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Theorem hoaddassi 23271
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 23235 . . . . . 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 6088 . . . 4  |-  ( x  e.  ~H  ->  (
( ( R  +op  S ) `  x )  +h  ( T `  x ) )  =  ( ( ( R `
 x )  +h  ( S `  x
) )  +h  ( T `  x )
) )
61, 2hoaddcli 23263 . . . . 5  |-  ( R 
+op  S ) : ~H --> ~H
7 hods.3 . . . . 5  |-  T : ~H
--> ~H
8 hosval 23235 . . . . 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 23235 . . . . . . 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 6089 . . . . 5  |-  ( x  e.  ~H  ->  (
( R `  x
)  +h  ( ( S  +op  T ) `
 x ) )  =  ( ( R `
 x )  +h  ( ( S `  x )  +h  ( T `  x )
) ) )
132, 7hoaddcli 23263 . . . . . 6  |-  ( S 
+op  T ) : ~H --> ~H
14 hosval 23235 . . . . . 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 5861 . . . . . 6  |-  ( x  e.  ~H  ->  ( R `  x )  e.  ~H )
172ffvelrni 5861 . . . . . 6  |-  ( x  e.  ~H  ->  ( S `  x )  e.  ~H )
187ffvelrni 5861 . . . . . 6  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
19 ax-hvass 22497 . . . . . 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 2477 . . . 4  |-  ( x  e.  ~H  ->  (
( R  +op  ( S  +op  T ) ) `
 x )  =  ( ( ( R `
 x )  +h  ( S `  x
) )  +h  ( T `  x )
) )
225, 9, 213eqtr4d 2477 . . 3  |-  ( x  e.  ~H  ->  (
( ( R  +op  S )  +op  T ) `
 x )  =  ( ( R  +op  ( S  +op  T ) ) `  x ) )
2322rgen 2763 . 2  |-  A. x  e.  ~H  ( ( ( R  +op  S ) 
+op  T ) `  x )  =  ( ( R  +op  ( S  +op  T ) ) `
 x )
246, 7hoaddcli 23263 . . 3  |-  ( ( R  +op  S ) 
+op  T ) : ~H --> ~H
251, 13hoaddcli 23263 . . 3  |-  ( R 
+op  ( S  +op  T ) ) : ~H --> ~H
2624, 25hoeqi 23256 . 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 1652    e. wcel 1725   A.wral 2697   -->wf 5442   ` cfv 5446  (class class class)co 6073   ~Hchil 22414    +h cva 22415    +op chos 22433
This theorem is referenced by:  hoadd12i  23272  hoadd32i  23273  hoaddass  23277  hosubeq0i  23321
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-hilex 22494  ax-hfvadd 22495  ax-hvass 22497
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-hosum 23225
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