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Theorem adjmul 23583
Description: The adjoint of the scalar product of an operator. Theorem 3.11(ii) of [Beran] p. 106. (Contributed by NM, 21-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
adjmul  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh )  ->  ( adjh `  ( A  .op  T ) )  =  ( ( * `
 A )  .op  ( adjh `  T )
) )

Proof of Theorem adjmul
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmadjop 23379 . . 3  |-  ( T  e.  dom  adjh  ->  T : ~H --> ~H )
2 homulcl 23250 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
31, 2sylan2 461 . 2  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh )  ->  ( A  .op  T
) : ~H --> ~H )
4 cjcl 11898 . . 3  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
5 dmadjrn 23386 . . . 4  |-  ( T  e.  dom  adjh  ->  (
adjh `  T )  e.  dom  adjh )
6 dmadjop 23379 . . . 4  |-  ( (
adjh `  T )  e.  dom  adjh  ->  ( adjh `  T ) : ~H --> ~H )
75, 6syl 16 . . 3  |-  ( T  e.  dom  adjh  ->  (
adjh `  T ) : ~H --> ~H )
8 homulcl 23250 . . 3  |-  ( ( ( * `  A
)  e.  CC  /\  ( adjh `  T ) : ~H --> ~H )  -> 
( ( * `  A )  .op  ( adjh `  T ) ) : ~H --> ~H )
94, 7, 8syl2an 464 . 2  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh )  ->  ( ( * `  A )  .op  ( adjh `  T ) ) : ~H --> ~H )
10 adj2 23425 . . . . . . . 8  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H  /\  y  e.  ~H )  ->  ( ( T `  x )  .ih  y
)  =  ( x 
.ih  ( ( adjh `  T ) `  y
) ) )
11103expb 1154 . . . . . . 7  |-  ( ( T  e.  dom  adjh  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( T `  x )  .ih  y )  =  ( x  .ih  ( (
adjh `  T ) `  y ) ) )
1211adantll 695 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( T `  x
)  .ih  y )  =  ( x  .ih  ( ( adjh `  T
) `  y )
) )
1312oveq2d 6088 . . . . 5  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  ( A  x.  ( ( T `  x )  .ih  y ) )  =  ( A  x.  (
x  .ih  ( ( adjh `  T ) `  y ) ) ) )
141ffvelrnda 5861 . . . . . . . . 9  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
15 ax-his3 22574 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( T `  x )  e.  ~H  /\  y  e.  ~H )  ->  (
( A  .h  ( T `  x )
)  .ih  y )  =  ( A  x.  ( ( T `  x )  .ih  y
) ) )
1614, 15syl3an2 1218 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( T  e.  dom  adjh  /\  x  e.  ~H )  /\  y  e.  ~H )  ->  ( ( A  .h  ( T `  x ) )  .ih  y )  =  ( A  x.  ( ( T `  x ) 
.ih  y ) ) )
17163exp 1152 . . . . . . 7  |-  ( A  e.  CC  ->  (
( T  e.  dom  adjh  /\  x  e.  ~H )  ->  ( y  e. 
~H  ->  ( ( A  .h  ( T `  x ) )  .ih  y )  =  ( A  x.  ( ( T `  x ) 
.ih  y ) ) ) ) )
1817exp3a 426 . . . . . 6  |-  ( A  e.  CC  ->  ( T  e.  dom  adjh  ->  ( x  e.  ~H  ->  ( y  e.  ~H  ->  ( ( A  .h  ( T `  x )
)  .ih  y )  =  ( A  x.  ( ( T `  x )  .ih  y
) ) ) ) ) )
1918imp43 579 . . . . 5  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( A  .h  ( T `  x )
)  .ih  y )  =  ( A  x.  ( ( T `  x )  .ih  y
) ) )
20 simpll 731 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  A  e.  CC )
21 simprl 733 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  x  e.  ~H )
22 adjcl 23423 . . . . . . 7  |-  ( ( T  e.  dom  adjh  /\  y  e.  ~H )  ->  ( ( adjh `  T
) `  y )  e.  ~H )
2322ad2ant2l 727 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( adjh `  T ) `  y )  e.  ~H )
24 his52 22577 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  ~H  /\  (
( adjh `  T ) `  y )  e.  ~H )  ->  ( x  .ih  ( ( * `  A )  .h  (
( adjh `  T ) `  y ) ) )  =  ( A  x.  ( x  .ih  ( (
adjh `  T ) `  y ) ) ) )
2520, 21, 23, 24syl3anc 1184 . . . . 5  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
x  .ih  ( (
* `  A )  .h  ( ( adjh `  T
) `  y )
) )  =  ( A  x.  ( x 
.ih  ( ( adjh `  T ) `  y
) ) ) )
2613, 19, 253eqtr4d 2477 . . . 4  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( A  .h  ( T `  x )
)  .ih  y )  =  ( x  .ih  ( ( * `  A )  .h  (
( adjh `  T ) `  y ) ) ) )
27 homval 23232 . . . . . . . 8  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  T ) `  x )  =  ( A  .h  ( T `  x ) ) )
281, 27syl3an2 1218 . . . . . . 7  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh  /\  x  e.  ~H )  ->  (
( A  .op  T
) `  x )  =  ( A  .h  ( T `  x ) ) )
29283expa 1153 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
3029adantrr 698 . . . . 5  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( A  .op  T
) `  x )  =  ( A  .h  ( T `  x ) ) )
3130oveq1d 6087 . . . 4  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( ( A  .op  T ) `  x ) 
.ih  y )  =  ( ( A  .h  ( T `  x ) )  .ih  y ) )
32 id 20 . . . . . . . 8  |-  ( y  e.  ~H  ->  y  e.  ~H )
33 homval 23232 . . . . . . . 8  |-  ( ( ( * `  A
)  e.  CC  /\  ( adjh `  T ) : ~H --> ~H  /\  y  e.  ~H )  ->  (
( ( * `  A )  .op  ( adjh `  T ) ) `
 y )  =  ( ( * `  A )  .h  (
( adjh `  T ) `  y ) ) )
344, 7, 32, 33syl3an 1226 . . . . . . 7  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh  /\  y  e.  ~H )  ->  (
( ( * `  A )  .op  ( adjh `  T ) ) `
 y )  =  ( ( * `  A )  .h  (
( adjh `  T ) `  y ) ) )
35343expa 1153 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  y  e.  ~H )  ->  ( ( ( * `  A ) 
.op  ( adjh `  T
) ) `  y
)  =  ( ( * `  A )  .h  ( ( adjh `  T ) `  y
) ) )
3635adantrl 697 . . . . 5  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( ( * `  A )  .op  ( adjh `  T ) ) `
 y )  =  ( ( * `  A )  .h  (
( adjh `  T ) `  y ) ) )
3736oveq2d 6088 . . . 4  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
x  .ih  ( (
( * `  A
)  .op  ( adjh `  T ) ) `  y ) )  =  ( x  .ih  (
( * `  A
)  .h  ( (
adjh `  T ) `  y ) ) ) )
3826, 31, 373eqtr4d 2477 . . 3  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( ( A  .op  T ) `  x ) 
.ih  y )  =  ( x  .ih  (
( ( * `  A )  .op  ( adjh `  T ) ) `
 y ) ) )
3938ralrimivva 2790 . 2  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh )  ->  A. x  e.  ~H  A. y  e.  ~H  (
( ( A  .op  T ) `  x ) 
.ih  y )  =  ( x  .ih  (
( ( * `  A )  .op  ( adjh `  T ) ) `
 y ) ) )
40 adjeq 23426 . 2  |-  ( ( ( A  .op  T
) : ~H --> ~H  /\  ( ( * `  A )  .op  ( adjh `  T ) ) : ~H --> ~H  /\  A. x  e.  ~H  A. y  e.  ~H  (
( ( A  .op  T ) `  x ) 
.ih  y )  =  ( x  .ih  (
( ( * `  A )  .op  ( adjh `  T ) ) `
 y ) ) )  ->  ( adjh `  ( A  .op  T
) )  =  ( ( * `  A
)  .op  ( adjh `  T ) ) )
413, 9, 39, 40syl3anc 1184 1  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh )  ->  ( adjh `  ( A  .op  T ) )  =  ( ( * `
 A )  .op  ( adjh `  T )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   dom cdm 4869   -->wf 5441   ` cfv 5445  (class class class)co 6072   CCcc 8977    x. cmul 8984   *ccj 11889   ~Hchil 22410    .h csm 22412    .ih csp 22413    .op chot 22430   adjhcado 22446
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-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-hilex 22490  ax-hfvadd 22491  ax-hvcom 22492  ax-hvass 22493  ax-hv0cl 22494  ax-hvaddid 22495  ax-hfvmul 22496  ax-hvmulid 22497  ax-hvdistr2 22500  ax-hvmul0 22501  ax-hfi 22569  ax-his1 22572  ax-his2 22573  ax-his3 22574  ax-his4 22575
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 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-riota 6540  df-er 6896  df-map 7011  df-en 7101  df-dom 7102  df-sdom 7103  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-2 10047  df-cj 11892  df-re 11893  df-im 11894  df-hvsub 22462  df-homul 23222  df-adjh 23340
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