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Theorem hoadd32i 31026
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 31020 . . 3 (𝑆 +op 𝑇) = (𝑇 +op 𝑆)
43oveq2i 7419 . 2 (𝑅 +op (𝑆 +op 𝑇)) = (𝑅 +op (𝑇 +op 𝑆))
5 hods.1 . . 3 𝑅: ℋ⟶ ℋ
65, 1, 2hoaddassi 31024 . 2 ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇))
75, 2, 1hoaddassi 31024 . 2 ((𝑅 +op 𝑇) +op 𝑆) = (𝑅 +op (𝑇 +op 𝑆))
84, 6, 73eqtr4i 2770 1 ((𝑅 +op 𝑆) +op 𝑇) = ((𝑅 +op 𝑇) +op 𝑆)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wf 6539  (class class class)co 7408  chba 30167   +op chos 30186
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-hilex 30247  ax-hfvadd 30248  ax-hvcom 30249  ax-hvass 30250
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-hosum 30978
This theorem is referenced by:  hosubeq0i  31074
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