Theorem hoaddass 9697

Description: Associativity of sum of Hilbert space operators.

Hypotheses

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

Assertion

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

Proof of Theorem hoaddass

Step	Hyp	Ref	Expression
1		ax-hvass 8867	. . . . 5 |- (((R` x) e. H~ /\ (S` x) e. H~ /\ (T` x) e. H~) -> (((R` x) +h (S` x)) +h (T` x)) = ((R` x) +h ((S` x) +h (T` x))))
2		hods.1	. . . . . 6 |- R:H~-->H~
3	2	ffvelrni 3821	. . . . 5 |- (x e. H~ -> (R` x) e. H~)
4		hods.2	. . . . . 6 |- S:H~-->H~
5	4	ffvelrni 3821	. . . . 5 |- (x e. H~ -> (S` x) e. H~)
6		hods.3	. . . . . 6 |- T:H~-->H~
7	6	ffvelrni 3821	. . . . 5 |- (x e. H~ -> (T` x) e. H~)
8	1, 3, 5, 7	syl3anc 860	. . . 4 |- (x e. H~ -> (((R` x) +h (S` x)) +h (T` x)) = ((R` x) +h ((S` x) +h (T` x))))
9	2, 4	hoaddcl 9689	. . . . . 6 |- (R +op S):H~-->H~
10		hosvaltOLD 9512	. . . . . 6 |- ((((R +op S):H~-->H~ /\ T:H~-->H~) /\ x e. H~) -> (((R +op S) +op T)` x) = (((R +op S)` x) +h (T` x)))
11	9, 6, 10	mpanl12 710	. . . . 5 |- (x e. H~ -> (((R +op S) +op T)` x) = (((R +op S)` x) +h (T` x)))
12		hosvaltOLD 9512	. . . . . . 7 |- (((R:H~-->H~ /\ S:H~-->H~) /\ x e. H~) -> ((R +op S)` x) = ((R` x) +h (S` x)))
13	2, 4, 12	mpanl12 710	. . . . . 6 |- (x e. H~ -> ((R +op S)` x) = ((R` x) +h (S` x)))
14	13	opreq1d 3981	. . . . 5 |- (x e. H~ -> (((R +op S)` x) +h (T` x)) = (((R` x) +h (S` x)) +h (T` x)))
15	11, 14	eqtrd 1510	. . . 4 |- (x e. H~ -> (((R +op S) +op T)` x) = (((R` x) +h (S` x)) +h (T` x)))
16	4, 6	hoaddcl 9689	. . . . . 6 |- (S +op T):H~-->H~
17		hosvaltOLD 9512	. . . . . 6 |- (((R:H~-->H~ /\ (S +op T):H~-->H~) /\ x e. H~) -> ((R +op (S +op T))` x) = ((R` x) +h ((S +op T)` x)))
18	2, 16, 17	mpanl12 710	. . . . 5 |- (x e. H~ -> ((R +op (S +op T))` x) = ((R` x) +h ((S +op T)` x)))
19		hosvaltOLD 9512	. . . . . . 7 |- (((S:H~-->H~ /\ T:H~-->H~) /\ x e. H~) -> ((S +op T)` x) = ((S` x) +h (T` x)))
20	4, 6, 19	mpanl12 710	. . . . . 6 |- (x e. H~ -> ((S +op T)` x) = ((S` x) +h (T` x)))
21	20	opreq2d 3982	. . . . 5 |- (x e. H~ -> ((R` x) +h ((S +op T)` x)) = ((R` x) +h ((S` x) +h (T` x))))
22	18, 21	eqtrd 1510	. . . 4 |- (x e. H~ -> ((R +op (S +op T))` x) = ((R` x) +h ((S` x) +h (T` x))))
23	8, 15, 22	3eqtr4d 1520	. . 3 |- (x e. H~ -> (((R +op S) +op T)` x) = ((R +op (S +op T))` x))
24	23	rgen 1701	. 2 |- A.x e. H~ (((R +op S) +op T)` x) = ((R +op (S +op T))` x)
25	9, 6	hoaddcl 9689	. . 3 |- ((R +op S) +op T):H~-->H~
26	2, 16	hoaddcl 9689	. . 3 |- (R +op (S +op T)):H~-->H~
27	25, 26	hoeq 9682	. 2 |- (A.x e. H~ (((R +op S) +op T)` x) = ((R +op (S +op T))` x) <-> ((R +op S) +op T) = (R +op (S +op T)))
28	24, 27	mpbi 189	1 |- ((R +op S) +op T) = (R +op (S +op T))

Syntax hints:   = wceq 958   e. wcel 960  A.wral 1648  -->wf 3184  ` cfv 3188  (class class class)co 3969  H~chil 8783   +h cva 8784   +op chos 8802

This theorem is referenced by:  hoadd12 9698  hoadd23 9699  hoaddasst 9703  hosubeq0 9747

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-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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