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Theorem adjmul 22672
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 22468 . . 3  |-  ( T  e.  dom  adjh  ->  T : ~H --> ~H )
2 homulcl 22339 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
31, 2sylan2 460 . 2  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh )  ->  ( A  .op  T
) : ~H --> ~H )
4 cjcl 11590 . . 3  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
5 dmadjrn 22475 . . . 4  |-  ( T  e.  dom  adjh  ->  (
adjh `  T )  e.  dom  adjh )
6 dmadjop 22468 . . . 4  |-  ( (
adjh `  T )  e.  dom  adjh  ->  ( adjh `  T ) : ~H --> ~H )
75, 6syl 15 . . 3  |-  ( T  e.  dom  adjh  ->  (
adjh `  T ) : ~H --> ~H )
8 homulcl 22339 . . 3  |-  ( ( ( * `  A
)  e.  CC  /\  ( adjh `  T ) : ~H --> ~H )  -> 
( ( * `  A )  .op  ( adjh `  T ) ) : ~H --> ~H )
94, 7, 8syl2an 463 . 2  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh )  ->  ( ( * `  A )  .op  ( adjh `  T ) ) : ~H --> ~H )
10 adj2 22514 . . . . . . . 8  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H  /\  y  e.  ~H )  ->  ( ( T `  x )  .ih  y
)  =  ( x 
.ih  ( ( adjh `  T ) `  y
) ) )
11103expb 1152 . . . . . . 7  |-  ( ( T  e.  dom  adjh  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( T `  x )  .ih  y )  =  ( x  .ih  ( (
adjh `  T ) `  y ) ) )
1211adantll 694 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( T `  x
)  .ih  y )  =  ( x  .ih  ( ( adjh `  T
) `  y )
) )
1312oveq2d 5874 . . . . 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 ) ) ) )
14 ffvelrn 5663 . . . . . . . . . 10  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
151, 14sylan 457 . . . . . . . . 9  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
16 ax-his3 21663 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( T `  x )  e.  ~H  /\  y  e.  ~H )  ->  (
( A  .h  ( T `  x )
)  .ih  y )  =  ( A  x.  ( ( T `  x )  .ih  y
) ) )
1715, 16syl3an2 1216 . . . . . . . 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 ) ) )
18173exp 1150 . . . . . . 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 ) ) ) ) )
1918exp3a 425 . . . . . 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
) ) ) ) ) )
2019imp43 578 . . . . 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
) ) )
21 simpll 730 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  A  e.  CC )
22 simprl 732 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  x  e.  ~H )
23 adjcl 22512 . . . . . . 7  |-  ( ( T  e.  dom  adjh  /\  y  e.  ~H )  ->  ( ( adjh `  T
) `  y )  e.  ~H )
2423ad2ant2l 726 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( adjh `  T ) `  y )  e.  ~H )
25 his52 21666 . . . . . 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 ) ) ) )
2621, 22, 24, 25syl3anc 1182 . . . . 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
) ) ) )
2713, 20, 263eqtr4d 2325 . . . 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 ) ) ) )
28 homval 22321 . . . . . . . 8  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  T ) `  x )  =  ( A  .h  ( T `  x ) ) )
291, 28syl3an2 1216 . . . . . . 7  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh  /\  x  e.  ~H )  ->  (
( A  .op  T
) `  x )  =  ( A  .h  ( T `  x ) ) )
30293expa 1151 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
3130adantrr 697 . . . . 5  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( A  .op  T
) `  x )  =  ( A  .h  ( T `  x ) ) )
3231oveq1d 5873 . . . 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 ) )
33 id 19 . . . . . . . 8  |-  ( y  e.  ~H  ->  y  e.  ~H )
34 homval 22321 . . . . . . . 8  |-  ( ( ( * `  A
)  e.  CC  /\  ( adjh `  T ) : ~H --> ~H  /\  y  e.  ~H )  ->  (
( ( * `  A )  .op  ( adjh `  T ) ) `
 y )  =  ( ( * `  A )  .h  (
( adjh `  T ) `  y ) ) )
354, 7, 33, 34syl3an 1224 . . . . . . 7  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh  /\  y  e.  ~H )  ->  (
( ( * `  A )  .op  ( adjh `  T ) ) `
 y )  =  ( ( * `  A )  .h  (
( adjh `  T ) `  y ) ) )
36353expa 1151 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  y  e.  ~H )  ->  ( ( ( * `  A ) 
.op  ( adjh `  T
) ) `  y
)  =  ( ( * `  A )  .h  ( ( adjh `  T ) `  y
) ) )
3736adantrl 696 . . . . 5  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( ( * `  A )  .op  ( adjh `  T ) ) `
 y )  =  ( ( * `  A )  .h  (
( adjh `  T ) `  y ) ) )
3837oveq2d 5874 . . . 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 ) ) ) )
3927, 32, 383eqtr4d 2325 . . 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 ) ) )
4039ralrimivva 2635 . 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 ) ) )
41 adjeq 22515 . 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 ) ) )
423, 9, 40, 41syl3anc 1182 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 358    = wceq 1623    e. wcel 1684   A.wral 2543   dom cdm 4689   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735    x. cmul 8742   *ccj 11581   ~Hchil 21499    .h csm 21501    .ih csp 21502    .op chot 21519   adjhcado 21535
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663  ax-his4 21664
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-cj 11584  df-re 11585  df-im 11586  df-hvsub 21551  df-homul 22311  df-adjh 22429
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