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

Theorem ho2times 28566
 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 11039 . . . 4 2 = (1 + 1)
21oveq1i 6625 . . 3 (2 ·op 𝑇) = ((1 + 1) ·op 𝑇)
3 ax-1cn 9954 . . . 4 1 ∈ ℂ
4 hoadddir 28551 . . . 4 ((1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((1 + 1) ·op 𝑇) = ((1 ·op 𝑇) +op (1 ·op 𝑇)))
53, 3, 4mp3an12 1411 . . 3 (𝑇: ℋ⟶ ℋ → ((1 + 1) ·op 𝑇) = ((1 ·op 𝑇) +op (1 ·op 𝑇)))
62, 5syl5eq 2667 . 2 (𝑇: ℋ⟶ ℋ → (2 ·op 𝑇) = ((1 ·op 𝑇) +op (1 ·op 𝑇)))
7 hoadddi 28550 . . . 4 ((1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (1 ·op (𝑇 +op 𝑇)) = ((1 ·op 𝑇) +op (1 ·op 𝑇)))
83, 7mp3an1 1408 . . 3 ((𝑇: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (1 ·op (𝑇 +op 𝑇)) = ((1 ·op 𝑇) +op (1 ·op 𝑇)))
98anidms 676 . 2 (𝑇: ℋ⟶ ℋ → (1 ·op (𝑇 +op 𝑇)) = ((1 ·op 𝑇) +op (1 ·op 𝑇)))
10 hoaddcl 28505 . . . 4 ((𝑇: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑇 +op 𝑇): ℋ⟶ ℋ)
1110anidms 676 . . 3 (𝑇: ℋ⟶ ℋ → (𝑇 +op 𝑇): ℋ⟶ ℋ)
12 homulid2 28547 . . 3 ((𝑇 +op 𝑇): ℋ⟶ ℋ → (1 ·op (𝑇 +op 𝑇)) = (𝑇 +op 𝑇))
1311, 12syl 17 . 2 (𝑇: ℋ⟶ ℋ → (1 ·op (𝑇 +op 𝑇)) = (𝑇 +op 𝑇))
146, 9, 133eqtr2d 2661 1 (𝑇: ℋ⟶ ℋ → (2 ·op 𝑇) = (𝑇 +op 𝑇))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987  ⟶wf 5853  (class class class)co 6615  ℂcc 9894  1c1 9897   + caddc 9899  2c2 11030   ℋchil 27664   +op chos 27683   ·op chot 27684 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-1cn 9954  ax-addcl 9956  ax-hilex 27744  ax-hfvadd 27745  ax-hfvmul 27750  ax-hvmulid 27751  ax-hvdistr1 27753  ax-hvdistr2 27754 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-2 11039  df-hosum 28477  df-homul 28478 This theorem is referenced by:  opsqrlem6  28892
 Copyright terms: Public domain W3C validator