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Theorem hoadd12i 29564
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 29559 . . 3 (𝑅 +op 𝑆) = (𝑆 +op 𝑅)
43oveq1i 7149 . 2 ((𝑅 +op 𝑆) +op 𝑇) = ((𝑆 +op 𝑅) +op 𝑇)
5 hods.3 . . 3 𝑇: ℋ⟶ ℋ
61, 2, 5hoaddassi 29563 . 2 ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇))
72, 1, 5hoaddassi 29563 . 2 ((𝑆 +op 𝑅) +op 𝑇) = (𝑆 +op (𝑅 +op 𝑇))
84, 6, 73eqtr3i 2832 1 (𝑅 +op (𝑆 +op 𝑇)) = (𝑆 +op (𝑅 +op 𝑇))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wf 6324  (class class class)co 7139  chba 28706   +op chos 28725
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-hilex 28786  ax-hfvadd 28787  ax-hvcom 28788  ax-hvass 28789
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-hosum 29517
This theorem is referenced by:  ho0subi  29582
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