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Theorem homulcl 29642
 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 6841 . . . . 5 ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇𝑥) ∈ ℋ)
2 hvmulcl 28896 . . . . 5 ((𝐴 ∈ ℂ ∧ (𝑇𝑥) ∈ ℋ) → (𝐴 · (𝑇𝑥)) ∈ ℋ)
31, 2sylan2 596 . . . 4 ((𝐴 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝐴 · (𝑇𝑥)) ∈ ℋ)
43anassrs 472 . . 3 (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 · (𝑇𝑥)) ∈ ℋ)
54fmpttd 6871 . 2 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))): ℋ⟶ ℋ)
6 hommval 29619 . . 3 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
76feq1d 6484 . 2 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇): ℋ⟶ ℋ ↔ (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))): ℋ⟶ ℋ))
85, 7mpbird 260 1 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2112   ↦ cmpt 5113  ⟶wf 6332  ‘cfv 6336  (class class class)co 7151  ℂcc 10574   ℋchba 28802   ·ℎ csm 28804   ·op chot 28822 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-hilex 28882  ax-hfvmul 28888 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-map 8419  df-homul 29614 This theorem is referenced by:  honegsubi  29679  homulid2  29683  homco1  29684  homulass  29685  hoadddi  29686  hoadddir  29687  hosubneg  29690  hosubdi  29691  honegsubdi  29693  honegsubdi2  29694  hosub4  29696  hosubsub4  29701  hosubeq0i  29709  nmopnegi  29848  homco2  29860  lnopmi  29883  hmopm  29904  nmophmi  29914  adjmul  29975  opsqrlem1  30023  opsqrlem6  30028
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