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Theorem adjmul 27137
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 26933 . . 3  |-  ( T  e.  dom  adjh  ->  T : ~H --> ~H )
2 homulcl 26804 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
31, 2sylan2 474 . 2  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh )  ->  ( A  .op  T
) : ~H --> ~H )
4 cjcl 12949 . . 3  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
5 dmadjrn 26940 . . . 4  |-  ( T  e.  dom  adjh  ->  (
adjh `  T )  e.  dom  adjh )
6 dmadjop 26933 . . . 4  |-  ( (
adjh `  T )  e.  dom  adjh  ->  ( adjh `  T ) : ~H --> ~H )
75, 6syl 16 . . 3  |-  ( T  e.  dom  adjh  ->  (
adjh `  T ) : ~H --> ~H )
8 homulcl 26804 . . 3  |-  ( ( ( * `  A
)  e.  CC  /\  ( adjh `  T ) : ~H --> ~H )  -> 
( ( * `  A )  .op  ( adjh `  T ) ) : ~H --> ~H )
94, 7, 8syl2an 477 . 2  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh )  ->  ( ( * `  A )  .op  ( adjh `  T ) ) : ~H --> ~H )
10 adj2 26979 . . . . . . . 8  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H  /\  y  e.  ~H )  ->  ( ( T `  x )  .ih  y
)  =  ( x 
.ih  ( ( adjh `  T ) `  y
) ) )
11103expb 1197 . . . . . . 7  |-  ( ( T  e.  dom  adjh  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( T `  x )  .ih  y )  =  ( x  .ih  ( (
adjh `  T ) `  y ) ) )
1211adantll 713 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( T `  x
)  .ih  y )  =  ( x  .ih  ( ( adjh `  T
) `  y )
) )
1312oveq2d 6312 . . . . 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 6032 . . . . . . . . 9  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
15 ax-his3 26127 . . . . . . . . 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 1262 . . . . . . . 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 1195 . . . . . . 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 ) ) ) ) )
1817expd 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 595 . . . . 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 753 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  A  e.  CC )
21 simprl 756 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  x  e.  ~H )
22 adjcl 26977 . . . . . . 7  |-  ( ( T  e.  dom  adjh  /\  y  e.  ~H )  ->  ( ( adjh `  T
) `  y )  e.  ~H )
2322ad2ant2l 745 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( adjh `  T ) `  y )  e.  ~H )
24 his52 26130 . . . . . 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 1228 . . . . 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 2508 . . . 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 26786 . . . . . . . 8  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  T ) `  x )  =  ( A  .h  ( T `  x ) ) )
281, 27syl3an2 1262 . . . . . . 7  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh  /\  x  e.  ~H )  ->  (
( A  .op  T
) `  x )  =  ( A  .h  ( T `  x ) ) )
29283expa 1196 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
3029adantrr 716 . . . . 5  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( A  .op  T
) `  x )  =  ( A  .h  ( T `  x ) ) )
3130oveq1d 6311 . . . 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 26786 . . . . . . . 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 1270 . . . . . . 7  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh  /\  y  e.  ~H )  ->  (
( ( * `  A )  .op  ( adjh `  T ) ) `
 y )  =  ( ( * `  A )  .h  (
( adjh `  T ) `  y ) ) )
35343expa 1196 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  y  e.  ~H )  ->  ( ( ( * `  A ) 
.op  ( adjh `  T
) ) `  y
)  =  ( ( * `  A )  .h  ( ( adjh `  T ) `  y
) ) )
3635adantrl 715 . . . . 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 6312 . . . 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 2508 . . 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 2878 . 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 26980 . 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 1228 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 1395    e. wcel 1819   A.wral 2807   dom cdm 5008   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507    x. cmul 9514   *ccj 12940   ~Hchil 25962    .h csm 25964    .ih csp 25965    .op chot 25982   adjhcado 25998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-hilex 26042  ax-hfvadd 26043  ax-hvcom 26044  ax-hvass 26045  ax-hv0cl 26046  ax-hvaddid 26047  ax-hfvmul 26048  ax-hvmulid 26049  ax-hvdistr2 26052  ax-hvmul0 26053  ax-hfi 26122  ax-his1 26125  ax-his2 26126  ax-his3 26127  ax-his4 26128
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-2 10615  df-cj 12943  df-re 12944  df-im 12945  df-hvsub 26014  df-homul 26776  df-adjh 26894
This theorem is referenced by: (None)
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