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Theorem hoaddcomi 29476
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 6842 . . . . 5 (𝑥 ∈ ℋ → (𝑆𝑥) ∈ ℋ)
3 hoeq.2 . . . . . 6 𝑇: ℋ⟶ ℋ
43ffvelrni 6842 . . . . 5 (𝑥 ∈ ℋ → (𝑇𝑥) ∈ ℋ)
5 ax-hvcom 28705 . . . . 5 (((𝑆𝑥) ∈ ℋ ∧ (𝑇𝑥) ∈ ℋ) → ((𝑆𝑥) + (𝑇𝑥)) = ((𝑇𝑥) + (𝑆𝑥)))
62, 4, 5syl2anc 584 . . . 4 (𝑥 ∈ ℋ → ((𝑆𝑥) + (𝑇𝑥)) = ((𝑇𝑥) + (𝑆𝑥)))
7 hosval 29444 . . . . 5 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
81, 3, 7mp3an12 1442 . . . 4 (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
9 hosval 29444 . . . . 5 ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇𝑥) + (𝑆𝑥)))
103, 1, 9mp3an12 1442 . . . 4 (𝑥 ∈ ℋ → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇𝑥) + (𝑆𝑥)))
116, 8, 103eqtr4d 2863 . . 3 (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥))
1211rgen 3145 . 2 𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥)
131, 3hoaddcli 29472 . . 3 (𝑆 +op 𝑇): ℋ⟶ ℋ
143, 1hoaddcli 29472 . . 3 (𝑇 +op 𝑆): ℋ⟶ ℋ
1513, 14hoeqi 29465 . 2 (∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥) ↔ (𝑆 +op 𝑇) = (𝑇 +op 𝑆))
1612, 15mpbi 231 1 (𝑆 +op 𝑇) = (𝑇 +op 𝑆)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  wral 3135  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  ax-hfvadd 28704  ax-hvcom 28705
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:  hoaddcom  29478  hoadd12i  29481  hoadd32i  29482  hoaddsubi  29525  hosd1i  29526  hosubeq0i  29530
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