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Theorem hoaddassi 23120
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 23084 . . . . . 6  |-  ( ( R : ~H --> ~H  /\  S : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( R  +op  S ) `  x )  =  ( ( R `
 x )  +h  ( S `  x
) ) )
41, 2, 3mp3an12 1269 . . . . 5  |-  ( x  e.  ~H  ->  (
( R  +op  S
) `  x )  =  ( ( R `
 x )  +h  ( S `  x
) ) )
54oveq1d 6028 . . . 4  |-  ( x  e.  ~H  ->  (
( ( R  +op  S ) `  x )  +h  ( T `  x ) )  =  ( ( ( R `
 x )  +h  ( S `  x
) )  +h  ( T `  x )
) )
61, 2hoaddcli 23112 . . . . 5  |-  ( R 
+op  S ) : ~H --> ~H
7 hods.3 . . . . 5  |-  T : ~H
--> ~H
8 hosval 23084 . . . . 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 1269 . . . 4  |-  ( x  e.  ~H  ->  (
( ( R  +op  S )  +op  T ) `
 x )  =  ( ( ( R 
+op  S ) `  x )  +h  ( T `  x )
) )
10 hosval 23084 . . . . . . 7  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( S  +op  T ) `  x )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
112, 7, 10mp3an12 1269 . . . . . 6  |-  ( x  e.  ~H  ->  (
( S  +op  T
) `  x )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
1211oveq2d 6029 . . . . 5  |-  ( x  e.  ~H  ->  (
( R `  x
)  +h  ( ( S  +op  T ) `
 x ) )  =  ( ( R `
 x )  +h  ( ( S `  x )  +h  ( T `  x )
) ) )
132, 7hoaddcli 23112 . . . . . 6  |-  ( S 
+op  T ) : ~H --> ~H
14 hosval 23084 . . . . . 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 1269 . . . . 5  |-  ( x  e.  ~H  ->  (
( R  +op  ( S  +op  T ) ) `
 x )  =  ( ( R `  x )  +h  (
( S  +op  T
) `  x )
) )
161ffvelrni 5801 . . . . . 6  |-  ( x  e.  ~H  ->  ( R `  x )  e.  ~H )
172ffvelrni 5801 . . . . . 6  |-  ( x  e.  ~H  ->  ( S `  x )  e.  ~H )
187ffvelrni 5801 . . . . . 6  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
19 ax-hvass 22346 . . . . . 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 1184 . . . . 5  |-  ( x  e.  ~H  ->  (
( ( R `  x )  +h  ( S `  x )
)  +h  ( T `
 x ) )  =  ( ( R `
 x )  +h  ( ( S `  x )  +h  ( T `  x )
) ) )
2112, 15, 203eqtr4d 2422 . . . 4  |-  ( x  e.  ~H  ->  (
( R  +op  ( S  +op  T ) ) `
 x )  =  ( ( ( R `
 x )  +h  ( S `  x
) )  +h  ( T `  x )
) )
225, 9, 213eqtr4d 2422 . . 3  |-  ( x  e.  ~H  ->  (
( ( R  +op  S )  +op  T ) `
 x )  =  ( ( R  +op  ( S  +op  T ) ) `  x ) )
2322rgen 2707 . 2  |-  A. x  e.  ~H  ( ( ( R  +op  S ) 
+op  T ) `  x )  =  ( ( R  +op  ( S  +op  T ) ) `
 x )
246, 7hoaddcli 23112 . . 3  |-  ( ( R  +op  S ) 
+op  T ) : ~H --> ~H
251, 13hoaddcli 23112 . . 3  |-  ( R 
+op  ( S  +op  T ) ) : ~H --> ~H
2624, 25hoeqi 23105 . 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 200 1  |-  ( ( R  +op  S ) 
+op  T )  =  ( R  +op  ( S  +op  T ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717   A.wral 2642   -->wf 5383   ` cfv 5387  (class class class)co 6013   ~Hchil 22263    +h cva 22264    +op chos 22282
This theorem is referenced by:  hoadd12i  23121  hoadd32i  23122  hoaddass  23126  hosubeq0i  23170
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-hilex 22343  ax-hfvadd 22344  ax-hvass 22346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-map 6949  df-hosum 23074
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