ( B  +h  (
_i  .h  A )
)
5251fveq2i 5789 . . . . . . . . . . . 12  |-  ( T `
 ( _i  .h  ( A  -h  (
_i  .h  B )
) ) )  =  ( T `  ( B  +h  ( _i  .h  A ) ) )
533lnopmuli 25508 . . . . . . . . . . . . 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 25509 . . . . . . . . . . . . 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 1315 . . . . . . . . . . . 12  |-  ( T `
 ( B  +h  ( _i  .h  A
) ) )  =  ( ( T `  B )  +h  (
_i  .h  ( T `  A ) ) )
5752, 54, 563eqtr3i 2487 . . . . . . . . . . 11  |-  ( _i  .h  ( T `  ( A  -h  (
_i  .h  B )
) ) )  =  ( ( T `  B )  +h  (
_i  .h  ( T `  A ) ) )
5851, 57oveq12i 6199 . . . . . . . . . 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 12747 . . . . . . . . . . . . . 14  |-  ( * `
 _i )  = 
-u _i
6059oveq2i 6198 . . . . . . . . . . . . 13  |-  ( _i  x.  ( * `  _i ) )  =  ( _i  x.  -u _i )
6122, 22mulneg2i 9889 . . . . . . . . . . . . 13  |-  ( _i  x.  -u _i )  = 
-u ( _i  x.  _i )
6234negeqi 9701 . . . . . . . . . . . . . 14  |-  -u (
_i  x.  _i )  =  -u -u 1
63 negneg1e1 10527 . . . . . . . . . . . . . 14  |-  -u -u 1  =  1
6462, 63eqtri 2479 . . . . . . . . . . . . 13  |-  -u (
_i  x.  _i )  =  1
6560, 61, 643eqtri 2483 . . . . . . . . . . . 12  |-  ( _i  x.  ( * `  _i ) )  =  1
6665oveq1i 6197 . . . . . . . . . . 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 25552 . . . . . . . . . . . . 13  |-  ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  e.  RR
6968recni 9496 . . . . . . . . . . . 12  |-  ( ( A  -h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  -h  (
_i  .h  B )
) ) )  e.  CC
7069mulid2i 9487 . . . . . . . . . . 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 2479 . . . . . . . . . 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 2487 . . . . . . . . 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 24548 . . . . . . . . . . . 12  |-  ( _i  .h  ( T `  A ) )  e. 
~H
741, 29, 8, 73hisubcomi 24638 . . . . . . . . . . 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 6197 . . . . . . . . . . . . . . 15  |-  ( ( _i  x.  _i )  .h  B )  =  ( -u 1  .h  B )
7632, 75eqtr3i 2481 . . . . . . . . . . . . . 14  |-  ( _i  .h  ( _i  .h  B ) )  =  ( -u 1  .h  B )
7776oveq2i 6198 . . . . . . . . . . . . 13  |-  ( ( _i  .h  A )  +h  ( _i  .h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  +h  ( -u 1  .h  B ) )
7822, 2, 23hvdistr1i 24585 . . . . . . . . . . . . 13  |-  ( _i  .h  ( A  +h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  +h  (
_i  .h  ( _i  .h  B ) ) )
7929, 1hvsubvali 24554 . . . . . . . . . . . . 13  |-  ( ( _i  .h  A )  -h  B )  =  ( ( _i  .h  A )  +h  ( -u 1  .h  B ) )
8077, 78, 793eqtr4i 2489 . . . . . . . . . . . 12  |-  ( _i  .h  ( A  +h  ( _i  .h  B
) ) )  =  ( ( _i  .h  A )  -h  B
)
8180fveq2i 5789 . . . . . . . . . . . . 13  |-  ( T `
 ( _i  .h  ( A  +h  (
_i  .h  B )
) ) )  =  ( T `  (
( _i  .h  A
)  -h  B ) )
822, 23hvaddcli 24552 . . . . . . . . . . . . . 14  |-  ( A  +h  ( _i  .h  B ) )  e. 
~H
833lnopmuli 25508 . . . . . . . . . . . . . 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 25512 . . . . . . . . . . . . . 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 1315 . . . . . . . . . . . . 13  |-  ( T `
 ( ( _i  .h  A )  -h  B ) )  =  ( ( _i  .h  ( T `  A ) )  -h  ( T `
 B ) )
8781, 84, 863eqtr3i 2487 . . . . . . . . . . . 12  |-  ( _i  .h  ( T `  ( A  +h  (
_i  .h  B )
) ) )  =  ( ( _i  .h  ( T `  A ) )  -h  ( T `
 B ) )
8880, 87oveq12i 6199 . . . . . . . . . . 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 2482 . . . . . . . . . 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 5938 . . . . . . . . . . . 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 24623 . . . . . . . . . 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 6197 . . . . . . . . . . 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 25552 . . . . . . . . . . . . 13  |-  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  e.  RR
9594recni 9496 . . . . . . . . . . . 12  |-  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  e.  CC
9695mulid2i 9487 . . . . . . . . . . 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 2479 . . . . . . . . . 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 2483 . . . . . . . . 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 6199 . . . . . . . 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 6198 . . . . . . 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 6199 . . . . . 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 6197 . . . . 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 2479 . . . 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 5789 . . 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 10516 . . . 4  |-  4  =/=  0
1062, 1hvaddcli 24552 . . . . . . . . 9  |-  ( A  +h  B )  e. 
~H
107106, 2, 3, 67lnophmlem1 25552 . . . . . . . 8  |-  ( ( A  +h  B ) 
.ih  ( T `  ( A  +h  B
) ) )  e.  RR
1082, 1hvsubcli 24555 . . . . . . . . 9  |-  ( A  -h  B )  e. 
~H
109108, 2, 3, 67lnophmlem1 25552 . . . . . . . 8  |-  ( ( A  -h  B ) 
.ih  ( T `  ( A  -h  B
) ) )  e.  RR
110107, 109resubcli 9769 . . . . . . 7  |-  ( ( ( A  +h  B
)  .ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  e.  RR
111110recni 9496 . . . . . 6  |-  ( ( ( A  +h  B
)  .ih  ( T `  ( A  +h  B
) ) )  -  ( ( A  -h  B )  .ih  ( T `  ( A  -h  B ) ) ) )  e.  CC
11268, 94resubcli 9769 . . . . . . . 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 9496 . . . . . . 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 9489 . . . . . 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 9488 . . . . 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 10496 . . . . . 6  |-  4  e.  RR
117116recni 9496 . . . . 5  |-  4  e.  CC
118115, 117cjdivi 12779 . . . 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 12748 . . . . . . 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 25552 . . . . . . . . . 10  |-  ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( T `  ( A  +h  (
_i  .h  B )
) ) )  e.  RR
12368, 122resubcli 9769 . . . . . . . . 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 9496 . . . . . . . 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 9489 . . . . . . 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 9784 . . . . . 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 2482 . . . . 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 9889 . . . . . . 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 9792 . . . . . . . 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 6198 . . . . . . 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 2481 . . . . . 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 6198 . . . . 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 6198 . . . . . . 7  |-  ( ( A  +h  B ) 
.ih  ( T `  ( A  +h  B
) ) )  =  ( ( A  +h  B )  .ih  (
( T `  A
)  +h  ( T `
 B ) ) )
134133, 19oveq12i 6199 . . . . . 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 25509 . . . . . . . . . 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 1315 . . . . . . . . 9  |-  ( T `
 ( A  +h  ( _i  .h  B
) ) )  =  ( ( T `  A )  +h  (
_i  .h  ( T `  B ) ) )
137136oveq2i 6198 . . . . . . . 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 25511 . . . . . . . . . 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 1315 . . . . . . . . 9  |-  ( T `
 ( A  -h  ( _i  .h  B
) ) )  =  ( ( T `  A )  -h  (
_i  .h  ( T `  B ) ) )
140139oveq2i 6198 . . . . . . . 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 6199 . . . . . . 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 6198 . . . . . 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 6199 . . . . 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 2483 . . . 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 12727 . . . . 5  |-  ( 4  e.  RR  ->  (
* `  4 )  =  4 )
146116, 145ax-mp 5 . . . 4  |-  ( * `
 4 )  =  4
147144, 146oveq12i 6199 . . 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 2484 . 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 24691 . 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 24634 . 2  |-  ( ( T `  A ) 
.ih  B )  =  ( * `  ( B  .ih  ( T `  A ) ) )
151148, 149, 1503eqtr4i 2489 1  |-  ( A 
.ih  ( T `  B ) )  =  ( ( T `  A )  .ih  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758    =/= wne 2642   A.wral 2793   ` cfv 5513  (class class class)co 6187   CCcc 9378   RRcr 9379   0cc0 9380   1c1 9381   _ici 9382    + caddc 9383    x. cmul 9385    - cmin 9693   -ucneg 9694    / cdiv 10091   4c4 10471   *ccj 12684   ~Hchil 24453    +h cva 24454    .h csm 24455    .ih csp 24456    -h cmv 24459   LinOpclo 24481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457  ax-hilex 24533  ax-hfvadd 24534  ax-hvcom 24535  ax-hvass 24536  ax-hv0cl 24537  ax-hvaddid 24538  ax-hfvmul 24539  ax-hvmulid 24540  ax-hvmulass 24541  ax-hvdistr1 24542  ax-hvdistr2 24543  ax-hvmul0 24544  ax-hfi 24613  ax-his1 24616  ax-his2 24617  ax-his3 24618
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-po 4736  df-so 4737  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-er 7198  df-map 7313  df-en 7408  df-dom 7409  df-sdom 7410  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-div 10092  df-2 10478  df-3 10479  df-4 10480  df-cj 12687  df-re 12688  df-im 12689  df-hvsub 24505  df-lnop 25377
This theorem is referenced by:  lnophmi  25554
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