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Theorem adjmul 22502
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
StepHypRef Expression
1 dmadjop 22298 . . 3  |-  ( T  e.  dom  adjh  ->  T : ~H --> ~H )
2 homulcl 22169 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
31, 2sylan2 462 . 2  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh )  ->  ( A  .op  T
) : ~H --> ~H )
4 cjcl 11467 . . 3  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
5 dmadjrn 22305 . . . 4  |-  ( T  e.  dom  adjh  ->  (
adjh `  T )  e.  dom  adjh )
6 dmadjop 22298 . . . 4  |-  ( (
adjh `  T )  e.  dom  adjh  ->  ( adjh `  T ) : ~H --> ~H )
75, 6syl 17 . . 3  |-  ( T  e.  dom  adjh  ->  (
adjh `  T ) : ~H --> ~H )
8 homulcl 22169 . . 3  |-  ( ( ( * `  A
)  e.  CC  /\  ( adjh `  T ) : ~H --> ~H )  -> 
( ( * `  A )  .op  ( adjh `  T ) ) : ~H --> ~H )
94, 7, 8syl2an 465 . 2  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh )  ->  ( ( * `  A )  .op  ( adjh `  T ) ) : ~H --> ~H )
10 adj2 22344 . . . . . . . 8  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H  /\  y  e.  ~H )  ->  ( ( T `  x )  .ih  y
)  =  ( x 
.ih  ( ( adjh `  T ) `  y
) ) )
11103expb 1157 . . . . . . 7  |-  ( ( T  e.  dom  adjh  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( T `  x )  .ih  y )  =  ( x  .ih  ( (
adjh `  T ) `  y ) ) )
1211adantll 697 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( T `  x
)  .ih  y )  =  ( x  .ih  ( ( adjh `  T
) `  y )
) )
1312oveq2d 5726 . . . . 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 5515 . . . . . . . . . 10  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
151, 14sylan 459 . . . . . . . . 9  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
16 ax-his3 21493 . . . . . . . . 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 1221 . . . . . . . 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 1155 . . . . . . 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 427 . . . . . 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 581 . . . . 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 733 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  A  e.  CC )
22 simprl 735 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  x  e.  ~H )
23 adjcl 22342 . . . . . . 7  |-  ( ( T  e.  dom  adjh  /\  y  e.  ~H )  ->  ( ( adjh `  T
) `  y )  e.  ~H )
2423ad2ant2l 729 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( adjh `  T ) `  y )  e.  ~H )
25 his52 21496 . . . . . 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 1187 . . . . 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 2295 . . . 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 22003 . . . . . . . 8  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  T ) `  x )  =  ( A  .h  ( T `  x ) ) )
291, 28syl3an2 1221 . . . . . . 7  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh  /\  x  e.  ~H )  ->  (
( A  .op  T
) `  x )  =  ( A  .h  ( T `  x ) ) )
30293expa 1156 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
3130adantrr 700 . . . . 5  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( A  .op  T
) `  x )  =  ( A  .h  ( T `  x ) ) )
3231oveq1d 5725 . . . 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 21 . . . . . . . 8  |-  ( y  e.  ~H  ->  y  e.  ~H )
34 homval 22003 . . . . . . . 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 1229 . . . . . . 7  |-  ( ( A  e.  CC  /\  T  e.  dom  adjh  /\  y  e.  ~H )  ->  (
( ( * `  A )  .op  ( adjh `  T ) ) `
 y )  =  ( ( * `  A )  .h  (
( adjh `  T ) `  y ) ) )
36353expa 1156 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T  e.  dom  adjh )  /\  y  e.  ~H )  ->  ( ( ( * `  A ) 
.op  ( adjh `  T
) ) `  y
)  =  ( ( * `  A )  .h  ( ( adjh `  T ) `  y
) ) )
3736adantrl 699 . . . . 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 5726 . . . 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 2295 . . 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 2597 . 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 22345 . 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 1187 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 6    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2509   dom cdm 4580   -->wf 4588   ` cfv 4592  (class class class)co 5710   CCcc 8615    x. cmul 8622   *ccj 11458   ~Hchil 21329    .h csm 21331    .ih csp 21332    .op chot 21349   adjhcado 21365
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-hilex 21409  ax-hfvadd 21410  ax-hvcom 21411  ax-hvass 21412  ax-hv0cl 21413  ax-hvaddid 21414  ax-hfvmul 21415  ax-hvmulid 21416  ax-hvdistr2 21419  ax-hvmul0 21420  ax-hfi 21488  ax-his1 21491  ax-his2 21492  ax-his3 21493  ax-his4 21494
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-iota 6143  df-riota 6190  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-2 9684  df-cj 11461  df-re 11462  df-im 11463  df-hvsub 21381  df-homul 21993  df-adjh 22259
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