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Theorem hoadddi 25126
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 986 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  A  e.  CC )
2 ffvelrn 5838 . . . . . . 7  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
323ad2antl2 1146 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( T `  x )  e.  ~H )
4 ffvelrn 5838 . . . . . . 7  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( U `  x
)  e.  ~H )
543ad2antl3 1147 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( U `  x )  e.  ~H )
6 ax-hvdistr1 24329 . . . . . 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 1213 . . . . 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 25063 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( T  +op  U ) `  x )  =  ( ( T `
 x )  +h  ( U `  x
) ) )
98oveq2d 6106 . . . . . . 7  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( A  .h  (
( T  +op  U
) `  x )
)  =  ( A  .h  ( ( T `
 x )  +h  ( U `  x
) ) ) )
1093expa 1182 . . . . . 6  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( ( T  +op  U ) `  x ) )  =  ( A  .h  (
( T `  x
)  +h  ( U `
 x ) ) ) )
11103adantl1 1139 . . . . 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 25064 . . . . . . . 8  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  T ) `  x )  =  ( A  .h  ( T `  x ) ) )
13123expa 1182 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
14133adantl3 1141 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
15 homval 25064 . . . . . . . 8  |-  ( ( A  e.  CC  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  U ) `  x )  =  ( A  .h  ( U `  x ) ) )
16153expa 1182 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  U ) `  x )  =  ( A  .h  ( U `
 x ) ) )
17163adantl2 1140 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  U ) `  x )  =  ( A  .h  ( U `
 x ) ) )
1814, 17oveq12d 6108 . . . . 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 2483 . . . 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 25081 . . . . . . 7  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( T  +op  U
) : ~H --> ~H )
2120anim2i 566 . . . . . 6  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  U : ~H --> ~H )
)  ->  ( A  e.  CC  /\  ( T 
+op  U ) : ~H --> ~H ) )
22213impb 1178 . . . . 5  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  e.  CC  /\  ( T  +op  U
) : ~H --> ~H )
)
23 homval 25064 . . . . . 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 1182 . . . . 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 468 . . . 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 25082 . . . . . . 7  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
27 homulcl 25082 . . . . . . 7  |-  ( ( A  e.  CC  /\  U : ~H --> ~H )  ->  ( A  .op  U
) : ~H --> ~H )
2826, 27anim12i 563 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( A  e.  CC  /\  U : ~H --> ~H )
)  ->  ( ( A  .op  T ) : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H ) )
29283impdi 1268 . . . . 5  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T ) : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H )
)
30 hosval 25063 . . . . . 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 1182 . . . . 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 468 . . . 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 2483 . . 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 2797 . 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 25082 . . . . 5  |-  ( ( A  e.  CC  /\  ( T  +op  U ) : ~H --> ~H )  ->  ( A  .op  ( T  +op  U ) ) : ~H --> ~H )
3620, 35sylan2 471 . . . 4  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  U : ~H --> ~H )
)  ->  ( A  .op  ( T  +op  U
) ) : ~H --> ~H )
37363impb 1178 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  .op  ( T  +op  U ) ) : ~H --> ~H )
38 hoaddcl 25081 . . . . 5  |-  ( ( ( A  .op  T
) : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H )  ->  ( ( A  .op  T )  +op  ( A 
.op  U ) ) : ~H --> ~H )
3926, 27, 38syl2an 474 . . . 4  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( A  e.  CC  /\  U : ~H --> ~H )
)  ->  ( ( A  .op  T )  +op  ( A  .op  U ) ) : ~H --> ~H )
40393impdi 1268 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T )  +op  ( A 
.op  U ) ) : ~H --> ~H )
41 hoeq 25083 . . 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 656 . 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 210 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713   -->wf 5411   ` cfv 5415  (class class class)co 6090   CCcc 9276   ~Hchil 24240    +h cva 24241    .h csm 24242    +op chos 24259    .op chot 24260
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-hilex 24320  ax-hfvadd 24321  ax-hfvmul 24326  ax-hvdistr1 24329
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  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-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-ov 6093  df-oprab 6094  df-mpt2 6095  df-map 7212  df-hosum 25053  df-homul 25054
This theorem is referenced by:  hosubdi  25131  honegdi  25132  ho2times  25142  opsqrlem6  25468
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