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Theorem his1i 21695
Description: Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. (Contributed by NM, 15-May-2005.) (New usage is discouraged.)
Hypotheses
Ref Expression
his1.1  |-  A  e. 
~H
his1.2  |-  B  e. 
~H
Assertion
Ref Expression
his1i  |-  ( A 
.ih  B )  =  ( * `  ( B  .ih  A ) )

Proof of Theorem his1i
StepHypRef Expression
1 his1.1 . 2  |-  A  e. 
~H
2 his1.2 . 2  |-  B  e. 
~H
3 ax-his1 21677 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  B
)  =  ( * `
 ( B  .ih  A ) ) )
41, 2, 3mp2an 653 1  |-  ( A 
.ih  B )  =  ( * `  ( B  .ih  A ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   *ccj 11597   ~Hchil 21515    .ih csp 21518
This theorem is referenced by:  normlem2  21706  bcseqi  21715  bcsiALT  21774  pjadjii  22269  lnopunilem1  22606  lnophmlem2  22613
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-his1 21677
This theorem depends on definitions:  df-bi 177  df-an 360
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