HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hoadd32i Structured version   Visualization version   GIF version

Theorem hoadd32i 31798
Description: Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
hods.1 𝑅: ℋ⟶ ℋ
hods.2 𝑆: ℋ⟶ ℋ
hods.3 𝑇: ℋ⟶ ℋ
Assertion
Ref Expression
hoadd32i ((𝑅 +op 𝑆) +op 𝑇) = ((𝑅 +op 𝑇) +op 𝑆)

Proof of Theorem hoadd32i
StepHypRef Expression
1 hods.2 . . . 4 𝑆: ℋ⟶ ℋ
2 hods.3 . . . 4 𝑇: ℋ⟶ ℋ
31, 2hoaddcomi 31792 . . 3 (𝑆 +op 𝑇) = (𝑇 +op 𝑆)
43oveq2i 7443 . 2 (𝑅 +op (𝑆 +op 𝑇)) = (𝑅 +op (𝑇 +op 𝑆))
5 hods.1 . . 3 𝑅: ℋ⟶ ℋ
65, 1, 2hoaddassi 31796 . 2 ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇))
75, 2, 1hoaddassi 31796 . 2 ((𝑅 +op 𝑇) +op 𝑆) = (𝑅 +op (𝑇 +op 𝑆))
84, 6, 73eqtr4i 2774 1 ((𝑅 +op 𝑆) +op 𝑇) = ((𝑅 +op 𝑇) +op 𝑆)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wf 6556  (class class class)co 7432  chba 30939   +op chos 30958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-hilex 31019  ax-hfvadd 31020  ax-hvcom 31021  ax-hvass 31022
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-hosum 31750
This theorem is referenced by:  hosubeq0i  31846
  Copyright terms: Public domain W3C validator