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Theorem homulcl 28958
Description: The scalar product of a Hilbert space operator is an operator. (Contributed by NM, 21-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
homulcl ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)

Proof of Theorem homulcl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ffvelrn 6500 . . . . 5 ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇𝑥) ∈ ℋ)
2 hvmulcl 28210 . . . . 5 ((𝐴 ∈ ℂ ∧ (𝑇𝑥) ∈ ℋ) → (𝐴 · (𝑇𝑥)) ∈ ℋ)
31, 2sylan2 580 . . . 4 ((𝐴 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝐴 · (𝑇𝑥)) ∈ ℋ)
43anassrs 458 . . 3 (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 · (𝑇𝑥)) ∈ ℋ)
5 eqid 2771 . . 3 (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥)))
64, 5fmptd 6527 . 2 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))): ℋ⟶ ℋ)
7 hommval 28935 . . 3 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
87feq1d 6170 . 2 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇): ℋ⟶ ℋ ↔ (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))): ℋ⟶ ℋ))
96, 8mpbird 247 1 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2145  cmpt 4863  wf 6027  cfv 6031  (class class class)co 6793  cc 10136  chil 28116   · csm 28118   ·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 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-hilex 28196  ax-hfvmul 28202
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  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 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-homul 28930
This theorem is referenced by:  honegsubi  28995  homulid2  28999  homco1  29000  homulass  29001  hoadddi  29002  hoadddir  29003  hosubneg  29006  hosubdi  29007  honegsubdi  29009  honegsubdi2  29010  hosub4  29012  hosubsub4  29017  hosubeq0i  29025  nmopnegi  29164  homco2  29176  lnopmi  29199  hmopm  29220  nmophmi  29230  adjmul  29291  opsqrlem1  29339  opsqrlem6  29344
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