Theorem hoaddassi 11339

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

Proof of Theorem hoaddassi

Step	Hyp	Ref	Expression
1		hods.1	. . . . . 6 |- R:~H-->~H
2	1	ffvelrni 4788	. . . . 5 |- (x e. ~H -> (R` x) e. ~H)
3		hods.2	. . . . . 6 |- S:~H-->~H
4	3	ffvelrni 4788	. . . . 5 |- (x e. ~H -> (S` x) e. ~H)
5		hods.3	. . . . . 6 |- T:~H-->~H
6	5	ffvelrni 4788	. . . . 5 |- (x e. ~H -> (T` x) e. ~H)
7		ax-hvass 10504	. . . . 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))))
8	2, 4, 6, 7	syl111anc 1100	. . . 4 |- (x e. ~H -> (((R` x) +h (S` x)) +h (T` x)) = ((R` x) +h ((S` x) +h (T` x))))
9	1, 3	hoaddcli 11331	. . . . . 6 |- (R +op S):~H-->~H
10		hosvalOLD 11150	. . . . . 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, 5, 10	mpanl12 773	. . . . 5 |- (x e. ~H -> (((R +op S) +op T)` x) = (((R +op S)` x) +h (T` x)))
12		hosvalOLD 11150	. . . . . . 7 |- (((R:~H-->~H /\ S:~H-->~H) /\ x e. ~H) -> ((R +op S)` x) = ((R` x) +h (S` x)))
13	1, 3, 12	mpanl12 773	. . . . . 6 |- (x e. ~H -> ((R +op S)` x) = ((R` x) +h (S` x)))
14	13	opreq1d 4897	. . . . 5 |- (x e. ~H -> (((R +op S)` x) +h (T` x)) = (((R` x) +h (S` x)) +h (T` x)))
15	11, 14	eqtrd 1925	. . . 4 |- (x e. ~H -> (((R +op S) +op T)` x) = (((R` x) +h (S` x)) +h (T` x)))
16	3, 5	hoaddcli 11331	. . . . . 6 |- (S +op T):~H-->~H
17		hosvalOLD 11150	. . . . . 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	1, 16, 17	mpanl12 773	. . . . 5 |- (x e. ~H -> ((R +op (S +op T))` x) = ((R` x) +h ((S +op T)` x)))
19		hosvalOLD 11150	. . . . . . 7 |- (((S:~H-->~H /\ T:~H-->~H) /\ x e. ~H) -> ((S +op T)` x) = ((S` x) +h (T` x)))
20	3, 5, 19	mpanl12 773	. . . . . 6 |- (x e. ~H -> ((S +op T)` x) = ((S` x) +h (T` x)))
21	20	opreq2d 4898	. . . . 5 |- (x e. ~H -> ((R` x) +h ((S +op T)` x)) = ((R` x) +h ((S` x) +h (T` x))))
22	18, 21	eqtrd 1925	. . . 4 |- (x e. ~H -> ((R +op (S +op T))` x) = ((R` x) +h ((S` x) +h (T` x))))
23	8, 15, 22	3eqtr4d 1937	. . 3 |- (x e. ~H -> (((R +op S) +op T)` x) = ((R +op (S +op T))` x))
24	23	rgen 2159	. 2 |- A.x e. ~H (((R +op S) +op T)` x) = ((R +op (S +op T))` x)
25	9, 5	hoaddcli 11331	. . 3 |- ((R +op S) +op T):~H-->~H
26	1, 16	hoaddcli 11331	. . 3 |- (R +op (S +op T)):~H-->~H
27	25, 26	hoeqi 11324	. 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 206	1 |- ((R +op S) +op T) = (R +op (S +op T))

Syntax hints:   = wceq 1298   e. wcel 1300  A.wral 2105  -->wf 3994  ` cfv 3998  (class class class)co 4884  ~Hchil 10420   +h cva 10421   +op chos 10439

This theorem is referenced by:  hoadd12i 11340  hoadd23i 11341  hoaddass 11345  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-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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