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

 Description: Associativity of sum of Hilbert space operators. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
Hypotheses
Ref Expression
hods.1 𝑅: ℋ⟶ ℋ
hods.2 𝑆: ℋ⟶ ℋ
hods.3 𝑇: ℋ⟶ ℋ
Assertion
Ref Expression
hoaddassi ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇))

Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 hods.1 . . . . . 6 𝑅: ℋ⟶ ℋ
2 hods.2 . . . . . 6 𝑆: ℋ⟶ ℋ
3 hosval 27983 . . . . . 6 ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑅 +op 𝑆)‘𝑥) = ((𝑅𝑥) + (𝑆𝑥)))
41, 2, 3mp3an12 1406 . . . . 5 (𝑥 ∈ ℋ → ((𝑅 +op 𝑆)‘𝑥) = ((𝑅𝑥) + (𝑆𝑥)))
54oveq1d 6564 . . . 4 (𝑥 ∈ ℋ → (((𝑅 +op 𝑆)‘𝑥) + (𝑇𝑥)) = (((𝑅𝑥) + (𝑆𝑥)) + (𝑇𝑥)))
61, 2hoaddcli 28011 . . . . 5 (𝑅 +op 𝑆): ℋ⟶ ℋ
7 hods.3 . . . . 5 𝑇: ℋ⟶ ℋ
8 hosval 27983 . . . . 5 (((𝑅 +op 𝑆): ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = (((𝑅 +op 𝑆)‘𝑥) + (𝑇𝑥)))
96, 7, 8mp3an12 1406 . . . 4 (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = (((𝑅 +op 𝑆)‘𝑥) + (𝑇𝑥)))
10 hosval 27983 . . . . . . 7 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
112, 7, 10mp3an12 1406 . . . . . 6 (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
1211oveq2d 6565 . . . . 5 (𝑥 ∈ ℋ → ((𝑅𝑥) + ((𝑆 +op 𝑇)‘𝑥)) = ((𝑅𝑥) + ((𝑆𝑥) + (𝑇𝑥))))
132, 7hoaddcli 28011 . . . . . 6 (𝑆 +op 𝑇): ℋ⟶ ℋ
14 hosval 27983 . . . . . 6 ((𝑅: ℋ⟶ ℋ ∧ (𝑆 +op 𝑇): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) = ((𝑅𝑥) + ((𝑆 +op 𝑇)‘𝑥)))
151, 13, 14mp3an12 1406 . . . . 5 (𝑥 ∈ ℋ → ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) = ((𝑅𝑥) + ((𝑆 +op 𝑇)‘𝑥)))
161ffvelrni 6266 . . . . . 6 (𝑥 ∈ ℋ → (𝑅𝑥) ∈ ℋ)
172ffvelrni 6266 . . . . . 6 (𝑥 ∈ ℋ → (𝑆𝑥) ∈ ℋ)
187ffvelrni 6266 . . . . . 6 (𝑥 ∈ ℋ → (𝑇𝑥) ∈ ℋ)
19 ax-hvass 27243 . . . . . 6 (((𝑅𝑥) ∈ ℋ ∧ (𝑆𝑥) ∈ ℋ ∧ (𝑇𝑥) ∈ ℋ) → (((𝑅𝑥) + (𝑆𝑥)) + (𝑇𝑥)) = ((𝑅𝑥) + ((𝑆𝑥) + (𝑇𝑥))))
2016, 17, 18, 19syl3anc 1318 . . . . 5 (𝑥 ∈ ℋ → (((𝑅𝑥) + (𝑆𝑥)) + (𝑇𝑥)) = ((𝑅𝑥) + ((𝑆𝑥) + (𝑇𝑥))))
2112, 15, 203eqtr4d 2654 . . . 4 (𝑥 ∈ ℋ → ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) = (((𝑅𝑥) + (𝑆𝑥)) + (𝑇𝑥)))
225, 9, 213eqtr4d 2654 . . 3 (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = ((𝑅 +op (𝑆 +op 𝑇))‘𝑥))
2322rgen 2906 . 2 𝑥 ∈ ℋ (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = ((𝑅 +op (𝑆 +op 𝑇))‘𝑥)
246, 7hoaddcli 28011 . . 3 ((𝑅 +op 𝑆) +op 𝑇): ℋ⟶ ℋ
251, 13hoaddcli 28011 . . 3 (𝑅 +op (𝑆 +op 𝑇)): ℋ⟶ ℋ
2624, 25hoeqi 28004 . 2 (∀𝑥 ∈ ℋ (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) ↔ ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇)))
2723, 26mpbi 219 1 ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ℋchil 27160   +ℎ cva 27161   +op chos 27179 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-hilex 27240  ax-hfvadd 27241  ax-hvass 27243 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-hosum 27973 This theorem is referenced by:  hoadd12i  28020  hoadd32i  28021  hoaddass  28025  hosubeq0i  28069
 Copyright terms: Public domain W3C validator