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Theorem hoaddcomi 28519
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)

Proof of Theorem hoaddcomi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 hoeq.1 . . . . . 6 𝑆: ℋ⟶ ℋ
21ffvelrni 6324 . . . . 5 (𝑥 ∈ ℋ → (𝑆𝑥) ∈ ℋ)
3 hoeq.2 . . . . . 6 𝑇: ℋ⟶ ℋ
43ffvelrni 6324 . . . . 5 (𝑥 ∈ ℋ → (𝑇𝑥) ∈ ℋ)
5 ax-hvcom 27746 . . . . 5 (((𝑆𝑥) ∈ ℋ ∧ (𝑇𝑥) ∈ ℋ) → ((𝑆𝑥) + (𝑇𝑥)) = ((𝑇𝑥) + (𝑆𝑥)))
62, 4, 5syl2anc 692 . . . 4 (𝑥 ∈ ℋ → ((𝑆𝑥) + (𝑇𝑥)) = ((𝑇𝑥) + (𝑆𝑥)))
7 hosval 28487 . . . . 5 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
81, 3, 7mp3an12 1411 . . . 4 (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
9 hosval 28487 . . . . 5 ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇𝑥) + (𝑆𝑥)))
103, 1, 9mp3an12 1411 . . . 4 (𝑥 ∈ ℋ → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇𝑥) + (𝑆𝑥)))
116, 8, 103eqtr4d 2665 . . 3 (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥))
1211rgen 2918 . 2 𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥)
131, 3hoaddcli 28515 . . 3 (𝑆 +op 𝑇): ℋ⟶ ℋ
143, 1hoaddcli 28515 . . 3 (𝑇 +op 𝑆): ℋ⟶ ℋ
1513, 14hoeqi 28508 . 2 (∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥) ↔ (𝑆 +op 𝑇) = (𝑇 +op 𝑆))
1612, 15mpbi 220 1 (𝑆 +op 𝑇) = (𝑇 +op 𝑆)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  wral 2908  wf 5853  cfv 5857  (class class class)co 6615  chil 27664   + cva 27665   +op chos 27683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-hilex 27744  ax-hfvadd 27745  ax-hvcom 27746
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-hosum 28477
This theorem is referenced by:  hoaddcom  28521  hoadd12i  28524  hoadd32i  28525  hoaddsubi  28568  hosd1i  28569  hosubeq0i  28573
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