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Theorem adj2 23429
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 23428 . . . 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 23383 . . . . . . 7  |-  ( T  e.  dom  adjh  ->  T : ~H --> ~H )
43ffvelrnda 5862 . . . . . 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 22576 . . . . 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 23427 . . . . . 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 22576 . . . . 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 2475 . . 3  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( * `  (
( T `  A
)  .ih  B )
)  =  ( * `
 ( A  .ih  ( ( adjh `  T
) `  B )
) ) )
14 hicl 22574 . . . . 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 22574 . . . . 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 11959 . . . 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 1652    e. wcel 1725   dom cdm 4870   ` cfv 5446  (class class class)co 6073   CCcc 8980   *ccj 11893   ~Hchil 22414    .ih csp 22417   adjhcado 22450
This theorem is referenced by:  adjadj  23431  adjvalval  23432  adjlnop  23581  adjmul  23587  adjadd  23588  adjcoi  23595  nmopcoadji  23596
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-hilex 22494  ax-hfvadd 22495  ax-hvcom 22496  ax-hvass 22497  ax-hv0cl 22498  ax-hvaddid 22499  ax-hfvmul 22500  ax-hvmulid 22501  ax-hvdistr2 22504  ax-hvmul0 22505  ax-hfi 22573  ax-his1 22576  ax-his2 22577  ax-his3 22578  ax-his4 22579
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-2 10050  df-cj 11896  df-re 11897  df-im 11898  df-hvsub 22466  df-adjh 23344
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