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Theorem ho2times 25370
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 10486 . . . 4  |-  2  =  ( 1  +  1 )
21oveq1i 6205 . . 3  |-  ( 2 
.op  T )  =  ( ( 1  +  1 )  .op  T
)
3 ax-1cn 9446 . . . 4  |-  1  e.  CC
4 hoadddir 25355 . . . 4  |-  ( ( 1  e.  CC  /\  1  e.  CC  /\  T : ~H --> ~H )  -> 
( ( 1  +  1 )  .op  T
)  =  ( ( 1  .op  T ) 
+op  ( 1  .op 
T ) ) )
53, 3, 4mp3an12 1305 . . 3  |-  ( T : ~H --> ~H  ->  ( ( 1  +  1 )  .op  T )  =  ( ( 1 
.op  T )  +op  ( 1  .op  T
) ) )
62, 5syl5eq 2505 . 2  |-  ( T : ~H --> ~H  ->  ( 2  .op  T )  =  ( ( 1 
.op  T )  +op  ( 1  .op  T
) ) )
7 hoadddi 25354 . . . 4  |-  ( ( 1  e.  CC  /\  T : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( 1  .op  ( T  +op  T ) )  =  ( ( 1 
.op  T )  +op  ( 1  .op  T
) ) )
83, 7mp3an1 1302 . . 3  |-  ( ( T : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( 1  .op  ( T  +op  T ) )  =  ( ( 1 
.op  T )  +op  ( 1  .op  T
) ) )
98anidms 645 . 2  |-  ( T : ~H --> ~H  ->  ( 1  .op  ( T 
+op  T ) )  =  ( ( 1 
.op  T )  +op  ( 1  .op  T
) ) )
10 hoaddcl 25309 . . . 4  |-  ( ( T : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( T  +op  T
) : ~H --> ~H )
1110anidms 645 . . 3  |-  ( T : ~H --> ~H  ->  ( T  +op  T ) : ~H --> ~H )
12 homulid2 25351 . . 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 2499 1  |-  ( T : ~H --> ~H  ->  ( 2  .op  T )  =  ( T  +op  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   -->wf 5517  (class class class)co 6195   CCcc 9386   1c1 9389    + caddc 9391   2c2 10477   ~Hchil 24468    +op chos 24487    .op chot 24488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-1cn 9446  ax-addcl 9448  ax-hilex 24548  ax-hfvadd 24549  ax-hfvmul 24554  ax-hvmulid 24555  ax-hvdistr1 24557  ax-hvdistr2 24558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-map 7321  df-2 10486  df-hosum 25281  df-homul 25282
This theorem is referenced by:  opsqrlem6  25696
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