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Theorem adj2 25357
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 25356 . . . 4  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( B  .ih  ( T `  A )
)  =  ( ( ( adjh `  T
) `  B )  .ih  A ) )
2 simp2 989 . . . . 5  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  B  e.  ~H )
3 dmadjop 25311 . . . . . . 7  |-  ( T  e.  dom  adjh  ->  T : ~H --> ~H )
43ffvelrnda 5862 . . . . . 6  |-  ( ( T  e.  dom  adjh  /\  A  e.  ~H )  ->  ( T `  A
)  e.  ~H )
543adant2 1007 . . . . 5  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( T `  A
)  e.  ~H )
6 ax-his1 24503 . . . . 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 25355 . . . . . 6  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H )  ->  ( ( adjh `  T
) `  B )  e.  ~H )
983adant3 1008 . . . . 5  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( ( adjh `  T
) `  B )  e.  ~H )
10 simp3 990 . . . . 5  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  A  e.  ~H )
11 ax-his1 24503 . . . . 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 2483 . . 3  |-  ( ( T  e.  dom  adjh  /\  B  e.  ~H  /\  A  e.  ~H )  ->  ( * `  (
( T `  A
)  .ih  B )
)  =  ( * `
 ( A  .ih  ( ( adjh `  T
) `  B )
) ) )
14 hicl 24501 . . . . 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 24501 . . . . 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 12670 . . . 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 1193 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 965    = wceq 1369    e. wcel 1756   dom cdm 4859   ` cfv 5437  (class class class)co 6110   CCcc 9299   *ccj 12604   ~Hchil 24340    .ih csp 24343   adjhcado 24376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378  ax-hilex 24420  ax-hfvadd 24421  ax-hvcom 24422  ax-hvass 24423  ax-hv0cl 24424  ax-hvaddid 24425  ax-hfvmul 24426  ax-hvmulid 24427  ax-hvdistr2 24430  ax-hvmul0 24431  ax-hfi 24500  ax-his1 24503  ax-his2 24504  ax-his3 24505  ax-his4 24506
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-po 4660  df-so 4661  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-er 7120  df-map 7235  df-en 7330  df-dom 7331  df-sdom 7332  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-div 10013  df-2 10399  df-cj 12607  df-re 12608  df-im 12609  df-hvsub 24392  df-adjh 25272
This theorem is referenced by:  adjadj  25359  adjvalval  25360  adjlnop  25509  adjmul  25515  adjadd  25516  adjcoi  25523  nmopcoadji  25524
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