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Theorem hmop 26819
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 26770 . . . 4  |-  ( T  e.  HrmOp 
<->  ( T : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( x  .ih  ( T `
 y ) )  =  ( ( T `
 x )  .ih  y ) ) )
21simprbi 464 . . 3  |-  ( T  e.  HrmOp  ->  A. x  e.  ~H  A. y  e. 
~H  ( x  .ih  ( T `  y ) )  =  ( ( T `  x ) 
.ih  y ) )
323ad2ant1 1018 . 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 6288 . . . . 5  |-  ( x  =  A  ->  (
x  .ih  ( T `  y ) )  =  ( A  .ih  ( T `  y )
) )
5 fveq2 5856 . . . . . 6  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
65oveq1d 6296 . . . . 5  |-  ( x  =  A  ->  (
( T `  x
)  .ih  y )  =  ( ( T `
 A )  .ih  y ) )
74, 6eqeq12d 2465 . . . 4  |-  ( x  =  A  ->  (
( x  .ih  ( T `  y )
)  =  ( ( T `  x ) 
.ih  y )  <->  ( A  .ih  ( T `  y
) )  =  ( ( T `  A
)  .ih  y )
) )
8 fveq2 5856 . . . . . 6  |-  ( y  =  B  ->  ( T `  y )  =  ( T `  B ) )
98oveq2d 6297 . . . . 5  |-  ( y  =  B  ->  ( A  .ih  ( T `  y ) )  =  ( A  .ih  ( T `  B )
) )
10 oveq2 6289 . . . . 5  |-  ( y  =  B  ->  (
( T `  A
)  .ih  y )  =  ( ( T `
 A )  .ih  B ) )
119, 10eqeq12d 2465 . . . 4  |-  ( y  =  B  ->  (
( A  .ih  ( T `  y )
)  =  ( ( T `  A ) 
.ih  y )  <->  ( A  .ih  ( T `  B
) )  =  ( ( T `  A
)  .ih  B )
) )
127, 11rspc2v 3205 . . 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 1015 . 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 setvar class
Syntax hints:    -> wi 4    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   -->wf 5574   ` cfv 5578  (class class class)co 6281   ~Hchil 25814    .ih csp 25817   HrmOpcho 25845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-hilex 25894
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-map 7424  df-hmop 26741
This theorem is referenced by:  hmopre  26820  hmopadj  26836  hmoplin  26839  eighmre  26860  eighmorth  26861  hmopbdoptHIL  26885  hmops  26917  hmopm  26918  hmopco  26920  leopsq  27026  hmopidmpji  27049
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