HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  ho2times Structured version   Unicode version

Theorem ho2times 26936
Description: Two times a Hilbert space operator. (Contributed by NM, 26-Aug-2006.) (New usage is discouraged.)
Assertion
Ref Expression
ho2times  |-  ( T : ~H --> ~H  ->  ( 2  .op  T )  =  ( T  +op  T ) )

Proof of Theorem ho2times
StepHypRef Expression
1 df-2 10590 . . . 4  |-  2  =  ( 1  +  1 )
21oveq1i 6280 . . 3  |-  ( 2 
.op  T )  =  ( ( 1  +  1 )  .op  T
)
3 ax-1cn 9539 . . . 4  |-  1  e.  CC
4 hoadddir 26921 . . . 4  |-  ( ( 1  e.  CC  /\  1  e.  CC  /\  T : ~H --> ~H )  -> 
( ( 1  +  1 )  .op  T
)  =  ( ( 1  .op  T ) 
+op  ( 1  .op 
T ) ) )
53, 3, 4mp3an12 1312 . . 3  |-  ( T : ~H --> ~H  ->  ( ( 1  +  1 )  .op  T )  =  ( ( 1 
.op  T )  +op  ( 1  .op  T
) ) )
62, 5syl5eq 2507 . 2  |-  ( T : ~H --> ~H  ->  ( 2  .op  T )  =  ( ( 1 
.op  T )  +op  ( 1  .op  T
) ) )
7 hoadddi 26920 . . . 4  |-  ( ( 1  e.  CC  /\  T : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( 1  .op  ( T  +op  T ) )  =  ( ( 1 
.op  T )  +op  ( 1  .op  T
) ) )
83, 7mp3an1 1309 . . 3  |-  ( ( T : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( 1  .op  ( T  +op  T ) )  =  ( ( 1 
.op  T )  +op  ( 1  .op  T
) ) )
98anidms 643 . 2  |-  ( T : ~H --> ~H  ->  ( 1  .op  ( T 
+op  T ) )  =  ( ( 1 
.op  T )  +op  ( 1  .op  T
) ) )
10 hoaddcl 26875 . . . 4  |-  ( ( T : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( T  +op  T
) : ~H --> ~H )
1110anidms 643 . . 3  |-  ( T : ~H --> ~H  ->  ( T  +op  T ) : ~H --> ~H )
12 homulid2 26917 . . 3  |-  ( ( T  +op  T ) : ~H --> ~H  ->  ( 1  .op  ( T 
+op  T ) )  =  ( T  +op  T ) )
1311, 12syl 16 . 2  |-  ( T : ~H --> ~H  ->  ( 1  .op  ( T 
+op  T ) )  =  ( T  +op  T ) )
146, 9, 133eqtr2d 2501 1  |-  ( T : ~H --> ~H  ->  ( 2  .op  T )  =  ( T  +op  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   -->wf 5566  (class class class)co 6270   CCcc 9479   1c1 9482    + caddc 9484   2c2 10581   ~Hchil 26034    +op chos 26053    .op chot 26054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-1cn 9539  ax-addcl 9541  ax-hilex 26114  ax-hfvadd 26115  ax-hfvmul 26120  ax-hvmulid 26121  ax-hvdistr1 26123  ax-hvdistr2 26124
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-2 10590  df-hosum 26847  df-homul 26848
This theorem is referenced by:  opsqrlem6  27262
  Copyright terms: Public domain W3C validator