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

Theorem hosval 29444
Description: Value of the sum of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
hosval ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝐴) = ((𝑆𝐴) + (𝑇𝐴)))

Proof of Theorem hosval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 hosmval 29439 . . . 4 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))))
21fveq1d 6665 . . 3 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((𝑆 +op 𝑇)‘𝐴) = ((𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥)))‘𝐴))
3 fveq2 6663 . . . . 5 (𝑥 = 𝐴 → (𝑆𝑥) = (𝑆𝐴))
4 fveq2 6663 . . . . 5 (𝑥 = 𝐴 → (𝑇𝑥) = (𝑇𝐴))
53, 4oveq12d 7163 . . . 4 (𝑥 = 𝐴 → ((𝑆𝑥) + (𝑇𝑥)) = ((𝑆𝐴) + (𝑇𝐴)))
6 eqid 2818 . . . 4 (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥)))
7 ovex 7178 . . . 4 ((𝑆𝐴) + (𝑇𝐴)) ∈ V
85, 6, 7fvmpt 6761 . . 3 (𝐴 ∈ ℋ → ((𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥)))‘𝐴) = ((𝑆𝐴) + (𝑇𝐴)))
92, 8sylan9eq 2873 . 2 (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝐴) = ((𝑆𝐴) + (𝑇𝐴)))
1093impa 1102 1 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝐴) = ((𝑆𝐴) + (𝑇𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  cmpt 5137  wf 6344  cfv 6348  (class class class)co 7145  chba 28623   + cva 28624   +op chos 28642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-hilex 28703
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-hosum 29434
This theorem is referenced by:  hoscl  29449  hoaddcomi  29476  hodsi  29479  hoaddassi  29480  hocadddiri  29483  hoaddid1i  29490  honegsubi  29500  hoadddi  29507  hoadddir  29508  lnophsi  29705  hmops  29724  adjadd  29797  nmoptrii  29798  leopadd  29836  pjsdii  29859  pjscji  29874  pjtoi  29883
  Copyright terms: Public domain W3C validator