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Theorem hoaddcomi 30811
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
21ffvelcdmi 7054 . . . . 5 (𝑥 ∈ ℋ → (𝑆𝑥) ∈ ℋ)
3 hoeq.2 . . . . . 6 𝑇: ℋ⟶ ℋ
43ffvelcdmi 7054 . . . . 5 (𝑥 ∈ ℋ → (𝑇𝑥) ∈ ℋ)
5 ax-hvcom 30040 . . . . 5 (((𝑆𝑥) ∈ ℋ ∧ (𝑇𝑥) ∈ ℋ) → ((𝑆𝑥) + (𝑇𝑥)) = ((𝑇𝑥) + (𝑆𝑥)))
62, 4, 5syl2anc 584 . . . 4 (𝑥 ∈ ℋ → ((𝑆𝑥) + (𝑇𝑥)) = ((𝑇𝑥) + (𝑆𝑥)))
7 hosval 30779 . . . . 5 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
81, 3, 7mp3an12 1451 . . . 4 (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
9 hosval 30779 . . . . 5 ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇𝑥) + (𝑆𝑥)))
103, 1, 9mp3an12 1451 . . . 4 (𝑥 ∈ ℋ → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇𝑥) + (𝑆𝑥)))
116, 8, 103eqtr4d 2781 . . 3 (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥))
1211rgen 3062 . 2 𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥)
131, 3hoaddcli 30807 . . 3 (𝑆 +op 𝑇): ℋ⟶ ℋ
143, 1hoaddcli 30807 . . 3 (𝑇 +op 𝑆): ℋ⟶ ℋ
1513, 14hoeqi 30800 . 2 (∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥) ↔ (𝑆 +op 𝑇) = (𝑇 +op 𝑆))
1612, 15mpbi 229 1 (𝑆 +op 𝑇) = (𝑇 +op 𝑆)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wcel 2106  wral 3060  wf 6512  cfv 6516  (class class class)co 7377  chba 29958   + cva 29959   +op chos 29977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-hilex 30038  ax-hfvadd 30039  ax-hvcom 30040
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7380  df-oprab 7381  df-mpo 7382  df-map 8789  df-hosum 30769
This theorem is referenced by:  hoaddcom  30813  hoadd12i  30816  hoadd32i  30817  hoaddsubi  30860  hosd1i  30861  hosubeq0i  30865
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