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Theorem adj2 26684
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 26683 . . . 4  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( B  .ih  ( T `  A )
)  =  ( ( ( adjh `  T
) `  B )  .ih  A ) )
2 simp2 997 . . . . 5  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  B  e.  ~H )
3 dmadjop 26638 . . . . . . 7  |-  ( T  e.  dom  adjh  ->  T : ~H --> ~H )
43ffvelrnda 6032 . . . . . 6  |-  ( ( T  e.  dom  adjh  /\  A  e.  ~H )  ->  ( T `  A
)  e.  ~H )
543adant2 1015 . . . . 5  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( T `  A
)  e.  ~H )
6 ax-his1 25830 . . . . 5  |-  ( ( B  e.  ~H  /\  ( T `  A )  e.  ~H )  -> 
( B  .ih  ( T `  A )
)  =  ( * `
 ( ( T `
 A )  .ih  B ) ) )
72, 5, 6syl2anc 661 . . . 4  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( B  .ih  ( T `  A )
)  =  ( * `
 ( ( T `
 A )  .ih  B ) ) )
8 adjcl 26682 . . . . . 6  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H )  ->  ( ( adjh `  T
) `  B )  e.  ~H )
983adant3 1016 . . . . 5  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( ( adjh `  T
) `  B )  e.  ~H )
10 simp3 998 . . . . 5  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  A  e.  ~H )
11 ax-his1 25830 . . . . 5  |-  ( ( ( ( adjh `  T
) `  B )  e.  ~H  /\  A  e. 
~H )  ->  (
( ( adjh `  T
) `  B )  .ih  A )  =  ( * `  ( A 
.ih  ( ( adjh `  T ) `  B
) ) ) )
129, 10, 11syl2anc 661 . . . 4  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( ( ( adjh `  T ) `  B
)  .ih  A )  =  ( * `  ( A  .ih  ( (
adjh `  T ) `  B ) ) ) )
131, 7, 123eqtr3d 2516 . . 3  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( * `  (
( T `  A
)  .ih  B )
)  =  ( * `
 ( A  .ih  ( ( adjh `  T
) `  B )
) ) )
14 hicl 25828 . . . . 5  |-  ( ( ( T `  A
)  e.  ~H  /\  B  e.  ~H )  ->  ( ( T `  A )  .ih  B
)  e.  CC )
155, 2, 14syl2anc 661 . . . 4  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( ( T `  A )  .ih  B
)  e.  CC )
16 hicl 25828 . . . . 5  |-  ( ( A  e.  ~H  /\  ( ( adjh `  T
) `  B )  e.  ~H )  ->  ( A  .ih  ( ( adjh `  T ) `  B
) )  e.  CC )
1710, 9, 16syl2anc 661 . . . 4  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( A  .ih  (
( adjh `  T ) `  B ) )  e.  CC )
18 cj11 12974 . . . 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 661 . . 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 210 . 2  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( ( T `  A )  .ih  B
)  =  ( A 
.ih  ( ( adjh `  T ) `  B
) ) )
21203com23 1202 1  |-  ( ( T  e.  dom  adjh  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( ( T `  A )  .ih  B
)  =  ( A 
.ih  ( ( adjh `  T ) `  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1379    e. wcel 1767   dom cdm 5005   ` cfv 5594  (class class class)co 6295   CCcc 9502   *ccj 12908   ~Hchil 25667    .ih csp 25670   adjhcado 25703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-hilex 25747  ax-hfvadd 25748  ax-hvcom 25749  ax-hvass 25750  ax-hv0cl 25751  ax-hvaddid 25752  ax-hfvmul 25753  ax-hvmulid 25754  ax-hvdistr2 25757  ax-hvmul0 25758  ax-hfi 25827  ax-his1 25830  ax-his2 25831  ax-his3 25832  ax-his4 25833
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-2 10606  df-cj 12911  df-re 12912  df-im 12913  df-hvsub 25719  df-adjh 26599
This theorem is referenced by:  adjadj  26686  adjvalval  26687  adjlnop  26836  adjmul  26842  adjadd  26843  adjcoi  26850  nmopcoadji  26851
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