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Theorem polidi 21729
Description: Polarization identity. Recovers inner product from norm. Exercise 4(a) of [ReedSimon] p. 63. The outermost operation is + instead of - due to our mathematicians' (rather than physicists') version of axiom ax-his3 21655. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
Hypotheses
Ref Expression
polid.1  |-  A  e. 
~H
polid.2  |-  B  e. 
~H
Assertion
Ref Expression
polidi  |-  ( A 
.ih  B )  =  ( ( ( ( ( normh `  ( A  +h  B ) ) ^
2 )  -  (
( normh `  ( A  -h  B ) ) ^
2 ) )  +  ( _i  x.  (
( ( normh `  ( A  +h  ( _i  .h  B ) ) ) ^ 2 )  -  ( ( normh `  ( A  -h  ( _i  .h  B ) ) ) ^ 2 ) ) ) )  /  4
)

Proof of Theorem polidi
StepHypRef Expression
1 polid.1 . . 3  |-  A  e. 
~H
2 polid.2 . . 3  |-  B  e. 
~H
31, 2, 2, 1polid2i 21728 . 2  |-  ( A 
.ih  B )  =  ( ( ( ( ( A  +h  B
)  .ih  ( A  +h  B ) )  -  ( ( A  -h  B )  .ih  ( A  -h  B ) ) )  +  ( _i  x.  ( ( ( A  +h  ( _i  .h  B ) ) 
.ih  ( A  +h  ( _i  .h  B
) ) )  -  ( ( A  -h  ( _i  .h  B
) )  .ih  ( A  -h  ( _i  .h  B ) ) ) ) ) )  / 
4 )
41, 2hvaddcli 21590 . . . . . 6  |-  ( A  +h  B )  e. 
~H
54normsqi 21703 . . . . 5  |-  ( (
normh `  ( A  +h  B ) ) ^
2 )  =  ( ( A  +h  B
)  .ih  ( A  +h  B ) )
61, 2hvsubcli 21593 . . . . . 6  |-  ( A  -h  B )  e. 
~H
76normsqi 21703 . . . . 5  |-  ( (
normh `  ( A  -h  B ) ) ^
2 )  =  ( ( A  -h  B
)  .ih  ( A  -h  B ) )
85, 7oveq12i 5831 . . . 4  |-  ( ( ( normh `  ( A  +h  B ) ) ^
2 )  -  (
( normh `  ( A  -h  B ) ) ^
2 ) )  =  ( ( ( A  +h  B )  .ih  ( A  +h  B
) )  -  (
( A  -h  B
)  .ih  ( A  -h  B ) ) )
9 ax-icn 8791 . . . . . . . . 9  |-  _i  e.  CC
109, 2hvmulcli 21586 . . . . . . . 8  |-  ( _i  .h  B )  e. 
~H
111, 10hvaddcli 21590 . . . . . . 7  |-  ( A  +h  ( _i  .h  B ) )  e. 
~H
1211normsqi 21703 . . . . . 6  |-  ( (
normh `  ( A  +h  ( _i  .h  B
) ) ) ^
2 )  =  ( ( A  +h  (
_i  .h  B )
)  .ih  ( A  +h  ( _i  .h  B
) ) )
131, 10hvsubcli 21593 . . . . . . 7  |-  ( A  -h  ( _i  .h  B ) )  e. 
~H
1413normsqi 21703 . . . . . 6  |-  ( (
normh `  ( A  -h  ( _i  .h  B
) ) ) ^
2 )  =  ( ( A  -h  (
_i  .h  B )
)  .ih  ( A  -h  ( _i  .h  B
) ) )
1512, 14oveq12i 5831 . . . . 5  |-  ( ( ( normh `  ( A  +h  ( _i  .h  B
) ) ) ^
2 )  -  (
( normh `  ( A  -h  ( _i  .h  B
) ) ) ^
2 ) )  =  ( ( ( A  +h  ( _i  .h  B ) )  .ih  ( A  +h  (
_i  .h  B )
) )  -  (
( A  -h  (
_i  .h  B )
)  .ih  ( A  -h  ( _i  .h  B
) ) ) )
1615oveq2i 5830 . . . 4  |-  ( _i  x.  ( ( (
normh `  ( A  +h  ( _i  .h  B
) ) ) ^
2 )  -  (
( normh `  ( A  -h  ( _i  .h  B
) ) ) ^
2 ) ) )  =  ( _i  x.  ( ( ( A  +h  ( _i  .h  B ) )  .ih  ( A  +h  (
_i  .h  B )
) )  -  (
( A  -h  (
_i  .h  B )
)  .ih  ( A  -h  ( _i  .h  B
) ) ) ) )
178, 16oveq12i 5831 . . 3  |-  ( ( ( ( normh `  ( A  +h  B ) ) ^ 2 )  -  ( ( normh `  ( A  -h  B ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( normh `  ( A  +h  (
_i  .h  B )
) ) ^ 2 )  -  ( (
normh `  ( A  -h  ( _i  .h  B
) ) ) ^
2 ) ) ) )  =  ( ( ( ( A  +h  B )  .ih  ( A  +h  B ) )  -  ( ( A  -h  B )  .ih  ( A  -h  B
) ) )  +  ( _i  x.  (
( ( A  +h  ( _i  .h  B
) )  .ih  ( A  +h  ( _i  .h  B ) ) )  -  ( ( A  -h  ( _i  .h  B ) )  .ih  ( A  -h  (
_i  .h  B )
) ) ) ) )
1817oveq1i 5829 . 2  |-  ( ( ( ( ( normh `  ( A  +h  B
) ) ^ 2 )  -  ( (
normh `  ( A  -h  B ) ) ^
2 ) )  +  ( _i  x.  (
( ( normh `  ( A  +h  ( _i  .h  B ) ) ) ^ 2 )  -  ( ( normh `  ( A  -h  ( _i  .h  B ) ) ) ^ 2 ) ) ) )  /  4
)  =  ( ( ( ( ( A  +h  B )  .ih  ( A  +h  B
) )  -  (
( A  -h  B
)  .ih  ( A  -h  B ) ) )  +  ( _i  x.  ( ( ( A  +h  ( _i  .h  B ) )  .ih  ( A  +h  (
_i  .h  B )
) )  -  (
( A  -h  (
_i  .h  B )
)  .ih  ( A  -h  ( _i  .h  B
) ) ) ) ) )  /  4
)
193, 18eqtr4i 2307 1  |-  ( A 
.ih  B )  =  ( ( ( ( ( normh `  ( A  +h  B ) ) ^
2 )  -  (
( normh `  ( A  -h  B ) ) ^
2 ) )  +  ( _i  x.  (
( ( normh `  ( A  +h  ( _i  .h  B ) ) ) ^ 2 )  -  ( ( normh `  ( A  -h  ( _i  .h  B ) ) ) ^ 2 ) ) ) )  /  4
)
Colors of variables: wff set class
Syntax hints:    = wceq 1624    e. wcel 1685   ` cfv 5221  (class class class)co 5819   _ici 8734    + caddc 8735    x. cmul 8737    - cmin 9032    / cdiv 9418   2c2 9790   4c4 9792   ^cexp 11098   ~Hchil 21491    +h cva 21492    .h csm 21493    .ih csp 21494   normhcno 21495    -h cmv 21497
This theorem is referenced by:  polid  21730
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810  ax-hfvadd 21572  ax-hv0cl 21575  ax-hfvmul 21577  ax-hvmul0 21582  ax-hfi 21650  ax-his1 21653  ax-his2 21654  ax-his3 21655  ax-his4 21656
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-sup 7189  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-n0 9961  df-z 10020  df-uz 10226  df-rp 10350  df-seq 11041  df-exp 11099  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-hnorm 21540  df-hvsub 21543
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