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Theorem ho2times 29606
 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 11692 . . . 4 2 = (1 + 1)
21oveq1i 7149 . . 3 (2 ·op 𝑇) = ((1 + 1) ·op 𝑇)
3 ax-1cn 10588 . . . 4 1 ∈ ℂ
4 hoadddir 29591 . . . 4 ((1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((1 + 1) ·op 𝑇) = ((1 ·op 𝑇) +op (1 ·op 𝑇)))
53, 3, 4mp3an12 1448 . . 3 (𝑇: ℋ⟶ ℋ → ((1 + 1) ·op 𝑇) = ((1 ·op 𝑇) +op (1 ·op 𝑇)))
62, 5syl5eq 2848 . 2 (𝑇: ℋ⟶ ℋ → (2 ·op 𝑇) = ((1 ·op 𝑇) +op (1 ·op 𝑇)))
7 hoadddi 29590 . . . 4 ((1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (1 ·op (𝑇 +op 𝑇)) = ((1 ·op 𝑇) +op (1 ·op 𝑇)))
83, 7mp3an1 1445 . . 3 ((𝑇: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (1 ·op (𝑇 +op 𝑇)) = ((1 ·op 𝑇) +op (1 ·op 𝑇)))
98anidms 570 . 2 (𝑇: ℋ⟶ ℋ → (1 ·op (𝑇 +op 𝑇)) = ((1 ·op 𝑇) +op (1 ·op 𝑇)))
10 hoaddcl 29545 . . . 4 ((𝑇: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑇 +op 𝑇): ℋ⟶ ℋ)
1110anidms 570 . . 3 (𝑇: ℋ⟶ ℋ → (𝑇 +op 𝑇): ℋ⟶ ℋ)
12 homulid2 29587 . . 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 2842 1 (𝑇: ℋ⟶ ℋ → (2 ·op 𝑇) = (𝑇 +op 𝑇))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2112  ⟶wf 6324  (class class class)co 7139  ℂcc 10528  1c1 10531   + caddc 10533  2c2 11684   ℋchba 28706   +op chos 28725   ·op chot 28726 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-1cn 10588  ax-addcl 10590  ax-hilex 28786  ax-hfvadd 28787  ax-hfvmul 28792  ax-hvmulid 28793  ax-hvdistr1 28795  ax-hvdistr2 28796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-2 11692  df-hosum 29517  df-homul 29518 This theorem is referenced by:  opsqrlem6  29932
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