HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hoadddi Unicode version

Theorem hoadddi 22383
Description: Scalar product distributive law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hoadddi  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  .op  ( T  +op  U ) )  =  ( ( A 
.op  T )  +op  ( A  .op  U ) ) )

Proof of Theorem hoadddi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl1 958 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  A  e.  CC )
2 ffvelrn 5663 . . . . . . 7  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
323ad2antl2 1118 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( T `  x )  e.  ~H )
4 ffvelrn 5663 . . . . . . 7  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( U `  x
)  e.  ~H )
543ad2antl3 1119 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( U `  x )  e.  ~H )
6 ax-hvdistr1 21588 . . . . . 6  |-  ( ( A  e.  CC  /\  ( T `  x )  e.  ~H  /\  ( U `  x )  e.  ~H )  ->  ( A  .h  ( ( T `  x )  +h  ( U `  x
) ) )  =  ( ( A  .h  ( T `  x ) )  +h  ( A  .h  ( U `  x ) ) ) )
71, 3, 5, 6syl3anc 1182 . . . . 5  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( ( T `  x )  +h  ( U `  x )
) )  =  ( ( A  .h  ( T `  x )
)  +h  ( A  .h  ( U `  x ) ) ) )
8 hosval 22320 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( T  +op  U ) `  x )  =  ( ( T `
 x )  +h  ( U `  x
) ) )
98oveq2d 5874 . . . . . . 7  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( A  .h  (
( T  +op  U
) `  x )
)  =  ( A  .h  ( ( T `
 x )  +h  ( U `  x
) ) ) )
1093expa 1151 . . . . . 6  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( ( T  +op  U ) `  x ) )  =  ( A  .h  (
( T `  x
)  +h  ( U `
 x ) ) ) )
11103adantl1 1111 . . . . 5  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( ( T  +op  U ) `  x ) )  =  ( A  .h  ( ( T `
 x )  +h  ( U `  x
) ) ) )
12 homval 22321 . . . . . . . 8  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  T ) `  x )  =  ( A  .h  ( T `  x ) ) )
13123expa 1151 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
14133adantl3 1113 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
15 homval 22321 . . . . . . . 8  |-  ( ( A  e.  CC  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  U ) `  x )  =  ( A  .h  ( U `  x ) ) )
16153expa 1151 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  U ) `  x )  =  ( A  .h  ( U `
 x ) ) )
17163adantl2 1112 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  U ) `  x )  =  ( A  .h  ( U `
 x ) ) )
1814, 17oveq12d 5876 . . . . 5  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  .op  T ) `
 x )  +h  ( ( A  .op  U ) `  x ) )  =  ( ( A  .h  ( T `
 x ) )  +h  ( A  .h  ( U `  x ) ) ) )
197, 11, 183eqtr4d 2325 . . . 4  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( ( T  +op  U ) `  x ) )  =  ( ( ( A  .op  T
) `  x )  +h  ( ( A  .op  U ) `  x ) ) )
20 hoaddcl 22338 . . . . . . 7  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( T  +op  U
) : ~H --> ~H )
2120anim2i 552 . . . . . 6  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  U : ~H --> ~H )
)  ->  ( A  e.  CC  /\  ( T 
+op  U ) : ~H --> ~H ) )
22213impb 1147 . . . . 5  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  e.  CC  /\  ( T  +op  U
) : ~H --> ~H )
)
23 homval 22321 . . . . . 6  |-  ( ( A  e.  CC  /\  ( T  +op  U ) : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  ( T  +op  U ) ) `  x )  =  ( A  .h  ( ( T  +op  U ) `  x ) ) )
24233expa 1151 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( T  +op  U
) : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  ( T  +op  U ) ) `  x
)  =  ( A  .h  ( ( T 
+op  U ) `  x ) ) )
2522, 24sylan 457 . . . 4  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  ( T  +op  U ) ) `  x
)  =  ( A  .h  ( ( T 
+op  U ) `  x ) ) )
26 homulcl 22339 . . . . . . 7  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
27 homulcl 22339 . . . . . . 7  |-  ( ( A  e.  CC  /\  U : ~H --> ~H )  ->  ( A  .op  U
) : ~H --> ~H )
2826, 27anim12i 549 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( A  e.  CC  /\  U : ~H --> ~H )
)  ->  ( ( A  .op  T ) : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H ) )
29283impdi 1237 . . . . 5  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T ) : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H )
)
30 hosval 22320 . . . . . 6  |-  ( ( ( A  .op  T
) : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( A 
.op  T )  +op  ( A  .op  U ) ) `  x )  =  ( ( ( A  .op  T ) `
 x )  +h  ( ( A  .op  U ) `  x ) ) )
31303expa 1151 . . . . 5  |-  ( ( ( ( A  .op  T ) : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  .op  T ) 
+op  ( A  .op  U ) ) `  x
)  =  ( ( ( A  .op  T
) `  x )  +h  ( ( A  .op  U ) `  x ) ) )
3229, 31sylan 457 . . . 4  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  .op  T ) 
+op  ( A  .op  U ) ) `  x
)  =  ( ( ( A  .op  T
) `  x )  +h  ( ( A  .op  U ) `  x ) ) )
3319, 25, 323eqtr4d 2325 . . 3  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  ( T  +op  U ) ) `  x
)  =  ( ( ( A  .op  T
)  +op  ( A  .op  U ) ) `  x ) )
3433ralrimiva 2626 . 2  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  A. x  e.  ~H  ( ( A  .op  ( T  +op  U ) ) `  x )  =  ( ( ( A  .op  T ) 
+op  ( A  .op  U ) ) `  x
) )
35 homulcl 22339 . . . . 5  |-  ( ( A  e.  CC  /\  ( T  +op  U ) : ~H --> ~H )  ->  ( A  .op  ( T  +op  U ) ) : ~H --> ~H )
3620, 35sylan2 460 . . . 4  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  U : ~H --> ~H )
)  ->  ( A  .op  ( T  +op  U
) ) : ~H --> ~H )
37363impb 1147 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  .op  ( T  +op  U ) ) : ~H --> ~H )
38 hoaddcl 22338 . . . . 5  |-  ( ( ( A  .op  T
) : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H )  ->  ( ( A  .op  T )  +op  ( A 
.op  U ) ) : ~H --> ~H )
3926, 27, 38syl2an 463 . . . 4  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( A  e.  CC  /\  U : ~H --> ~H )
)  ->  ( ( A  .op  T )  +op  ( A  .op  U ) ) : ~H --> ~H )
40393impdi 1237 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T )  +op  ( A 
.op  U ) ) : ~H --> ~H )
41 hoeq 22340 . . 3  |-  ( ( ( A  .op  ( T  +op  U ) ) : ~H --> ~H  /\  ( ( A  .op  T )  +op  ( A 
.op  U ) ) : ~H --> ~H )  ->  ( A. x  e. 
~H  ( ( A 
.op  ( T  +op  U ) ) `  x
)  =  ( ( ( A  .op  T
)  +op  ( A  .op  U ) ) `  x )  <->  ( A  .op  ( T  +op  U
) )  =  ( ( A  .op  T
)  +op  ( A  .op  U ) ) ) )
4237, 40, 41syl2anc 642 . 2  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A. x  e. 
~H  ( ( A 
.op  ( T  +op  U ) ) `  x
)  =  ( ( ( A  .op  T
)  +op  ( A  .op  U ) ) `  x )  <->  ( A  .op  ( T  +op  U
) )  =  ( ( A  .op  T
)  +op  ( A  .op  U ) ) ) )
4334, 42mpbid 201 1  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  .op  ( T  +op  U ) )  =  ( ( A 
.op  T )  +op  ( A  .op  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   ~Hchil 21499    +h cva 21500    .h csm 21501    +op chos 21518    .op chot 21519
This theorem is referenced by:  hosubdi  22388  honegdi  22389  ho2times  22399  opsqrlem6  22725
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-hilex 21579  ax-hfvadd 21580  ax-hfvmul 21585  ax-hvdistr1 21588
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-reu 2550  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-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-map 6774  df-hosum 22310  df-homul 22311
  Copyright terms: Public domain W3C validator