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 Description: Commutativity of sum of Hilbert space operators. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
Hypotheses
Ref Expression
hoeq.1 𝑆: ℋ⟶ ℋ
hoeq.2 𝑇: ℋ⟶ ℋ
Assertion
Ref Expression
hoaddcomi (𝑆 +op 𝑇) = (𝑇 +op 𝑆)

Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 hoeq.1 . . . . . 6 𝑆: ℋ⟶ ℋ
21ffvelrni 6266 . . . . 5 (𝑥 ∈ ℋ → (𝑆𝑥) ∈ ℋ)
3 hoeq.2 . . . . . 6 𝑇: ℋ⟶ ℋ
43ffvelrni 6266 . . . . 5 (𝑥 ∈ ℋ → (𝑇𝑥) ∈ ℋ)
5 ax-hvcom 27242 . . . . 5 (((𝑆𝑥) ∈ ℋ ∧ (𝑇𝑥) ∈ ℋ) → ((𝑆𝑥) + (𝑇𝑥)) = ((𝑇𝑥) + (𝑆𝑥)))
62, 4, 5syl2anc 691 . . . 4 (𝑥 ∈ ℋ → ((𝑆𝑥) + (𝑇𝑥)) = ((𝑇𝑥) + (𝑆𝑥)))
7 hosval 27983 . . . . 5 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
81, 3, 7mp3an12 1406 . . . 4 (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
9 hosval 27983 . . . . 5 ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇𝑥) + (𝑆𝑥)))
103, 1, 9mp3an12 1406 . . . 4 (𝑥 ∈ ℋ → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇𝑥) + (𝑆𝑥)))
116, 8, 103eqtr4d 2654 . . 3 (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥))
1211rgen 2906 . 2 𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥)
131, 3hoaddcli 28011 . . 3 (𝑆 +op 𝑇): ℋ⟶ ℋ
143, 1hoaddcli 28011 . . 3 (𝑇 +op 𝑆): ℋ⟶ ℋ
1513, 14hoeqi 28004 . 2 (∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥) ↔ (𝑆 +op 𝑇) = (𝑇 +op 𝑆))
1612, 15mpbi 219 1 (𝑆 +op 𝑇) = (𝑇 +op 𝑆)
 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-hvcom 27242 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:  hoaddcom  28017  hoadd12i  28020  hoadd32i  28021  hoaddsubi  28064  hosd1i  28065  hosubeq0i  28069
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