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Theorem hocadddiri 23283
Description: Distributive law for Hilbert space operator sum. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
Hypotheses
Ref Expression
hods.1  |-  R : ~H
--> ~H
hods.2  |-  S : ~H
--> ~H
hods.3  |-  T : ~H
--> ~H
Assertion
Ref Expression
hocadddiri  |-  ( ( R  +op  S )  o.  T )  =  ( ( R  o.  T )  +op  ( S  o.  T )
)

Proof of Theorem hocadddiri
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hods.1 . . . . . 6  |-  R : ~H
--> ~H
2 hods.2 . . . . . 6  |-  S : ~H
--> ~H
31, 2hoaddcli 23272 . . . . 5  |-  ( R 
+op  S ) : ~H --> ~H
4 hods.3 . . . . 5  |-  T : ~H
--> ~H
53, 4hocoi 23268 . . . 4  |-  ( x  e.  ~H  ->  (
( ( R  +op  S )  o.  T ) `
 x )  =  ( ( R  +op  S ) `  ( T `
 x ) ) )
61, 4hocofi 23270 . . . . . 6  |-  ( R  o.  T ) : ~H --> ~H
72, 4hocofi 23270 . . . . . 6  |-  ( S  o.  T ) : ~H --> ~H
8 hosval 23244 . . . . . 6  |-  ( ( ( R  o.  T
) : ~H --> ~H  /\  ( S  o.  T
) : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( R  o.  T )  +op  ( S  o.  T
) ) `  x
)  =  ( ( ( R  o.  T
) `  x )  +h  ( ( S  o.  T ) `  x
) ) )
96, 7, 8mp3an12 1270 . . . . 5  |-  ( x  e.  ~H  ->  (
( ( R  o.  T )  +op  ( S  o.  T )
) `  x )  =  ( ( ( R  o.  T ) `
 x )  +h  ( ( S  o.  T ) `  x
) ) )
104ffvelrni 5870 . . . . . . 7  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
11 hosval 23244 . . . . . . . 8  |-  ( ( R : ~H --> ~H  /\  S : ~H --> ~H  /\  ( T `  x )  e.  ~H )  -> 
( ( R  +op  S ) `  ( T `
 x ) )  =  ( ( R `
 ( T `  x ) )  +h  ( S `  ( T `  x )
) ) )
121, 2, 11mp3an12 1270 . . . . . . 7  |-  ( ( T `  x )  e.  ~H  ->  (
( R  +op  S
) `  ( T `  x ) )  =  ( ( R `  ( T `  x ) )  +h  ( S `
 ( T `  x ) ) ) )
1310, 12syl 16 . . . . . 6  |-  ( x  e.  ~H  ->  (
( R  +op  S
) `  ( T `  x ) )  =  ( ( R `  ( T `  x ) )  +h  ( S `
 ( T `  x ) ) ) )
141, 4hocoi 23268 . . . . . . 7  |-  ( x  e.  ~H  ->  (
( R  o.  T
) `  x )  =  ( R `  ( T `  x ) ) )
152, 4hocoi 23268 . . . . . . 7  |-  ( x  e.  ~H  ->  (
( S  o.  T
) `  x )  =  ( S `  ( T `  x ) ) )
1614, 15oveq12d 6100 . . . . . 6  |-  ( x  e.  ~H  ->  (
( ( R  o.  T ) `  x
)  +h  ( ( S  o.  T ) `
 x ) )  =  ( ( R `
 ( T `  x ) )  +h  ( S `  ( T `  x )
) ) )
1713, 16eqtr4d 2472 . . . . 5  |-  ( x  e.  ~H  ->  (
( R  +op  S
) `  ( T `  x ) )  =  ( ( ( R  o.  T ) `  x )  +h  (
( S  o.  T
) `  x )
) )
189, 17eqtr4d 2472 . . . 4  |-  ( x  e.  ~H  ->  (
( ( R  o.  T )  +op  ( S  o.  T )
) `  x )  =  ( ( R 
+op  S ) `  ( T `  x ) ) )
195, 18eqtr4d 2472 . . 3  |-  ( x  e.  ~H  ->  (
( ( R  +op  S )  o.  T ) `
 x )  =  ( ( ( R  o.  T )  +op  ( S  o.  T
) ) `  x
) )
2019rgen 2772 . 2  |-  A. x  e.  ~H  ( ( ( R  +op  S )  o.  T ) `  x )  =  ( ( ( R  o.  T )  +op  ( S  o.  T )
) `  x )
213, 4hocofi 23270 . . 3  |-  ( ( R  +op  S )  o.  T ) : ~H --> ~H
226, 7hoaddcli 23272 . . 3  |-  ( ( R  o.  T ) 
+op  ( S  o.  T ) ) : ~H --> ~H
2321, 22hoeqi 23265 . 2  |-  ( A. x  e.  ~H  (
( ( R  +op  S )  o.  T ) `
 x )  =  ( ( ( R  o.  T )  +op  ( S  o.  T
) ) `  x
)  <->  ( ( R 
+op  S )  o.  T )  =  ( ( R  o.  T
)  +op  ( S  o.  T ) ) )
2420, 23mpbi 201 1  |-  ( ( R  +op  S )  o.  T )  =  ( ( R  o.  T )  +op  ( S  o.  T )
)
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   A.wral 2706    o. ccom 4883   -->wf 5451   ` cfv 5455  (class class class)co 6082   ~Hchil 22423    +h cva 22424    +op chos 22442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-hilex 22503  ax-hfvadd 22504
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-map 7021  df-hosum 23234
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