Theorem hoadd23 9695

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
hoadd23	|- ((R +op S) +op T) = ((R +op T) +op S)

Proof of Theorem hoadd23

Step	Hyp	Ref	Expression
1		hods.2	. . . 4 |- S:H~-->H~
2		hods.3	. . . 4 |- T:H~-->H~
3	1, 2	hoaddcom 9689	. . 3 |- (S +op T) = (T +op S)
4	3	opreq2i 3969	. 2 |- (R +op (S +op T)) = (R +op (T +op S))
5		hods.1	. . 3 |- R:H~-->H~
6	5, 1, 2	hoaddass 9693	. 2 |- ((R +op S) +op T) = (R +op (S +op T))
7	5, 2, 1	hoaddass 9693	. 2 |- ((R +op T) +op S) = (R +op (T +op S))
8	4, 6, 7	3eqtr4 1504	1 |- ((R +op S) +op T) = ((R +op T) +op S)

Syntax hints:   = wceq 955  -->wf 3175  (class class class)co 3960  H~chil 8772   +op chos 8791

This theorem is referenced by:  hosubeq0 9743

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-pow 2739  ax-pr 2776  ax-un 2863  ax-hilex 8853  ax-hfvadd 8854  ax-hvcom 8855  ax-hvass 8856

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-fv 3195  df-opr 3962  df-oprab 3963  df-map 4321  df-hosum 9496

Copyright terms: Public domain