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Theorem adj2 23285
Description: Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
adj2  |-  ( ( T  e.  dom  adjh  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( ( T `  A )  .ih  B
)  =  ( A 
.ih  ( ( adjh `  T ) `  B
) ) )

Proof of Theorem adj2
StepHypRef Expression
1 adj1 23284 . . . 4  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( B  .ih  ( T `  A )
)  =  ( ( ( adjh `  T
) `  B )  .ih  A ) )
2 simp2 958 . . . . 5  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  B  e.  ~H )
3 dmadjop 23239 . . . . . . 7  |-  ( T  e.  dom  adjh  ->  T : ~H --> ~H )
43ffvelrnda 5809 . . . . . 6  |-  ( ( T  e.  dom  adjh  /\  A  e.  ~H )  ->  ( T `  A
)  e.  ~H )
543adant2 976 . . . . 5  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( T `  A
)  e.  ~H )
6 ax-his1 22432 . . . . 5  |-  ( ( B  e.  ~H  /\  ( T `  A )  e.  ~H )  -> 
( B  .ih  ( T `  A )
)  =  ( * `
 ( ( T `
 A )  .ih  B ) ) )
72, 5, 6syl2anc 643 . . . 4  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( B  .ih  ( T `  A )
)  =  ( * `
 ( ( T `
 A )  .ih  B ) ) )
8 adjcl 23283 . . . . . 6  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H )  ->  ( ( adjh `  T
) `  B )  e.  ~H )
983adant3 977 . . . . 5  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( ( adjh `  T
) `  B )  e.  ~H )
10 simp3 959 . . . . 5  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  A  e.  ~H )
11 ax-his1 22432 . . . . 5  |-  ( ( ( ( adjh `  T
) `  B )  e.  ~H  /\  A  e. 
~H )  ->  (
( ( adjh `  T
) `  B )  .ih  A )  =  ( * `  ( A 
.ih  ( ( adjh `  T ) `  B
) ) ) )
129, 10, 11syl2anc 643 . . . 4  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( ( ( adjh `  T ) `  B
)  .ih  A )  =  ( * `  ( A  .ih  ( (
adjh `  T ) `  B ) ) ) )
131, 7, 123eqtr3d 2427 . . 3  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( * `  (
( T `  A
)  .ih  B )
)  =  ( * `
 ( A  .ih  ( ( adjh `  T
) `  B )
) ) )
14 hicl 22430 . . . . 5  |-  ( ( ( T `  A
)  e.  ~H  /\  B  e.  ~H )  ->  ( ( T `  A )  .ih  B
)  e.  CC )
155, 2, 14syl2anc 643 . . . 4  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( ( T `  A )  .ih  B
)  e.  CC )
16 hicl 22430 . . . . 5  |-  ( ( A  e.  ~H  /\  ( ( adjh `  T
) `  B )  e.  ~H )  ->  ( A  .ih  ( ( adjh `  T ) `  B
) )  e.  CC )
1710, 9, 16syl2anc 643 . . . 4  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( A  .ih  (
( adjh `  T ) `  B ) )  e.  CC )
18 cj11 11894 . . . 4  |-  ( ( ( ( T `  A )  .ih  B
)  e.  CC  /\  ( A  .ih  ( (
adjh `  T ) `  B ) )  e.  CC )  ->  (
( * `  (
( T `  A
)  .ih  B )
)  =  ( * `
 ( A  .ih  ( ( adjh `  T
) `  B )
) )  <->  ( ( T `  A )  .ih  B )  =  ( A  .ih  ( (
adjh `  T ) `  B ) ) ) )
1915, 17, 18syl2anc 643 . . 3  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( ( * `  ( ( T `  A )  .ih  B
) )  =  ( * `  ( A 
.ih  ( ( adjh `  T ) `  B
) ) )  <->  ( ( T `  A )  .ih  B )  =  ( A  .ih  ( (
adjh `  T ) `  B ) ) ) )
2013, 19mpbid 202 . 2  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( ( T `  A )  .ih  B
)  =  ( A 
.ih  ( ( adjh `  T ) `  B
) ) )
21203com23 1159 1  |-  ( ( T  e.  dom  adjh  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( ( T `  A )  .ih  B
)  =  ( A 
.ih  ( ( adjh `  T ) `  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1717   dom cdm 4818   ` cfv 5394  (class class class)co 6020   CCcc 8921   *ccj 11828   ~Hchil 22270    .ih csp 22273   adjhcado 22306
This theorem is referenced by:  adjadj  23287  adjvalval  23288  adjlnop  23437  adjmul  23443  adjadd  23444  adjcoi  23451  nmopcoadji  23452
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 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-hilex 22350  ax-hfvadd 22351  ax-hvcom 22352  ax-hvass 22353  ax-hv0cl 22354  ax-hvaddid 22355  ax-hfvmul 22356  ax-hvmulid 22357  ax-hvdistr2 22360  ax-hvmul0 22361  ax-hfi 22429  ax-his1 22432  ax-his2 22433  ax-his3 22434  ax-his4 22435
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-2 9990  df-cj 11831  df-re 11832  df-im 11833  df-hvsub 22322  df-adjh 23200
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