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Theorem hocadddiri 22359
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 22348 . . . . 5  |-  ( R 
+op  S ) : ~H --> ~H
4 hods.3 . . . . 5  |-  T : ~H
--> ~H
53, 4hocoi 22344 . . . 4  |-  ( x  e.  ~H  ->  (
( ( R  +op  S )  o.  T ) `
 x )  =  ( ( R  +op  S ) `  ( T `
 x ) ) )
61, 4hocofi 22346 . . . . . 6  |-  ( R  o.  T ) : ~H --> ~H
72, 4hocofi 22346 . . . . . 6  |-  ( S  o.  T ) : ~H --> ~H
8 hosval 22320 . . . . . 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 5664 . . . . . . 7  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
11 hosval 22320 . . . . . . . 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 22344 . . . . . . 7  |-  ( x  e.  ~H  ->  (
( R  o.  T
) `  x )  =  ( R `  ( T `  x ) ) )
152, 4hocoi 22344 . . . . . . 7  |-  ( x  e.  ~H  ->  (
( S  o.  T
) `  x )  =  ( S `  ( T `  x ) ) )
1614, 15oveq12d 5876 . . . . . 6  |-  ( x  e.  ~H  ->  (
( ( R  o.  T ) `  x
)  +h  ( ( S  o.  T ) `
 x ) )  =  ( ( R `
 ( T `  x ) )  +h  ( S `  ( T `  x )
) ) )
1713, 16eqtr4d 2318 . . . . 5  |-  ( x  e.  ~H  ->  (
( R  +op  S
) `  ( T `  x ) )  =  ( ( ( R  o.  T ) `  x )  +h  (
( S  o.  T
) `  x )
) )
189, 17eqtr4d 2318 . . . 4  |-  ( x  e.  ~H  ->  (
( ( R  o.  T )  +op  ( S  o.  T )
) `  x )  =  ( ( R 
+op  S ) `  ( T `  x ) ) )
195, 18eqtr4d 2318 . . 3  |-  ( x  e.  ~H  ->  (
( ( R  +op  S )  o.  T ) `
 x )  =  ( ( ( R  o.  T )  +op  ( S  o.  T
) ) `  x
) )
2019rgen 2608 . 2  |-  A. x  e.  ~H  ( ( ( R  +op  S )  o.  T ) `  x )  =  ( ( ( R  o.  T )  +op  ( S  o.  T )
) `  x )
213, 4hocofi 22346 . . 3  |-  ( ( R  +op  S )  o.  T ) : ~H --> ~H
226, 7hoaddcli 22348 . . 3  |-  ( ( R  o.  T ) 
+op  ( S  o.  T ) ) : ~H --> ~H
2321, 22hoeqi 22341 . 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 1623    e. wcel 1684   A.wral 2543    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   ~Hchil 21499    +h cva 21500    +op chos 21518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-hilex 21579  ax-hfvadd 21580
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-hosum 22310
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