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Theorem hoadddi 23259
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 960 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  A  e.  CC )
2 ffvelrn 5827 . . . . . . 7  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
323ad2antl2 1120 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( T `  x )  e.  ~H )
4 ffvelrn 5827 . . . . . . 7  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( U `  x
)  e.  ~H )
543ad2antl3 1121 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( U `  x )  e.  ~H )
6 ax-hvdistr1 22464 . . . . . 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 1184 . . . . 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 23196 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( T  +op  U ) `  x )  =  ( ( T `
 x )  +h  ( U `  x
) ) )
98oveq2d 6056 . . . . . . 7  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( A  .h  (
( T  +op  U
) `  x )
)  =  ( A  .h  ( ( T `
 x )  +h  ( U `  x
) ) ) )
1093expa 1153 . . . . . 6  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( ( T  +op  U ) `  x ) )  =  ( A  .h  (
( T `  x
)  +h  ( U `
 x ) ) ) )
11103adantl1 1113 . . . . 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 23197 . . . . . . . 8  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  T ) `  x )  =  ( A  .h  ( T `  x ) ) )
13123expa 1153 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
14133adantl3 1115 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
15 homval 23197 . . . . . . . 8  |-  ( ( A  e.  CC  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  U ) `  x )  =  ( A  .h  ( U `  x ) ) )
16153expa 1153 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  U ) `  x )  =  ( A  .h  ( U `
 x ) ) )
17163adantl2 1114 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  U ) `  x )  =  ( A  .h  ( U `
 x ) ) )
1814, 17oveq12d 6058 . . . . 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 2446 . . . 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 23214 . . . . . . 7  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( T  +op  U
) : ~H --> ~H )
2120anim2i 553 . . . . . 6  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  U : ~H --> ~H )
)  ->  ( A  e.  CC  /\  ( T 
+op  U ) : ~H --> ~H ) )
22213impb 1149 . . . . 5  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  e.  CC  /\  ( T  +op  U
) : ~H --> ~H )
)
23 homval 23197 . . . . . 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 1153 . . . . 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 458 . . . 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 23215 . . . . . . 7  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
27 homulcl 23215 . . . . . . 7  |-  ( ( A  e.  CC  /\  U : ~H --> ~H )  ->  ( A  .op  U
) : ~H --> ~H )
2826, 27anim12i 550 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( A  e.  CC  /\  U : ~H --> ~H )
)  ->  ( ( A  .op  T ) : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H ) )
29283impdi 1239 . . . . 5  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T ) : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H )
)
30 hosval 23196 . . . . . 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 1153 . . . . 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 458 . . . 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 2446 . . 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 2749 . 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 23215 . . . . 5  |-  ( ( A  e.  CC  /\  ( T  +op  U ) : ~H --> ~H )  ->  ( A  .op  ( T  +op  U ) ) : ~H --> ~H )
3620, 35sylan2 461 . . . 4  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  U : ~H --> ~H )
)  ->  ( A  .op  ( T  +op  U
) ) : ~H --> ~H )
37363impb 1149 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  .op  ( T  +op  U ) ) : ~H --> ~H )
38 hoaddcl 23214 . . . . 5  |-  ( ( ( A  .op  T
) : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H )  ->  ( ( A  .op  T )  +op  ( A 
.op  U ) ) : ~H --> ~H )
3926, 27, 38syl2an 464 . . . 4  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( A  e.  CC  /\  U : ~H --> ~H )
)  ->  ( ( A  .op  T )  +op  ( A  .op  U ) ) : ~H --> ~H )
40393impdi 1239 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T )  +op  ( A 
.op  U ) ) : ~H --> ~H )
41 hoeq 23216 . . 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 643 . 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 202 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   ~Hchil 22375    +h cva 22376    .h csm 22377    +op chos 22394    .op chot 22395
This theorem is referenced by:  hosubdi  23264  honegdi  23265  ho2times  23275  opsqrlem6  23601
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-hilex 22455  ax-hfvadd 22456  ax-hfvmul 22461  ax-hvdistr1 22464
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-map 6979  df-hosum 23186  df-homul 23187
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