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Theorem ho2times 22454
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 9849 . . . 4  |-  2  =  ( 1  +  1 )
21oveq1i 5910 . . 3  |-  ( 2 
.op  T )  =  ( ( 1  +  1 )  .op  T
)
3 ax-1cn 8840 . . . 4  |-  1  e.  CC
4 hoadddir 22439 . . . 4  |-  ( ( 1  e.  CC  /\  1  e.  CC  /\  T : ~H --> ~H )  -> 
( ( 1  +  1 )  .op  T
)  =  ( ( 1  .op  T ) 
+op  ( 1  .op 
T ) ) )
53, 3, 4mp3an12 1267 . . 3  |-  ( T : ~H --> ~H  ->  ( ( 1  +  1 )  .op  T )  =  ( ( 1 
.op  T )  +op  ( 1  .op  T
) ) )
62, 5syl5eq 2360 . 2  |-  ( T : ~H --> ~H  ->  ( 2  .op  T )  =  ( ( 1 
.op  T )  +op  ( 1  .op  T
) ) )
7 hoadddi 22438 . . . 4  |-  ( ( 1  e.  CC  /\  T : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( 1  .op  ( T  +op  T ) )  =  ( ( 1 
.op  T )  +op  ( 1  .op  T
) ) )
83, 7mp3an1 1264 . . 3  |-  ( ( T : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( 1  .op  ( T  +op  T ) )  =  ( ( 1 
.op  T )  +op  ( 1  .op  T
) ) )
98anidms 626 . 2  |-  ( T : ~H --> ~H  ->  ( 1  .op  ( T 
+op  T ) )  =  ( ( 1 
.op  T )  +op  ( 1  .op  T
) ) )
10 hoaddcl 22393 . . . 4  |-  ( ( T : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( T  +op  T
) : ~H --> ~H )
1110anidms 626 . . 3  |-  ( T : ~H --> ~H  ->  ( T  +op  T ) : ~H --> ~H )
12 homulid2 22435 . . 3  |-  ( ( T  +op  T ) : ~H --> ~H  ->  ( 1  .op  ( T 
+op  T ) )  =  ( T  +op  T ) )
1311, 12syl 15 . 2  |-  ( T : ~H --> ~H  ->  ( 1  .op  ( T 
+op  T ) )  =  ( T  +op  T ) )
146, 9, 133eqtr2d 2354 1  |-  ( T : ~H --> ~H  ->  ( 2  .op  T )  =  ( T  +op  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    e. wcel 1701   -->wf 5288  (class class class)co 5900   CCcc 8780   1c1 8783    + caddc 8785   2c2 9840   ~Hchil 21554    +op chos 21573    .op chot 21574
This theorem is referenced by:  opsqrlem6  22780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-1cn 8840  ax-addcl 8842  ax-hilex 21634  ax-hfvadd 21635  ax-hfvmul 21640  ax-hvmulid 21641  ax-hvdistr1 21643  ax-hvdistr2 21644
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-map 6817  df-2 9849  df-hosum 22365  df-homul 22366
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