Theorem hoadd23i 11341

Description: Commutative/associative law for Hilbert space operator sum that swaps the second and third terms.

Hypotheses

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

Assertion

Ref	Expression
hoadd23i	|- ((R +op S) +op T) = ((R +op T) +op S)

Proof of Theorem hoadd23i

Step	Hyp	Ref	Expression
1		hods.2	. . . 4 |- S:~H-->~H
2		hods.3	. . . 4 |- T:~H-->~H
3	1, 2	hoaddcomi 11335	. . 3 |- (S +op T) = (T +op S)
4	3	opreq2i 4893	. 2 |- (R +op (S +op T)) = (R +op (T +op S))
5		hods.1	. . 3 |- R:~H-->~H
6	5, 1, 2	hoaddassi 11339	. 2 |- ((R +op S) +op T) = (R +op (S +op T))
7	5, 2, 1	hoaddassi 11339	. 2 |- ((R +op T) +op S) = (R +op (T +op S))
8	4, 6, 7	3eqtr4i 1921	1 |- ((R +op S) +op T) = ((R +op T) +op S)

Syntax hints:   = wceq 1298  -->wf 3994  (class class class)co 4884  ~Hchil 10420   +op chos 10439

This theorem is referenced by:  hosubeq0i 11389

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-hilex 10501  ax-hfvadd 10502  ax-hvcom 10503  ax-hvass 10504

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-map 5383  df-hosum 11139

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