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Theorem hmop 23266
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 23217 . . . 4  |-  ( T  e.  HrmOp 
<->  ( T : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( x  .ih  ( T `
 y ) )  =  ( ( T `
 x )  .ih  y ) ) )
21simprbi 451 . . 3  |-  ( T  e.  HrmOp  ->  A. x  e.  ~H  A. y  e. 
~H  ( x  .ih  ( T `  y ) )  =  ( ( T `  x ) 
.ih  y ) )
323ad2ant1 978 . 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 6020 . . . . 5  |-  ( x  =  A  ->  (
x  .ih  ( T `  y ) )  =  ( A  .ih  ( T `  y )
) )
5 fveq2 5661 . . . . . 6  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
65oveq1d 6028 . . . . 5  |-  ( x  =  A  ->  (
( T `  x
)  .ih  y )  =  ( ( T `
 A )  .ih  y ) )
74, 6eqeq12d 2394 . . . 4  |-  ( x  =  A  ->  (
( x  .ih  ( T `  y )
)  =  ( ( T `  x ) 
.ih  y )  <->  ( A  .ih  ( T `  y
) )  =  ( ( T `  A
)  .ih  y )
) )
8 fveq2 5661 . . . . . 6  |-  ( y  =  B  ->  ( T `  y )  =  ( T `  B ) )
98oveq2d 6029 . . . . 5  |-  ( y  =  B  ->  ( A  .ih  ( T `  y ) )  =  ( A  .ih  ( T `  B )
) )
10 oveq2 6021 . . . . 5  |-  ( y  =  B  ->  (
( T `  A
)  .ih  y )  =  ( ( T `
 A )  .ih  B ) )
119, 10eqeq12d 2394 . . . 4  |-  ( y  =  B  ->  (
( A  .ih  ( T `  y )
)  =  ( ( T `  A ) 
.ih  y )  <->  ( A  .ih  ( T `  B
) )  =  ( ( T `  A
)  .ih  B )
) )
127, 11rspc2v 2994 . . 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 975 . 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 15 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 936    = wceq 1649    e. wcel 1717   A.wral 2642   -->wf 5383   ` cfv 5387  (class class class)co 6013   ~Hchil 22263    .ih csp 22266   HrmOpcho 22294
This theorem is referenced by:  hmopre  23267  hmopadj  23283  hmoplin  23286  eighmre  23307  eighmorth  23308  hmopbdoptHIL  23332  hmops  23364  hmopm  23365  hmopco  23367  leopsq  23473  hmopidmpji  23496
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-hilex 22343
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-map 6949  df-hmop 23188
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