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Theorem lnopeq0lem1 22601
Description: Lemma for lnopeq0i 22603. Apply the generalized polarization identity polid2i 21752 to the quadratic form  (
( T `  x
) ,  x ). (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnopeq0.1  |-  T  e. 
LinOp
lnopeq0lem1.2  |-  A  e. 
~H
lnopeq0lem1.3  |-  B  e. 
~H
Assertion
Ref Expression
lnopeq0lem1  |-  ( ( T `  A ) 
.ih  B )  =  ( ( ( ( ( T `  ( A  +h  B ) ) 
.ih  ( A  +h  B ) )  -  ( ( T `  ( A  -h  B
) )  .ih  ( A  -h  B ) ) )  +  ( _i  x.  ( ( ( T `  ( A  +h  ( _i  .h  B ) ) ) 
.ih  ( A  +h  ( _i  .h  B
) ) )  -  ( ( T `  ( A  -h  (
_i  .h  B )
) )  .ih  ( A  -h  ( _i  .h  B ) ) ) ) ) )  / 
4 )

Proof of Theorem lnopeq0lem1
StepHypRef Expression
1 lnopeq0lem1.2 . . . 4  |-  A  e. 
~H
2 lnopeq0.1 . . . . . 6  |-  T  e. 
LinOp
32lnopfi 22565 . . . . 5  |-  T : ~H
--> ~H
43ffvelrni 5680 . . . 4  |-  ( A  e.  ~H  ->  ( T `  A )  e.  ~H )
51, 4ax-mp 8 . . 3  |-  ( T `
 A )  e. 
~H
6 lnopeq0lem1.3 . . 3  |-  B  e. 
~H
73ffvelrni 5680 . . . 4  |-  ( B  e.  ~H  ->  ( T `  B )  e.  ~H )
86, 7ax-mp 8 . . 3  |-  ( T `
 B )  e. 
~H
95, 6, 8, 1polid2i 21752 . 2  |-  ( ( T `  A ) 
.ih  B )  =  ( ( ( ( ( ( T `  A )  +h  ( T `  B )
)  .ih  ( A  +h  B ) )  -  ( ( ( T `
 A )  -h  ( T `  B
) )  .ih  ( A  -h  B ) ) )  +  ( _i  x.  ( ( ( ( T `  A
)  +h  ( _i  .h  ( T `  B ) ) ) 
.ih  ( A  +h  ( _i  .h  B
) ) )  -  ( ( ( T `
 A )  -h  ( _i  .h  ( T `  B )
) )  .ih  ( A  -h  ( _i  .h  B ) ) ) ) ) )  / 
4 )
102lnopaddi 22567 . . . . . . 7  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  B ) )  =  ( ( T `
 A )  +h  ( T `  B
) ) )
111, 6, 10mp2an 653 . . . . . 6  |-  ( T `
 ( A  +h  B ) )  =  ( ( T `  A )  +h  ( T `  B )
)
1211oveq1i 5884 . . . . 5  |-  ( ( T `  ( A  +h  B ) ) 
.ih  ( A  +h  B ) )  =  ( ( ( T `
 A )  +h  ( T `  B
) )  .ih  ( A  +h  B ) )
132lnopsubi 22570 . . . . . . 7  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  -h  B ) )  =  ( ( T `
 A )  -h  ( T `  B
) ) )
141, 6, 13mp2an 653 . . . . . 6  |-  ( T `
 ( A  -h  B ) )  =  ( ( T `  A )  -h  ( T `  B )
)
1514oveq1i 5884 . . . . 5  |-  ( ( T `  ( A  -h  B ) ) 
.ih  ( A  -h  B ) )  =  ( ( ( T `
 A )  -h  ( T `  B
) )  .ih  ( A  -h  B ) )
1612, 15oveq12i 5886 . . . 4  |-  ( ( ( T `  ( A  +h  B ) ) 
.ih  ( A  +h  B ) )  -  ( ( T `  ( A  -h  B
) )  .ih  ( A  -h  B ) ) )  =  ( ( ( ( T `  A )  +h  ( T `  B )
)  .ih  ( A  +h  B ) )  -  ( ( ( T `
 A )  -h  ( T `  B
) )  .ih  ( A  -h  B ) ) )
17 ax-icn 8812 . . . . . . . 8  |-  _i  e.  CC
182lnopaddmuli 22569 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  ( _i  .h  B
) ) )  =  ( ( T `  A )  +h  (
_i  .h  ( T `  B ) ) ) )
1917, 1, 6, 18mp3an 1277 . . . . . . 7  |-  ( T `
 ( A  +h  ( _i  .h  B
) ) )  =  ( ( T `  A )  +h  (
_i  .h  ( T `  B ) ) )
2019oveq1i 5884 . . . . . 6  |-  ( ( T `  ( A  +h  ( _i  .h  B ) ) ) 
.ih  ( A  +h  ( _i  .h  B
) ) )  =  ( ( ( T `
 A )  +h  ( _i  .h  ( T `  B )
) )  .ih  ( A  +h  ( _i  .h  B ) ) )
212lnopsubmuli 22571 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  -h  ( _i  .h  B
) ) )  =  ( ( T `  A )  -h  (
_i  .h  ( T `  B ) ) ) )
2217, 1, 6, 21mp3an 1277 . . . . . . 7  |-  ( T `
 ( A  -h  ( _i  .h  B
) ) )  =  ( ( T `  A )  -h  (
_i  .h  ( T `  B ) ) )
2322oveq1i 5884 . . . . . 6  |-  ( ( T `  ( A  -h  ( _i  .h  B ) ) ) 
.ih  ( A  -h  ( _i  .h  B
) ) )  =  ( ( ( T `
 A )  -h  ( _i  .h  ( T `  B )
) )  .ih  ( A  -h  ( _i  .h  B ) ) )
2420, 23oveq12i 5886 . . . . 5  |-  ( ( ( T `  ( A  +h  ( _i  .h  B ) ) ) 
.ih  ( A  +h  ( _i  .h  B
) ) )  -  ( ( T `  ( A  -h  (
_i  .h  B )
) )  .ih  ( A  -h  ( _i  .h  B ) ) ) )  =  ( ( ( ( T `  A )  +h  (
_i  .h  ( T `  B ) ) ) 
.ih  ( A  +h  ( _i  .h  B
) ) )  -  ( ( ( T `
 A )  -h  ( _i  .h  ( T `  B )
) )  .ih  ( A  -h  ( _i  .h  B ) ) ) )
2524oveq2i 5885 . . . 4  |-  ( _i  x.  ( ( ( T `  ( A  +h  ( _i  .h  B ) ) ) 
.ih  ( A  +h  ( _i  .h  B
) ) )  -  ( ( T `  ( A  -h  (
_i  .h  B )
) )  .ih  ( A  -h  ( _i  .h  B ) ) ) ) )  =  ( _i  x.  ( ( ( ( T `  A )  +h  (
_i  .h  ( T `  B ) ) ) 
.ih  ( A  +h  ( _i  .h  B
) ) )  -  ( ( ( T `
 A )  -h  ( _i  .h  ( T `  B )
) )  .ih  ( A  -h  ( _i  .h  B ) ) ) ) )
2616, 25oveq12i 5886 . . 3  |-  ( ( ( ( T `  ( A  +h  B
) )  .ih  ( A  +h  B ) )  -  ( ( T `
 ( A  -h  B ) )  .ih  ( A  -h  B
) ) )  +  ( _i  x.  (
( ( T `  ( A  +h  (
_i  .h  B )
) )  .ih  ( A  +h  ( _i  .h  B ) ) )  -  ( ( T `
 ( A  -h  ( _i  .h  B
) ) )  .ih  ( A  -h  (
_i  .h  B )
) ) ) ) )  =  ( ( ( ( ( T `
 A )  +h  ( T `  B
) )  .ih  ( A  +h  B ) )  -  ( ( ( T `  A )  -h  ( T `  B ) )  .ih  ( A  -h  B
) ) )  +  ( _i  x.  (
( ( ( T `
 A )  +h  ( _i  .h  ( T `  B )
) )  .ih  ( A  +h  ( _i  .h  B ) ) )  -  ( ( ( T `  A )  -h  ( _i  .h  ( T `  B ) ) )  .ih  ( A  -h  ( _i  .h  B ) ) ) ) ) )
2726oveq1i 5884 . 2  |-  ( ( ( ( ( T `
 ( A  +h  B ) )  .ih  ( A  +h  B
) )  -  (
( T `  ( A  -h  B ) ) 
.ih  ( A  -h  B ) ) )  +  ( _i  x.  ( ( ( T `
 ( A  +h  ( _i  .h  B
) ) )  .ih  ( A  +h  (
_i  .h  B )
) )  -  (
( T `  ( A  -h  ( _i  .h  B ) ) ) 
.ih  ( A  -h  ( _i  .h  B
) ) ) ) ) )  /  4
)  =  ( ( ( ( ( ( T `  A )  +h  ( T `  B ) )  .ih  ( A  +h  B
) )  -  (
( ( T `  A )  -h  ( T `  B )
)  .ih  ( A  -h  B ) ) )  +  ( _i  x.  ( ( ( ( T `  A )  +h  ( _i  .h  ( T `  B ) ) )  .ih  ( A  +h  ( _i  .h  B ) ) )  -  ( ( ( T `  A )  -h  ( _i  .h  ( T `  B ) ) )  .ih  ( A  -h  ( _i  .h  B ) ) ) ) ) )  / 
4 )
289, 27eqtr4i 2319 1  |-  ( ( T `  A ) 
.ih  B )  =  ( ( ( ( ( T `  ( A  +h  B ) ) 
.ih  ( A  +h  B ) )  -  ( ( T `  ( A  -h  B
) )  .ih  ( A  -h  B ) ) )  +  ( _i  x.  ( ( ( T `  ( A  +h  ( _i  .h  B ) ) ) 
.ih  ( A  +h  ( _i  .h  B
) ) )  -  ( ( T `  ( A  -h  (
_i  .h  B )
) )  .ih  ( A  -h  ( _i  .h  B ) ) ) ) ) )  / 
4 )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   CCcc 8751   _ici 8755    + caddc 8756    x. cmul 8758    - cmin 9053    / cdiv 9439   4c4 9813   ~Hchil 21515    +h cva 21516    .h csm 21517    .ih csp 21518    -h cmv 21521   LinOpclo 21543
This theorem is referenced by:  lnopeq0lem2  22602
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-hilex 21595  ax-hfvadd 21596  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvdistr2 21605  ax-hvmul0 21606  ax-hfi 21674  ax-his1 21677  ax-his2 21678  ax-his3 21679
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-2 9820  df-3 9821  df-4 9822  df-cj 11600  df-re 11601  df-im 11602  df-hvsub 21567  df-lnop 22437
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