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Theorem hoaddassi 25115
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 25079 . . . . . 6  |-  ( ( R : ~H --> ~H  /\  S : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( R  +op  S ) `  x )  =  ( ( R `
 x )  +h  ( S `  x
) ) )
41, 2, 3mp3an12 1299 . . . . 5  |-  ( x  e.  ~H  ->  (
( R  +op  S
) `  x )  =  ( ( R `
 x )  +h  ( S `  x
) ) )
54oveq1d 6105 . . . 4  |-  ( x  e.  ~H  ->  (
( ( R  +op  S ) `  x )  +h  ( T `  x ) )  =  ( ( ( R `
 x )  +h  ( S `  x
) )  +h  ( T `  x )
) )
61, 2hoaddcli 25107 . . . . 5  |-  ( R 
+op  S ) : ~H --> ~H
7 hods.3 . . . . 5  |-  T : ~H
--> ~H
8 hosval 25079 . . . . 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 1299 . . . 4  |-  ( x  e.  ~H  ->  (
( ( R  +op  S )  +op  T ) `
 x )  =  ( ( ( R 
+op  S ) `  x )  +h  ( T `  x )
) )
10 hosval 25079 . . . . . . 7  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( S  +op  T ) `  x )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
112, 7, 10mp3an12 1299 . . . . . 6  |-  ( x  e.  ~H  ->  (
( S  +op  T
) `  x )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
1211oveq2d 6106 . . . . 5  |-  ( x  e.  ~H  ->  (
( R `  x
)  +h  ( ( S  +op  T ) `
 x ) )  =  ( ( R `
 x )  +h  ( ( S `  x )  +h  ( T `  x )
) ) )
132, 7hoaddcli 25107 . . . . . 6  |-  ( S 
+op  T ) : ~H --> ~H
14 hosval 25079 . . . . . 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 1299 . . . . 5  |-  ( x  e.  ~H  ->  (
( R  +op  ( S  +op  T ) ) `
 x )  =  ( ( R `  x )  +h  (
( S  +op  T
) `  x )
) )
161ffvelrni 5839 . . . . . 6  |-  ( x  e.  ~H  ->  ( R `  x )  e.  ~H )
172ffvelrni 5839 . . . . . 6  |-  ( x  e.  ~H  ->  ( S `  x )  e.  ~H )
187ffvelrni 5839 . . . . . 6  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
19 ax-hvass 24339 . . . . . 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 1213 . . . . 5  |-  ( x  e.  ~H  ->  (
( ( R `  x )  +h  ( S `  x )
)  +h  ( T `
 x ) )  =  ( ( R `
 x )  +h  ( ( S `  x )  +h  ( T `  x )
) ) )
2112, 15, 203eqtr4d 2483 . . . 4  |-  ( x  e.  ~H  ->  (
( R  +op  ( S  +op  T ) ) `
 x )  =  ( ( ( R `
 x )  +h  ( S `  x
) )  +h  ( T `  x )
) )
225, 9, 213eqtr4d 2483 . . 3  |-  ( x  e.  ~H  ->  (
( ( R  +op  S )  +op  T ) `
 x )  =  ( ( R  +op  ( S  +op  T ) ) `  x ) )
2322rgen 2779 . 2  |-  A. x  e.  ~H  ( ( ( R  +op  S ) 
+op  T ) `  x )  =  ( ( R  +op  ( S  +op  T ) ) `
 x )
246, 7hoaddcli 25107 . . 3  |-  ( ( R  +op  S ) 
+op  T ) : ~H --> ~H
251, 13hoaddcli 25107 . . 3  |-  ( R 
+op  ( S  +op  T ) ) : ~H --> ~H
2624, 25hoeqi 25100 . 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 208 1  |-  ( ( R  +op  S ) 
+op  T )  =  ( R  +op  ( S  +op  T ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1364    e. wcel 1761   A.wral 2713   -->wf 5411   ` cfv 5415  (class class class)co 6090   ~Hchil 24256    +h cva 24257    +op chos 24275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-hilex 24336  ax-hfvadd 24337  ax-hvass 24339
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-map 7212  df-hosum 25069
This theorem is referenced by:  hoadd12i  25116  hoadd32i  25117  hoaddass  25121  hosubeq0i  25165
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