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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 6502 . . . . 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 453 . . 3 (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 · (𝑇𝑥)) ∈ ℋ)
54fmpttd 6530 . 2 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))): ℋ⟶ ℋ)
6 hommval 28935 . . 3 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
76feq1d 6169 . 2 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇): ℋ⟶ ℋ ↔ (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))): ℋ⟶ ℋ))
85, 7mpbird 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 4864  wf 6026  cfv 6030  (class class class)co 6796  cc 10140  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 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-hilex 28196  ax-hfvmul 28202
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-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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