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Theorem hmop 22518
Description: Basic inner product property of a Hermitian operator. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hmop  |-  ( ( T  e.  HrmOp  /\  A  e.  ~H  /\  B  e. 
~H )  ->  ( A  .ih  ( T `  B ) )  =  ( ( T `  A )  .ih  B
) )

Proof of Theorem hmop
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elhmop 22469 . . . 4  |-  ( T  e.  HrmOp 
<->  ( T : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( x  .ih  ( T `
 y ) )  =  ( ( T `
 x )  .ih  y ) ) )
21simprbi 450 . . 3  |-  ( T  e.  HrmOp  ->  A. x  e.  ~H  A. y  e. 
~H  ( x  .ih  ( T `  y ) )  =  ( ( T `  x ) 
.ih  y ) )
323ad2ant1 976 . 2  |-  ( ( T  e.  HrmOp  /\  A  e.  ~H  /\  B  e. 
~H )  ->  A. x  e.  ~H  A. y  e. 
~H  ( x  .ih  ( T `  y ) )  =  ( ( T `  x ) 
.ih  y ) )
4 oveq1 5881 . . . . 5  |-  ( x  =  A  ->  (
x  .ih  ( T `  y ) )  =  ( A  .ih  ( T `  y )
) )
5 fveq2 5541 . . . . . 6  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
65oveq1d 5889 . . . . 5  |-  ( x  =  A  ->  (
( T `  x
)  .ih  y )  =  ( ( T `
 A )  .ih  y ) )
74, 6eqeq12d 2310 . . . 4  |-  ( x  =  A  ->  (
( x  .ih  ( T `  y )
)  =  ( ( T `  x ) 
.ih  y )  <->  ( A  .ih  ( T `  y
) )  =  ( ( T `  A
)  .ih  y )
) )
8 fveq2 5541 . . . . . 6  |-  ( y  =  B  ->  ( T `  y )  =  ( T `  B ) )
98oveq2d 5890 . . . . 5  |-  ( y  =  B  ->  ( A  .ih  ( T `  y ) )  =  ( A  .ih  ( T `  B )
) )
10 oveq2 5882 . . . . 5  |-  ( y  =  B  ->  (
( T `  A
)  .ih  y )  =  ( ( T `
 A )  .ih  B ) )
119, 10eqeq12d 2310 . . . 4  |-  ( y  =  B  ->  (
( A  .ih  ( T `  y )
)  =  ( ( T `  A ) 
.ih  y )  <->  ( A  .ih  ( T `  B
) )  =  ( ( T `  A
)  .ih  B )
) )
127, 11rspc2v 2903 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A. x  e. 
~H  A. y  e.  ~H  ( x  .ih  ( T `
 y ) )  =  ( ( T `
 x )  .ih  y )  ->  ( A  .ih  ( T `  B ) )  =  ( ( T `  A )  .ih  B
) ) )
13123adant1 973 . 2  |-  ( ( T  e.  HrmOp  /\  A  e.  ~H  /\  B  e. 
~H )  ->  ( A. x  e.  ~H  A. y  e.  ~H  (
x  .ih  ( T `  y ) )  =  ( ( T `  x )  .ih  y
)  ->  ( A  .ih  ( T `  B
) )  =  ( ( T `  A
)  .ih  B )
) )
143, 13mpd 14 1  |-  ( ( T  e.  HrmOp  /\  A  e.  ~H  /\  B  e. 
~H )  ->  ( A  .ih  ( T `  B ) )  =  ( ( T `  A )  .ih  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   -->wf 5267   ` cfv 5271  (class class class)co 5874   ~Hchil 21515    .ih csp 21518   HrmOpcho 21546
This theorem is referenced by:  hmopre  22519  hmopadj  22535  hmoplin  22538  eighmre  22559  eighmorth  22560  hmopbdoptHIL  22584  hmops  22616  hmopm  22617  hmopco  22619  leopsq  22725  hmopidmpji  22748
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-hilex 21595
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-hmop 22440
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