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Theorem hocadddiri 22375
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 22364 . . . . 5  |-  ( R 
+op  S ) : ~H --> ~H
4 hods.3 . . . . 5  |-  T : ~H
--> ~H
53, 4hocoi 22360 . . . 4  |-  ( x  e.  ~H  ->  (
( ( R  +op  S )  o.  T ) `
 x )  =  ( ( R  +op  S ) `  ( T `
 x ) ) )
61, 4hocofi 22362 . . . . . 6  |-  ( R  o.  T ) : ~H --> ~H
72, 4hocofi 22362 . . . . . 6  |-  ( S  o.  T ) : ~H --> ~H
8 hosval 22336 . . . . . 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 1267 . . . . 5  |-  ( x  e.  ~H  ->  (
( ( R  o.  T )  +op  ( S  o.  T )
) `  x )  =  ( ( ( R  o.  T ) `
 x )  +h  ( ( S  o.  T ) `  x
) ) )
104ffvelrni 5680 . . . . . . 7  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
11 hosval 22336 . . . . . . . 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 1267 . . . . . . 7  |-  ( ( T `  x )  e.  ~H  ->  (
( R  +op  S
) `  ( T `  x ) )  =  ( ( R `  ( T `  x ) )  +h  ( S `
 ( T `  x ) ) ) )
1310, 12syl 15 . . . . . 6  |-  ( x  e.  ~H  ->  (
( R  +op  S
) `  ( T `  x ) )  =  ( ( R `  ( T `  x ) )  +h  ( S `
 ( T `  x ) ) ) )
141, 4hocoi 22360 . . . . . . 7  |-  ( x  e.  ~H  ->  (
( R  o.  T
) `  x )  =  ( R `  ( T `  x ) ) )
152, 4hocoi 22360 . . . . . . 7  |-  ( x  e.  ~H  ->  (
( S  o.  T
) `  x )  =  ( S `  ( T `  x ) ) )
1614, 15oveq12d 5892 . . . . . 6  |-  ( x  e.  ~H  ->  (
( ( R  o.  T ) `  x
)  +h  ( ( S  o.  T ) `
 x ) )  =  ( ( R `
 ( T `  x ) )  +h  ( S `  ( T `  x )
) ) )
1713, 16eqtr4d 2331 . . . . 5  |-  ( x  e.  ~H  ->  (
( R  +op  S
) `  ( T `  x ) )  =  ( ( ( R  o.  T ) `  x )  +h  (
( S  o.  T
) `  x )
) )
189, 17eqtr4d 2331 . . . 4  |-  ( x  e.  ~H  ->  (
( ( R  o.  T )  +op  ( S  o.  T )
) `  x )  =  ( ( R 
+op  S ) `  ( T `  x ) ) )
195, 18eqtr4d 2331 . . 3  |-  ( x  e.  ~H  ->  (
( ( R  +op  S )  o.  T ) `
 x )  =  ( ( ( R  o.  T )  +op  ( S  o.  T
) ) `  x
) )
2019rgen 2621 . 2  |-  A. x  e.  ~H  ( ( ( R  +op  S )  o.  T ) `  x )  =  ( ( ( R  o.  T )  +op  ( S  o.  T )
) `  x )
213, 4hocofi 22362 . . 3  |-  ( ( R  +op  S )  o.  T ) : ~H --> ~H
226, 7hoaddcli 22364 . . 3  |-  ( ( R  o.  T ) 
+op  ( S  o.  T ) ) : ~H --> ~H
2321, 22hoeqi 22357 . 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 199 1  |-  ( ( R  +op  S )  o.  T )  =  ( ( R  o.  T )  +op  ( S  o.  T )
)
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   A.wral 2556    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   ~Hchil 21515    +h cva 21516    +op chos 21534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-hilex 21595  ax-hfvadd 21596
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-hosum 22326
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