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Theorem unop 23418
Description: Basic inner product property of a unitary operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
Assertion
Ref Expression
unop  |-  ( ( T  e.  UniOp  /\  A  e.  ~H  /\  B  e. 
~H )  ->  (
( T `  A
)  .ih  ( T `  B ) )  =  ( A  .ih  B
) )

Proof of Theorem unop
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elunop 23375 . . . 4  |-  ( T  e.  UniOp 
<->  ( T : ~H -onto-> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( ( T `  x )  .ih  ( T `  y )
)  =  ( x 
.ih  y ) ) )
21simprbi 451 . . 3  |-  ( T  e.  UniOp  ->  A. x  e.  ~H  A. y  e. 
~H  ( ( T `
 x )  .ih  ( T `  y ) )  =  ( x 
.ih  y ) )
323ad2ant1 978 . 2  |-  ( ( T  e.  UniOp  /\  A  e.  ~H  /\  B  e. 
~H )  ->  A. x  e.  ~H  A. y  e. 
~H  ( ( T `
 x )  .ih  ( T `  y ) )  =  ( x 
.ih  y ) )
4 fveq2 5728 . . . . . 6  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
54oveq1d 6096 . . . . 5  |-  ( x  =  A  ->  (
( T `  x
)  .ih  ( T `  y ) )  =  ( ( T `  A )  .ih  ( T `  y )
) )
6 oveq1 6088 . . . . 5  |-  ( x  =  A  ->  (
x  .ih  y )  =  ( A  .ih  y ) )
75, 6eqeq12d 2450 . . . 4  |-  ( x  =  A  ->  (
( ( T `  x )  .ih  ( T `  y )
)  =  ( x 
.ih  y )  <->  ( ( T `  A )  .ih  ( T `  y
) )  =  ( A  .ih  y ) ) )
8 fveq2 5728 . . . . . 6  |-  ( y  =  B  ->  ( T `  y )  =  ( T `  B ) )
98oveq2d 6097 . . . . 5  |-  ( y  =  B  ->  (
( T `  A
)  .ih  ( T `  y ) )  =  ( ( T `  A )  .ih  ( T `  B )
) )
10 oveq2 6089 . . . . 5  |-  ( y  =  B  ->  ( A  .ih  y )  =  ( A  .ih  B
) )
119, 10eqeq12d 2450 . . . 4  |-  ( y  =  B  ->  (
( ( T `  A )  .ih  ( T `  y )
)  =  ( A 
.ih  y )  <->  ( ( T `  A )  .ih  ( T `  B
) )  =  ( A  .ih  B ) ) )
127, 11rspc2v 3058 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A. x  e. 
~H  A. y  e.  ~H  ( ( T `  x )  .ih  ( T `  y )
)  =  ( x 
.ih  y )  -> 
( ( T `  A )  .ih  ( T `  B )
)  =  ( A 
.ih  B ) ) )
13123adant1 975 . 2  |-  ( ( T  e.  UniOp  /\  A  e.  ~H  /\  B  e. 
~H )  ->  ( A. x  e.  ~H  A. y  e.  ~H  (
( T `  x
)  .ih  ( T `  y ) )  =  ( x  .ih  y
)  ->  ( ( T `  A )  .ih  ( T `  B
) )  =  ( A  .ih  B ) ) )
143, 13mpd 15 1  |-  ( ( T  e.  UniOp  /\  A  e.  ~H  /\  B  e. 
~H )  ->  (
( T `  A
)  .ih  ( T `  B ) )  =  ( A  .ih  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   -onto->wfo 5452   ` cfv 5454  (class class class)co 6081   ~Hchil 22422    .ih csp 22425   UniOpcuo 22452
This theorem is referenced by:  unopf1o  23419  unopnorm  23420  cnvunop  23421  unopadj  23422  counop  23424
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-hilex 22502
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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-ral 2710  df-rex 2711  df-reu 2712  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-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-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-unop 23346
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