Theorem hoadd12 9698

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
hoadd12	|- (R +op (S +op T)) = (S +op (R +op T))

Proof of Theorem hoadd12

Step	Hyp	Ref	Expression
1		hods.1	. . . 4 |- R:H~-->H~
2		hods.2	. . . 4 |- S:H~-->H~
3	1, 2	hoaddcom 9693	. . 3 |- (R +op S) = (S +op R)
4	3	opreq1i 3977	. 2 |- ((R +op S) +op T) = ((S +op R) +op T)
5		hods.3	. . 3 |- T:H~-->H~
6	1, 2, 5	hoaddass 9697	. 2 |- ((R +op S) +op T) = (R +op (S +op T))
7	2, 1, 5	hoaddass 9697	. 2 |- ((S +op R) +op T) = (S +op (R +op T))
8	4, 6, 7	3eqtr3 1506	1 |- (R +op (S +op T)) = (S +op (R +op T))

Syntax hints:   = wceq 958  -->wf 3184  (class class class)co 3969  H~chil 8783   +op chos 8802

This theorem is referenced by:  ho0sub 9716

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-hilex 8864  ax-hfvadd 8865  ax-hvcom 8866  ax-hvass 8867

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-oprab 3972  df-map 4330  df-hosum 9501

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