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Theorem hoadd32i 31009
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 31003 . . 3 (𝑆 +op 𝑇) = (𝑇 +op 𝑆)
43oveq2i 7415 . 2 (𝑅 +op (𝑆 +op 𝑇)) = (𝑅 +op (𝑇 +op 𝑆))
5 hods.1 . . 3 𝑅: ℋ⟶ ℋ
65, 1, 2hoaddassi 31007 . 2 ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇))
75, 2, 1hoaddassi 31007 . 2 ((𝑅 +op 𝑇) +op 𝑆) = (𝑅 +op (𝑇 +op 𝑆))
84, 6, 73eqtr4i 2771 1 ((𝑅 +op 𝑆) +op 𝑇) = ((𝑅 +op 𝑇) +op 𝑆)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wf 6536  (class class class)co 7404  chba 30150   +op chos 30169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-hilex 30230  ax-hfvadd 30231  ax-hvcom 30232  ax-hvass 30233
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8818  df-hosum 30961
This theorem is referenced by:  hosubeq0i  31057
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