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

Theorem ho2times 29018
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 11285 . . . 4 2 = (1 + 1)
21oveq1i 6806 . . 3 (2 ·op 𝑇) = ((1 + 1) ·op 𝑇)
3 ax-1cn 10200 . . . 4 1 ∈ ℂ
4 hoadddir 29003 . . . 4 ((1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((1 + 1) ·op 𝑇) = ((1 ·op 𝑇) +op (1 ·op 𝑇)))
53, 3, 4mp3an12 1562 . . 3 (𝑇: ℋ⟶ ℋ → ((1 + 1) ·op 𝑇) = ((1 ·op 𝑇) +op (1 ·op 𝑇)))
62, 5syl5eq 2817 . 2 (𝑇: ℋ⟶ ℋ → (2 ·op 𝑇) = ((1 ·op 𝑇) +op (1 ·op 𝑇)))
7 hoadddi 29002 . . . 4 ((1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (1 ·op (𝑇 +op 𝑇)) = ((1 ·op 𝑇) +op (1 ·op 𝑇)))
83, 7mp3an1 1559 . . 3 ((𝑇: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (1 ·op (𝑇 +op 𝑇)) = ((1 ·op 𝑇) +op (1 ·op 𝑇)))
98anidms 556 . 2 (𝑇: ℋ⟶ ℋ → (1 ·op (𝑇 +op 𝑇)) = ((1 ·op 𝑇) +op (1 ·op 𝑇)))
10 hoaddcl 28957 . . . 4 ((𝑇: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑇 +op 𝑇): ℋ⟶ ℋ)
1110anidms 556 . . 3 (𝑇: ℋ⟶ ℋ → (𝑇 +op 𝑇): ℋ⟶ ℋ)
12 homulid2 28999 . . 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 2811 1 (𝑇: ℋ⟶ ℋ → (2 ·op 𝑇) = (𝑇 +op 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  wf 6026  (class class class)co 6796  cc 10140  1c1 10143   + caddc 10145  2c2 11276  chil 28116   +op chos 28135   ·op chot 28136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-1cn 10200  ax-addcl 10202  ax-hilex 28196  ax-hfvadd 28197  ax-hfvmul 28202  ax-hvmulid 28203  ax-hvdistr1 28205  ax-hvdistr2 28206
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-map 8015  df-2 11285  df-hosum 28929  df-homul 28930
This theorem is referenced by:  opsqrlem6  29344
  Copyright terms: Public domain W3C validator