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Theorem hoadddi 27121
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 1000 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  A  e.  CC )
2 ffvelrn 6006 . . . . . . 7  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
323ad2antl2 1160 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( T `  x )  e.  ~H )
4 ffvelrn 6006 . . . . . . 7  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( U `  x
)  e.  ~H )
543ad2antl3 1161 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( U `  x )  e.  ~H )
6 ax-hvdistr1 26325 . . . . . 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 1230 . . . . 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 27058 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( T  +op  U ) `  x )  =  ( ( T `
 x )  +h  ( U `  x
) ) )
98oveq2d 6293 . . . . . . 7  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( A  .h  (
( T  +op  U
) `  x )
)  =  ( A  .h  ( ( T `
 x )  +h  ( U `  x
) ) ) )
1093expa 1197 . . . . . 6  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( ( T  +op  U ) `  x ) )  =  ( A  .h  (
( T `  x
)  +h  ( U `
 x ) ) ) )
11103adantl1 1153 . . . . 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 27059 . . . . . . . 8  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  T ) `  x )  =  ( A  .h  ( T `  x ) ) )
13123expa 1197 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
14133adantl3 1155 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
15 homval 27059 . . . . . . . 8  |-  ( ( A  e.  CC  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  U ) `  x )  =  ( A  .h  ( U `  x ) ) )
16153expa 1197 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  U ) `  x )  =  ( A  .h  ( U `
 x ) ) )
17163adantl2 1154 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  U ) `  x )  =  ( A  .h  ( U `
 x ) ) )
1814, 17oveq12d 6295 . . . . 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 2453 . . . 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 27076 . . . . . . 7  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( T  +op  U
) : ~H --> ~H )
2120anim2i 567 . . . . . 6  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  U : ~H --> ~H )
)  ->  ( A  e.  CC  /\  ( T 
+op  U ) : ~H --> ~H ) )
22213impb 1193 . . . . 5  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  e.  CC  /\  ( T  +op  U
) : ~H --> ~H )
)
23 homval 27059 . . . . . 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 1197 . . . . 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 469 . . . 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 27077 . . . . . . 7  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
27 homulcl 27077 . . . . . . 7  |-  ( ( A  e.  CC  /\  U : ~H --> ~H )  ->  ( A  .op  U
) : ~H --> ~H )
2826, 27anim12i 564 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( A  e.  CC  /\  U : ~H --> ~H )
)  ->  ( ( A  .op  T ) : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H ) )
29283impdi 1285 . . . . 5  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T ) : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H )
)
30 hosval 27058 . . . . . 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 1197 . . . . 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 469 . . . 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 2453 . . 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 2817 . 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 27077 . . . . 5  |-  ( ( A  e.  CC  /\  ( T  +op  U ) : ~H --> ~H )  ->  ( A  .op  ( T  +op  U ) ) : ~H --> ~H )
3620, 35sylan2 472 . . . 4  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  U : ~H --> ~H )
)  ->  ( A  .op  ( T  +op  U
) ) : ~H --> ~H )
37363impb 1193 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  .op  ( T  +op  U ) ) : ~H --> ~H )
38 hoaddcl 27076 . . . . 5  |-  ( ( ( A  .op  T
) : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H )  ->  ( ( A  .op  T )  +op  ( A 
.op  U ) ) : ~H --> ~H )
3926, 27, 38syl2an 475 . . . 4  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( A  e.  CC  /\  U : ~H --> ~H )
)  ->  ( ( A  .op  T )  +op  ( A  .op  U ) ) : ~H --> ~H )
40393impdi 1285 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T )  +op  ( A 
.op  U ) ) : ~H --> ~H )
41 hoeq 27078 . . 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 659 . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2753   -->wf 5564   ` cfv 5568  (class class class)co 6277   CCcc 9519   ~Hchil 26236    +h cva 26237    .h csm 26238    +op chos 26255    .op chot 26256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-hilex 26316  ax-hfvadd 26317  ax-hfvmul 26322  ax-hvdistr1 26325
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7458  df-hosum 27048  df-homul 27049
This theorem is referenced by:  hosubdi  27126  honegdi  27127  ho2times  27137  opsqrlem6  27463
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