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Theorem lnopeq0lem1 23508
Description: Lemma for lnopeq0i 23510. Apply the generalized polarization identity polid2i 22659 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 23472 . . . . 5  |-  T : ~H
--> ~H
43ffvelrni 5869 . . . 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 5869 . . . 4  |-  ( B  e.  ~H  ->  ( T `  B )  e.  ~H )
86, 7ax-mp 8 . . 3  |-  ( T `
 B )  e. 
~H
95, 6, 8, 1polid2i 22659 . 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 23474 . . . . . . 7  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  B ) )  =  ( ( T `
 A )  +h  ( T `  B
) ) )
111, 6, 10mp2an 654 . . . . . 6  |-  ( T `
 ( A  +h  B ) )  =  ( ( T `  A )  +h  ( T `  B )
)
1211oveq1i 6091 . . . . 5  |-  ( ( T `  ( A  +h  B ) ) 
.ih  ( A  +h  B ) )  =  ( ( ( T `
 A )  +h  ( T `  B
) )  .ih  ( A  +h  B ) )
132lnopsubi 23477 . . . . . . 7  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  -h  B ) )  =  ( ( T `
 A )  -h  ( T `  B
) ) )
141, 6, 13mp2an 654 . . . . . 6  |-  ( T `
 ( A  -h  B ) )  =  ( ( T `  A )  -h  ( T `  B )
)
1514oveq1i 6091 . . . . 5  |-  ( ( T `  ( A  -h  B ) ) 
.ih  ( A  -h  B ) )  =  ( ( ( T `
 A )  -h  ( T `  B
) )  .ih  ( A  -h  B ) )
1612, 15oveq12i 6093 . . . 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 9049 . . . . . . . 8  |-  _i  e.  CC
182lnopaddmuli 23476 . . . . . . . 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 1279 . . . . . . 7  |-  ( T `
 ( A  +h  ( _i  .h  B
) ) )  =  ( ( T `  A )  +h  (
_i  .h  ( T `  B ) ) )
2019oveq1i 6091 . . . . . 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 23478 . . . . . . . 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 1279 . . . . . . 7  |-  ( T `
 ( A  -h  ( _i  .h  B
) ) )  =  ( ( T `  A )  -h  (
_i  .h  ( T `  B ) ) )
2322oveq1i 6091 . . . . . 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 6093 . . . . 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 6092 . . . 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 6093 . . 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 6091 . 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 2459 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 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   CCcc 8988   _ici 8992    + caddc 8993    x. cmul 8995    - cmin 9291    / cdiv 9677   4c4 10051   ~Hchil 22422    +h cva 22423    .h csm 22424    .ih csp 22425    -h cmv 22428   LinOpclo 22450
This theorem is referenced by:  lnopeq0lem2  23509
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-hilex 22502  ax-hfvadd 22503  ax-hvass 22505  ax-hv0cl 22506  ax-hvaddid 22507  ax-hfvmul 22508  ax-hvmulid 22509  ax-hvdistr2 22512  ax-hvmul0 22513  ax-hfi 22581  ax-his1 22584  ax-his2 22585  ax-his3 22586
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-2 10058  df-3 10059  df-4 10060  df-cj 11904  df-re 11905  df-im 11906  df-hvsub 22474  df-lnop 23344
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