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Theorem adjmul 25431
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 25227 . . 3  |-  ( T  e.  dom  adjh  ->  T : ~H --> ~H )
2 homulcl 25098 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
31, 2sylan2 471 . 2  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh )  ->  ( A  .op  T
) : ~H --> ~H )
4 cjcl 12590 . . 3  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
5 dmadjrn 25234 . . . 4  |-  ( T  e.  dom  adjh  ->  (
adjh `  T )  e.  dom  adjh )
6 dmadjop 25227 . . . 4  |-  ( (
adjh `  T )  e.  dom  adjh  ->  ( adjh `  T ) : ~H --> ~H )
75, 6syl 16 . . 3  |-  ( T  e.  dom  adjh  ->  (
adjh `  T ) : ~H --> ~H )
8 homulcl 25098 . . 3  |-  ( ( ( * `  A
)  e.  CC  /\  ( adjh `  T ) : ~H --> ~H )  -> 
( ( * `  A )  .op  ( adjh `  T ) ) : ~H --> ~H )
94, 7, 8syl2an 474 . 2  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh )  ->  ( ( * `  A )  .op  ( adjh `  T ) ) : ~H --> ~H )
10 adj2 25273 . . . . . . . 8  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H  /\  y  e.  ~H )  ->  ( ( T `  x )  .ih  y
)  =  ( x 
.ih  ( ( adjh `  T ) `  y
) ) )
11103expb 1183 . . . . . . 7  |-  ( ( T  e.  dom  adjh  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( T `  x )  .ih  y )  =  ( x  .ih  ( (
adjh `  T ) `  y ) ) )
1211adantll 708 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( T `  x
)  .ih  y )  =  ( x  .ih  ( ( adjh `  T
) `  y )
) )
1312oveq2d 6106 . . . . 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 5840 . . . . . . . . 9  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
15 ax-his3 24421 . . . . . . . . 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 1247 . . . . . . . 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 1181 . . . . . . 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 436 . . . . . 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 592 . . . . 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 748 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  A  e.  CC )
21 simprl 750 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  x  e.  ~H )
22 adjcl 25271 . . . . . . 7  |-  ( ( T  e.  dom  adjh  /\  y  e.  ~H )  ->  ( ( adjh `  T
) `  y )  e.  ~H )
2322ad2ant2l 740 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( adjh `  T ) `  y )  e.  ~H )
24 his52 24424 . . . . . 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 1213 . . . . 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 2483 . . . 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 25080 . . . . . . . 8  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  T ) `  x )  =  ( A  .h  ( T `  x ) ) )
281, 27syl3an2 1247 . . . . . . 7  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh  /\  x  e.  ~H )  ->  (
( A  .op  T
) `  x )  =  ( A  .h  ( T `  x ) ) )
29283expa 1182 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
3029adantrr 711 . . . . 5  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( A  .op  T
) `  x )  =  ( A  .h  ( T `  x ) ) )
3130oveq1d 6105 . . . 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 22 . . . . . . . 8  |-  ( y  e.  ~H  ->  y  e.  ~H )
33 homval 25080 . . . . . . . 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 1255 . . . . . . 7  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh  /\  y  e.  ~H )  ->  (
( ( * `  A )  .op  ( adjh `  T ) ) `
 y )  =  ( ( * `  A )  .h  (
( adjh `  T ) `  y ) ) )
35343expa 1182 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  y  e.  ~H )  ->  ( ( ( * `  A ) 
.op  ( adjh `  T
) ) `  y
)  =  ( ( * `  A )  .h  ( ( adjh `  T ) `  y
) ) )
3635adantrl 710 . . . . 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 6106 . . . 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 2483 . . 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 2806 . 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 25274 . 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 1213 1  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh )  ->  ( adjh `  ( A  .op  T ) )  =  ( ( * `
 A )  .op  ( adjh `  T )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   dom cdm 4836   -->wf 5411   ` cfv 5415  (class class class)co 6090   CCcc 9276    x. cmul 9283   *ccj 12581   ~Hchil 24256    .h csm 24258    .ih csp 24259    .op chot 24276   adjhcado 24292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-hilex 24336  ax-hfvadd 24337  ax-hvcom 24338  ax-hvass 24339  ax-hv0cl 24340  ax-hvaddid 24341  ax-hfvmul 24342  ax-hvmulid 24343  ax-hvdistr2 24346  ax-hvmul0 24347  ax-hfi 24416  ax-his1 24419  ax-his2 24420  ax-his3 24421  ax-his4 24422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-2 10376  df-cj 12584  df-re 12585  df-im 12586  df-hvsub 24308  df-homul 25070  df-adjh 25188
This theorem is referenced by: (None)
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