Theorem hos12 5608

Description: Commutative/associative law for Hilbert space operator sum that swaps the first two terms.

Hypotheses

Ref	Expression
hods.1	|- R:H~-->H~
hods.2	|- S:H~-->H~
hods.3	|- T:H~-->H~

Assertion

Ref	Expression
hos12	|- (R +P (S +P T)) = (S +P (R +P T))

Proof of Theorem hos12

Step	Hyp	Ref	Expression
1		hods.1	. . . 4 |- R:H~-->H~
2		hods.2	. . . 4 |- S:H~-->H~
3	1, 2	hoscom 5605	. . 3 |- (R +P S) = (S +P R)
4	3	opreq1i 3009	. 2 |- ((R +P S) +P T) = ((S +P R) +P T)
5		hods.3	. . 3 |- T:H~-->H~
6	1, 2, 5	hosass 5607	. 2 |- ((R +P S) +P T) = (R +P (S +P T))
7	2, 1, 5	hosass 5607	. 2 |- ((S +P R) +P T) = (S +P (R +P T))
8	4, 6, 7	3eqtr3 1124	1 |- (R +P (S +P T)) = (S +P (R +P T))

Syntax hints:   = wceq 1091  -->wf 2418  (class class class)co 3001  H~chil 4958   +P chos 4977

This theorem is referenced by:  hods0 5620

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-hilex 4983  ax-hvaddcl 4984  ax-hvcom 4985  ax-hvass 4986

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fv 2438  df-opr 3003  df-oprab 3004  df-hosum 5485

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