Theorem hoaddass 9619

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 8793	. . . . 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 3800	. . . . 5 |- (x e. H~ -> (R` x) e. H~)
4		hods.2	. . . . . 6 |- S:H~-->H~
5	4	ffvelrni 3800	. . . . 5 |- (x e. H~ -> (S` x) e. H~)
6		hods.3	. . . . . 6 |- T:H~-->H~
7	6	ffvelrni 3800	. . . . 5 |- (x e. H~ -> (T` x) e. H~)
8	1, 3, 5, 7	syl3anc 856	. . . 4 |- (x e. H~ -> (((R` x) +h (S` x)) +h (T` x)) = ((R` x) +h ((S` x) +h (T` x))))
9	2, 4	hoaddcl 9611	. . . . . 6 |- (R +op S):H~-->H~
10		hosvaltOLD 9434	. . . . . 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 706	. . . . 5 |- (x e. H~ -> (((R +op S) +op T)` x) = (((R +op S)` x) +h (T` x)))
12		hosvaltOLD 9434	. . . . . . 7 |- (((R:H~-->H~ /\ S:H~-->H~) /\ x e. H~) -> ((R +op S)` x) = ((R` x) +h (S` x)))
13	2, 4, 12	mpanl12 706	. . . . . 6 |- (x e. H~ -> ((R +op S)` x) = ((R` x) +h (S` x)))
14	13	opreq1d 3960	. . . . 5 |- (x e. H~ -> (((R +op S)` x) +h (T` x)) = (((R` x) +h (S` x)) +h (T` x)))
15	11, 14	eqtrd 1499	. . . 4 |- (x e. H~ -> (((R +op S) +op T)` x) = (((R` x) +h (S` x)) +h (T` x)))
16	4, 6	hoaddcl 9611	. . . . . 6 |- (S +op T):H~-->H~
17		hosvaltOLD 9434	. . . . . 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 706	. . . . 5 |- (x e. H~ -> ((R +op (S +op T))` x) = ((R` x) +h ((S +op T)` x)))
19		hosvaltOLD 9434	. . . . . . 7 |- (((S:H~-->H~ /\ T:H~-->H~) /\ x e. H~) -> ((S +op T)` x) = ((S` x) +h (T` x)))
20	4, 6, 19	mpanl12 706	. . . . . 6 |- (x e. H~ -> ((S +op T)` x) = ((S` x) +h (T` x)))
21	20	opreq2d 3961	. . . . 5 |- (x e. H~ -> ((R` x) +h ((S +op T)` x)) = ((R` x) +h ((S` x) +h (T` x))))
22	18, 21	eqtrd 1499	. . . 4 |- (x e. H~ -> ((R +op (S +op T))` x) = ((R` x) +h ((S` x) +h (T` x))))
23	8, 15, 22	3eqtr4d 1509	. . 3 |- (x e. H~ -> (((R +op S) +op T)` x) = ((R +op (S +op T))` x))
24	23	rgen 1690	. 2 |- A.x e. H~ (((R +op S) +op T)` x) = ((R +op (S +op T))` x)
25	9, 6	hoaddcl 9611	. . 3 |- ((R +op S) +op T):H~-->H~
26	2, 16	hoaddcl 9611	. . 3 |- (R +op (S +op T)):H~-->H~
27	25, 26	hoeq 9604	. 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 953   e. wcel 955  A.wral 1637  -->wf 3168  ` cfv 3172  (class class class)co 3948  H~chil 8727   +h cva 8728   +op chos 8746

This theorem is referenced by:  hoadd12 9620  hoadd23 9621  hoaddasst 9625  hosubeq0 9669

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-hilex 8790  ax-hfvadd 8791  ax-hvass 8793

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-opr 3950  df-oprab 3951  df-map 4308  df-hosum 9423

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