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Theorem hoaddassi 22372
Description: Associativity of sum of Hilbert space operators. (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
hoaddassi  |-  ( ( R  +op  S ) 
+op  T )  =  ( R  +op  ( S  +op  T ) )

Proof of Theorem hoaddassi
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
3 hosval 22336 . . . . . 6  |-  ( ( R : ~H --> ~H  /\  S : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( R  +op  S ) `  x )  =  ( ( R `
 x )  +h  ( S `  x
) ) )
41, 2, 3mp3an12 1267 . . . . 5  |-  ( x  e.  ~H  ->  (
( R  +op  S
) `  x )  =  ( ( R `
 x )  +h  ( S `  x
) ) )
54oveq1d 5889 . . . 4  |-  ( x  e.  ~H  ->  (
( ( R  +op  S ) `  x )  +h  ( T `  x ) )  =  ( ( ( R `
 x )  +h  ( S `  x
) )  +h  ( T `  x )
) )
61, 2hoaddcli 22364 . . . . 5  |-  ( R 
+op  S ) : ~H --> ~H
7 hods.3 . . . . 5  |-  T : ~H
--> ~H
8 hosval 22336 . . . . 5  |-  ( ( ( R  +op  S
) : ~H --> ~H  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( R 
+op  S )  +op  T ) `  x )  =  ( ( ( R  +op  S ) `
 x )  +h  ( T `  x
) ) )
96, 7, 8mp3an12 1267 . . . 4  |-  ( x  e.  ~H  ->  (
( ( R  +op  S )  +op  T ) `
 x )  =  ( ( ( R 
+op  S ) `  x )  +h  ( T `  x )
) )
10 hosval 22336 . . . . . . 7  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( S  +op  T ) `  x )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
112, 7, 10mp3an12 1267 . . . . . 6  |-  ( x  e.  ~H  ->  (
( S  +op  T
) `  x )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
1211oveq2d 5890 . . . . 5  |-  ( x  e.  ~H  ->  (
( R `  x
)  +h  ( ( S  +op  T ) `
 x ) )  =  ( ( R `
 x )  +h  ( ( S `  x )  +h  ( T `  x )
) ) )
132, 7hoaddcli 22364 . . . . . 6  |-  ( S 
+op  T ) : ~H --> ~H
14 hosval 22336 . . . . . 6  |-  ( ( R : ~H --> ~H  /\  ( S  +op  T ) : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( R  +op  ( S  +op  T ) ) `  x )  =  ( ( R `
 x )  +h  ( ( S  +op  T ) `  x ) ) )
151, 13, 14mp3an12 1267 . . . . 5  |-  ( x  e.  ~H  ->  (
( R  +op  ( S  +op  T ) ) `
 x )  =  ( ( R `  x )  +h  (
( S  +op  T
) `  x )
) )
161ffvelrni 5680 . . . . . 6  |-  ( x  e.  ~H  ->  ( R `  x )  e.  ~H )
172ffvelrni 5680 . . . . . 6  |-  ( x  e.  ~H  ->  ( S `  x )  e.  ~H )
187ffvelrni 5680 . . . . . 6  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
19 ax-hvass 21598 . . . . . 6  |-  ( ( ( R `  x
)  e.  ~H  /\  ( S `  x )  e.  ~H  /\  ( T `  x )  e.  ~H )  ->  (
( ( R `  x )  +h  ( S `  x )
)  +h  ( T `
 x ) )  =  ( ( R `
 x )  +h  ( ( S `  x )  +h  ( T `  x )
) ) )
2016, 17, 18, 19syl3anc 1182 . . . . 5  |-  ( x  e.  ~H  ->  (
( ( R `  x )  +h  ( S `  x )
)  +h  ( T `
 x ) )  =  ( ( R `
 x )  +h  ( ( S `  x )  +h  ( T `  x )
) ) )
2112, 15, 203eqtr4d 2338 . . . 4  |-  ( x  e.  ~H  ->  (
( R  +op  ( S  +op  T ) ) `
 x )  =  ( ( ( R `
 x )  +h  ( S `  x
) )  +h  ( T `  x )
) )
225, 9, 213eqtr4d 2338 . . 3  |-  ( x  e.  ~H  ->  (
( ( R  +op  S )  +op  T ) `
 x )  =  ( ( R  +op  ( S  +op  T ) ) `  x ) )
2322rgen 2621 . 2  |-  A. x  e.  ~H  ( ( ( R  +op  S ) 
+op  T ) `  x )  =  ( ( R  +op  ( S  +op  T ) ) `
 x )
246, 7hoaddcli 22364 . . 3  |-  ( ( R  +op  S ) 
+op  T ) : ~H --> ~H
251, 13hoaddcli 22364 . . 3  |-  ( R 
+op  ( S  +op  T ) ) : ~H --> ~H
2624, 25hoeqi 22357 . 2  |-  ( A. x  e.  ~H  (
( ( R  +op  S )  +op  T ) `
 x )  =  ( ( R  +op  ( S  +op  T ) ) `  x )  <-> 
( ( R  +op  S )  +op  T )  =  ( R  +op  ( S  +op  T ) ) )
2723, 26mpbi 199 1  |-  ( ( R  +op  S ) 
+op  T )  =  ( R  +op  ( S  +op  T ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   A.wral 2556   -->wf 5267   ` cfv 5271  (class class class)co 5874   ~Hchil 21515    +h cva 21516    +op chos 21534
This theorem is referenced by:  hoadd12i  22373  hoadd32i  22374  hoaddass  22378  hosubeq0i  22422
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  ax-hvass 21598
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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