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Theorem unop 22495
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 22452 . . . 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 450 . . 3  |-  ( T  e.  UniOp  ->  A. x  e.  ~H  A. y  e. 
~H  ( ( T `
 x )  .ih  ( T `  y ) )  =  ( x 
.ih  y ) )
323ad2ant1 976 . 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 5525 . . . . . 6  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
54oveq1d 5873 . . . . 5  |-  ( x  =  A  ->  (
( T `  x
)  .ih  ( T `  y ) )  =  ( ( T `  A )  .ih  ( T `  y )
) )
6 oveq1 5865 . . . . 5  |-  ( x  =  A  ->  (
x  .ih  y )  =  ( A  .ih  y ) )
75, 6eqeq12d 2297 . . . 4  |-  ( x  =  A  ->  (
( ( T `  x )  .ih  ( T `  y )
)  =  ( x 
.ih  y )  <->  ( ( T `  A )  .ih  ( T `  y
) )  =  ( A  .ih  y ) ) )
8 fveq2 5525 . . . . . 6  |-  ( y  =  B  ->  ( T `  y )  =  ( T `  B ) )
98oveq2d 5874 . . . . 5  |-  ( y  =  B  ->  (
( T `  A
)  .ih  ( T `  y ) )  =  ( ( T `  A )  .ih  ( T `  B )
) )
10 oveq2 5866 . . . . 5  |-  ( y  =  B  ->  ( A  .ih  y )  =  ( A  .ih  B
) )
119, 10eqeq12d 2297 . . . 4  |-  ( y  =  B  ->  (
( ( T `  A )  .ih  ( T `  y )
)  =  ( A 
.ih  y )  <->  ( ( T `  A )  .ih  ( T `  B
) )  =  ( A  .ih  B ) ) )
127, 11rspc2v 2890 . . 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 973 . 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 14 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 934    = wceq 1623    e. wcel 1684   A.wral 2543   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   ~Hchil 21499    .ih csp 21502   UniOpcuo 21529
This theorem is referenced by:  unopf1o  22496  unopnorm  22497  cnvunop  22498  unopadj  22499  counop  22501
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-unop 22423
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