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Theorem hoaddcomi 29182
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 6612 . . . . 5 (𝑥 ∈ ℋ → (𝑆𝑥) ∈ ℋ)
3 hoeq.2 . . . . . 6 𝑇: ℋ⟶ ℋ
43ffvelrni 6612 . . . . 5 (𝑥 ∈ ℋ → (𝑇𝑥) ∈ ℋ)
5 ax-hvcom 28409 . . . . 5 (((𝑆𝑥) ∈ ℋ ∧ (𝑇𝑥) ∈ ℋ) → ((𝑆𝑥) + (𝑇𝑥)) = ((𝑇𝑥) + (𝑆𝑥)))
62, 4, 5syl2anc 579 . . . 4 (𝑥 ∈ ℋ → ((𝑆𝑥) + (𝑇𝑥)) = ((𝑇𝑥) + (𝑆𝑥)))
7 hosval 29150 . . . . 5 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
81, 3, 7mp3an12 1579 . . . 4 (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
9 hosval 29150 . . . . 5 ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇𝑥) + (𝑆𝑥)))
103, 1, 9mp3an12 1579 . . . 4 (𝑥 ∈ ℋ → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇𝑥) + (𝑆𝑥)))
116, 8, 103eqtr4d 2871 . . 3 (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥))
1211rgen 3131 . 2 𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥)
131, 3hoaddcli 29178 . . 3 (𝑆 +op 𝑇): ℋ⟶ ℋ
143, 1hoaddcli 29178 . . 3 (𝑇 +op 𝑆): ℋ⟶ ℋ
1513, 14hoeqi 29171 . 2 (∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥) ↔ (𝑆 +op 𝑇) = (𝑇 +op 𝑆))
1612, 15mpbi 222 1 (𝑆 +op 𝑇) = (𝑇 +op 𝑆)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1656  wcel 2164  wral 3117  wf 6123  cfv 6127  (class class class)co 6910  chba 28327   + cva 28328   +op chos 28346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-hilex 28407  ax-hfvadd 28408  ax-hvcom 28409
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-map 8129  df-hosum 29140
This theorem is referenced by:  hoaddcom  29184  hoadd12i  29187  hoadd32i  29188  hoaddsubi  29231  hosd1i  29232  hosubeq0i  29236
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