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Theorem hoadd12i 31806
Description: Commutative/associative law for Hilbert space operator sum that swaps the first two terms. (Contributed by NM, 27-Aug-2004.) (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
hoadd12i (𝑅 +op (𝑆 +op 𝑇)) = (𝑆 +op (𝑅 +op 𝑇))

Proof of Theorem hoadd12i
StepHypRef Expression
1 hods.1 . . . 4 𝑅: ℋ⟶ ℋ
2 hods.2 . . . 4 𝑆: ℋ⟶ ℋ
31, 2hoaddcomi 31801 . . 3 (𝑅 +op 𝑆) = (𝑆 +op 𝑅)
43oveq1i 7441 . 2 ((𝑅 +op 𝑆) +op 𝑇) = ((𝑆 +op 𝑅) +op 𝑇)
5 hods.3 . . 3 𝑇: ℋ⟶ ℋ
61, 2, 5hoaddassi 31805 . 2 ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇))
72, 1, 5hoaddassi 31805 . 2 ((𝑆 +op 𝑅) +op 𝑇) = (𝑆 +op (𝑅 +op 𝑇))
84, 6, 73eqtr3i 2771 1 (𝑅 +op (𝑆 +op 𝑇)) = (𝑆 +op (𝑅 +op 𝑇))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wf 6559  (class class class)co 7431  chba 30948   +op chos 30967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-hilex 31028  ax-hfvadd 31029  ax-hvcom 31030  ax-hvass 31031
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-hosum 31759
This theorem is referenced by:  ho0subi  31824
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