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Theorem lnopeq0lem1 27122
Description: Lemma for lnopeq0i 27124. Apply the generalized polarization identity polid2i 26272 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 27086 . . . . 5  |-  T : ~H
--> ~H
43ffvelrni 6006 . . . 4  |-  ( A  e.  ~H  ->  ( T `  A )  e.  ~H )
51, 4ax-mp 5 . . 3  |-  ( T `
 A )  e. 
~H
6 lnopeq0lem1.3 . . 3  |-  B  e. 
~H
73ffvelrni 6006 . . . 4  |-  ( B  e.  ~H  ->  ( T `  B )  e.  ~H )
86, 7ax-mp 5 . . 3  |-  ( T `
 B )  e. 
~H
95, 6, 8, 1polid2i 26272 . 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 27088 . . . . . . 7  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  B ) )  =  ( ( T `
 A )  +h  ( T `  B
) ) )
111, 6, 10mp2an 670 . . . . . 6  |-  ( T `
 ( A  +h  B ) )  =  ( ( T `  A )  +h  ( T `  B )
)
1211oveq1i 6280 . . . . 5  |-  ( ( T `  ( A  +h  B ) ) 
.ih  ( A  +h  B ) )  =  ( ( ( T `
 A )  +h  ( T `  B
) )  .ih  ( A  +h  B ) )
132lnopsubi 27091 . . . . . . 7  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  -h  B ) )  =  ( ( T `
 A )  -h  ( T `  B
) ) )
141, 6, 13mp2an 670 . . . . . 6  |-  ( T `
 ( A  -h  B ) )  =  ( ( T `  A )  -h  ( T `  B )
)
1514oveq1i 6280 . . . . 5  |-  ( ( T `  ( A  -h  B ) ) 
.ih  ( A  -h  B ) )  =  ( ( ( T `
 A )  -h  ( T `  B
) )  .ih  ( A  -h  B ) )
1612, 15oveq12i 6282 . . . 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 9540 . . . . . . . 8  |-  _i  e.  CC
182lnopaddmuli 27090 . . . . . . . 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 1322 . . . . . . 7  |-  ( T `
 ( A  +h  ( _i  .h  B
) ) )  =  ( ( T `  A )  +h  (
_i  .h  ( T `  B ) ) )
2019oveq1i 6280 . . . . . 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 27092 . . . . . . . 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 1322 . . . . . . 7  |-  ( T `
 ( A  -h  ( _i  .h  B
) ) )  =  ( ( T `  A )  -h  (
_i  .h  ( T `  B ) ) )
2322oveq1i 6280 . . . . . 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 6282 . . . . 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 6281 . . . 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 6282 . . 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 6280 . 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 2486 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 setvar class
Syntax hints:    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270   CCcc 9479   _ici 9483    + caddc 9484    x. cmul 9486    - cmin 9796    / cdiv 10202   4c4 10583   ~Hchil 26034    +h cva 26035    .h csm 26036    .ih csp 26037    -h cmv 26040   LinOpclo 26062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-hilex 26114  ax-hfvadd 26115  ax-hvass 26117  ax-hv0cl 26118  ax-hvaddid 26119  ax-hfvmul 26120  ax-hvmulid 26121  ax-hvdistr2 26124  ax-hvmul0 26125  ax-hfi 26194  ax-his1 26197  ax-his2 26198  ax-his3 26199
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-2 10590  df-3 10591  df-4 10592  df-cj 13014  df-re 13015  df-im 13016  df-hvsub 26086  df-lnop 26958
This theorem is referenced by:  lnopeq0lem2  27123
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