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Theorem hmop 23418
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 23369 . . . 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 6081 . . . . 5  |-  ( x  =  A  ->  (
x  .ih  ( T `  y ) )  =  ( A  .ih  ( T `  y )
) )
5 fveq2 5721 . . . . . 6  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
65oveq1d 6089 . . . . 5  |-  ( x  =  A  ->  (
( T `  x
)  .ih  y )  =  ( ( T `
 A )  .ih  y ) )
74, 6eqeq12d 2450 . . . 4  |-  ( x  =  A  ->  (
( x  .ih  ( T `  y )
)  =  ( ( T `  x ) 
.ih  y )  <->  ( A  .ih  ( T `  y
) )  =  ( ( T `  A
)  .ih  y )
) )
8 fveq2 5721 . . . . . 6  |-  ( y  =  B  ->  ( T `  y )  =  ( T `  B ) )
98oveq2d 6090 . . . . 5  |-  ( y  =  B  ->  ( A  .ih  ( T `  y ) )  =  ( A  .ih  ( T `  B )
) )
10 oveq2 6082 . . . . 5  |-  ( y  =  B  ->  (
( T `  A
)  .ih  y )  =  ( ( T `
 A )  .ih  B ) )
119, 10eqeq12d 2450 . . . 4  |-  ( y  =  B  ->  (
( A  .ih  ( T `  y )
)  =  ( ( T `  A ) 
.ih  y )  <->  ( A  .ih  ( T `  B
) )  =  ( ( T `  A
)  .ih  B )
) )
127, 11rspc2v 3051 . . 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 1652    e. wcel 1725   A.wral 2698   -->wf 5443   ` cfv 5447  (class class class)co 6074   ~Hchil 22415    .ih csp 22418   HrmOpcho 22446
This theorem is referenced by:  hmopre  23419  hmopadj  23435  hmoplin  23438  eighmre  23459  eighmorth  23460  hmopbdoptHIL  23484  hmops  23516  hmopm  23517  hmopco  23519  leopsq  23625  hmopidmpji  23648
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-hilex 22495
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 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-map 7013  df-hmop 23340
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