Theorem hoscom 5605

Description: Commutativity of sum of Hilbert space operators.

Hypotheses

Ref	Expression
hoeq.1	|- S:H~-->H~
hoeq.2	|- T:H~-->H~

Assertion

Ref	Expression
hoscom	|- (S +P T) = (T +P S)

Proof of Theorem hoscom

Step	Hyp	Ref	Expression
1		ax-hvcom 4985	. . . . 5 |- (((S` x) e. H~ /\ (T` x) e. H~) -> ((S` x) +v (T` x)) = ((T` x) +v (S` x)))
2		hoeq.1	. . . . . 6 |- S:H~-->H~
3	2	hocl 5594	. . . . 5 |- (x e. H~ -> (S` x) e. H~)
4		hoeq.2	. . . . . 6 |- T:H~-->H~
5	4	hocl 5594	. . . . 5 |- (x e. H~ -> (T` x) e. H~)
6	1, 3, 5	sylanc 361	. . . 4 |- (x e. H~ -> ((S` x) +v (T` x)) = ((T` x) +v (S` x)))
7	2, 4	pm3.2i 234	. . . . 5 |- (S:H~-->H~ /\ T:H~-->H~)
8		hosvalt 5489	. . . . 5 |- (((S:H~-->H~ /\ T:H~-->H~) /\ x e. H~) -> ((S +P T)` x) = ((S` x) +v (T` x)))
9	7, 8	mpan 518	. . . 4 |- (x e. H~ -> ((S +P T)` x) = ((S` x) +v (T` x)))
10	4, 2	pm3.2i 234	. . . . 5 |- (T:H~-->H~ /\ S:H~-->H~)
11		hosvalt 5489	. . . . 5 |- (((T:H~-->H~ /\ S:H~-->H~) /\ x e. H~) -> ((T +P S)` x) = ((T` x) +v (S` x)))
12	10, 11	mpan 518	. . . 4 |- (x e. H~ -> ((T +P S)` x) = ((T` x) +v (S` x)))
13	6, 9, 12	3eqtr4d 1134	. . 3 |- (x e. H~ -> ((S +P T)` x) = ((T +P S)` x))
14	13	rgen 1247	. 2 |- A.x e. H~ ((S +P T)` x) = ((T +P S)` x)
15	2, 4	hosf 5602	. . 3 |- (S +P T):H~-->H~
16	4, 2	hosf 5602	. . 3 |- (T +P S):H~-->H~
17	15, 16	hoeq 5595	. 2 |- (A.x e. H~ ((S +P T)` x) = ((T +P S)` x) <-> (S +P T) = (T +P S))
18	14, 17	mpbi 164	1 |- (S +P T) = (T +P S)

Syntax hints:   /\ wa 196   = wceq 1091   e. wcel 1092  A.wral 1201  -->wf 2418  ` cfv 2422  (class class class)co 3001  H~chil 4958   +v cva 4959   +P chos 4977

This theorem is referenced by:  hos12 5608  hoid0r 5615  hosd 5622  hosd1 5623

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-hilex 4983  ax-hvaddcl 4984  ax-hvcom 4985

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fv 2438  df-opr 3003  df-oprab 3004  df-hosum 5485

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