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Metamath Proof Explorer

Theorem polidi

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 . (Contributed by NM, 30-Jun-2005) (New usage is discouraged.)

Ref Expression
Hypotheses polid.1 
|- A e. ~H
polid.2 
|- B e. ~H
Assertion 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

Step Hyp Ref Expression
1 polid.1 
 |-  A e. ~H
2 polid.2 
 |-  B e. ~H
3 1 2 2 1 polid2i 
 |-  ( 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 )
4 1 2 hvaddcli 
 |-  ( A +h B ) e. ~H
5 4 normsqi 
 |-  ( ( normh ` ( A +h B ) ) ^ 2 ) = ( ( A +h B ) .ih ( A +h B ) )
6 1 2 hvsubcli 
 |-  ( A -h B ) e. ~H
7 6 normsqi 
 |-  ( ( normh ` ( A -h B ) ) ^ 2 ) = ( ( A -h B ) .ih ( A -h B ) )
8 5 7 oveq12i 
 |-  ( ( ( 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 
 |-  _i e. CC
10 9 2 hvmulcli 
 |-  ( _i .h B ) e. ~H
11 1 10 hvaddcli 
 |-  ( A +h ( _i .h B ) ) e. ~H
12 11 normsqi 
 |-  ( ( normh ` ( A +h ( _i .h B ) ) ) ^ 2 ) = ( ( A +h ( _i .h B ) ) .ih ( A +h ( _i .h B ) ) )
13 1 10 hvsubcli 
 |-  ( A -h ( _i .h B ) ) e. ~H
14 13 normsqi 
 |-  ( ( normh ` ( A -h ( _i .h B ) ) ) ^ 2 ) = ( ( A -h ( _i .h B ) ) .ih ( A -h ( _i .h B ) ) )
15 12 14 oveq12i 
 |-  ( ( ( 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 ) ) ) )
16 15 oveq2i 
 |-  ( _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 ) ) ) ) )
17 8 16 oveq12i 
 |-  ( ( ( ( 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 ) ) ) ) ) )
18 17 oveq1i 
 |-  ( ( ( ( ( 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 )
19 3 18 eqtr4i 
 |-  ( 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 )