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Theorem hoaddcomi 23277
Description: Commutativity of sum of Hilbert space operators. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
Hypotheses
Ref Expression
hoeq.1  |-  S : ~H
--> ~H
hoeq.2  |-  T : ~H
--> ~H
Assertion
Ref Expression
hoaddcomi  |-  ( S 
+op  T )  =  ( T  +op  S
)

Proof of Theorem hoaddcomi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hoeq.1 . . . . . 6  |-  S : ~H
--> ~H
21ffvelrni 5871 . . . . 5  |-  ( x  e.  ~H  ->  ( S `  x )  e.  ~H )
3 hoeq.2 . . . . . 6  |-  T : ~H
--> ~H
43ffvelrni 5871 . . . . 5  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
5 ax-hvcom 22506 . . . . 5  |-  ( ( ( S `  x
)  e.  ~H  /\  ( T `  x )  e.  ~H )  -> 
( ( S `  x )  +h  ( T `  x )
)  =  ( ( T `  x )  +h  ( S `  x ) ) )
62, 4, 5syl2anc 644 . . . 4  |-  ( x  e.  ~H  ->  (
( S `  x
)  +h  ( T `
 x ) )  =  ( ( T `
 x )  +h  ( S `  x
) ) )
7 hosval 23245 . . . . 5  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( S  +op  T ) `  x )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
81, 3, 7mp3an12 1270 . . . 4  |-  ( x  e.  ~H  ->  (
( S  +op  T
) `  x )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
9 hosval 23245 . . . . 5  |-  ( ( T : ~H --> ~H  /\  S : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( T  +op  S ) `  x )  =  ( ( T `
 x )  +h  ( S `  x
) ) )
103, 1, 9mp3an12 1270 . . . 4  |-  ( x  e.  ~H  ->  (
( T  +op  S
) `  x )  =  ( ( T `
 x )  +h  ( S `  x
) ) )
116, 8, 103eqtr4d 2480 . . 3  |-  ( x  e.  ~H  ->  (
( S  +op  T
) `  x )  =  ( ( T 
+op  S ) `  x ) )
1211rgen 2773 . 2  |-  A. x  e.  ~H  ( ( S 
+op  T ) `  x )  =  ( ( T  +op  S
) `  x )
131, 3hoaddcli 23273 . . 3  |-  ( S 
+op  T ) : ~H --> ~H
143, 1hoaddcli 23273 . . 3  |-  ( T 
+op  S ) : ~H --> ~H
1513, 14hoeqi 23266 . 2  |-  ( A. x  e.  ~H  (
( S  +op  T
) `  x )  =  ( ( T 
+op  S ) `  x )  <->  ( S  +op  T )  =  ( T  +op  S ) )
1612, 15mpbi 201 1  |-  ( S 
+op  T )  =  ( T  +op  S
)
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   A.wral 2707   -->wf 5452   ` cfv 5456  (class class class)co 6083   ~Hchil 22424    +h cva 22425    +op chos 22443
This theorem is referenced by:  hoaddcom  23279  hoadd12i  23282  hoadd32i  23283  hoaddsubi  23326  hosd1i  23327  hosubeq0i  23331
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-hilex 22504  ax-hfvadd 22505  ax-hvcom 22506
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-hosum 23235
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