Theorem hoaddcom 9693

Description: Commutativity of sum of Hilbert space operators.

Hypotheses

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

Assertion

Ref	Expression
hoaddcom	|- (S +op T) = (T +op S)

Proof of Theorem hoaddcom

Step	Hyp	Ref	Expression
1		ax-hvcom 8866	. . . . 5 |- (((S` x) e. H~ /\ (T` x) e. H~) -> ((S` x) +h (T` x)) = ((T` x) +h (S` x)))
2		hoeq.1	. . . . . 6 |- S:H~-->H~
3	2	ffvelrni 3821	. . . . 5 |- (x e. H~ -> (S` x) e. H~)
4		hoeq.2	. . . . . 6 |- T:H~-->H~
5	4	ffvelrni 3821	. . . . 5 |- (x e. H~ -> (T` x) e. H~)
6	1, 3, 5	sylanc 473	. . . 4 |- (x e. H~ -> ((S` x) +h (T` x)) = ((T` x) +h (S` x)))
7		hosvaltOLD 9512	. . . . 5 |- (((S:H~-->H~ /\ T:H~-->H~) /\ x e. H~) -> ((S +op T)` x) = ((S` x) +h (T` x)))
8	2, 4, 7	mpanl12 710	. . . 4 |- (x e. H~ -> ((S +op T)` x) = ((S` x) +h (T` x)))
9		hosvaltOLD 9512	. . . . 5 |- (((T:H~-->H~ /\ S:H~-->H~) /\ x e. H~) -> ((T +op S)` x) = ((T` x) +h (S` x)))
10	4, 2, 9	mpanl12 710	. . . 4 |- (x e. H~ -> ((T +op S)` x) = ((T` x) +h (S` x)))
11	6, 8, 10	3eqtr4d 1520	. . 3 |- (x e. H~ -> ((S +op T)` x) = ((T +op S)` x))
12	11	rgen 1701	. 2 |- A.x e. H~ ((S +op T)` x) = ((T +op S)` x)
13	2, 4	hoaddcl 9689	. . 3 |- (S +op T):H~-->H~
14	4, 2	hoaddcl 9689	. . 3 |- (T +op S):H~-->H~
15	13, 14	hoeq 9682	. 2 |- (A.x e. H~ ((S +op T)` x) = ((T +op S)` x) <-> (S +op T) = (T +op S))
16	12, 15	mpbi 189	1 |- (S +op T) = (T +op S)

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:  hoaddcomt 9695  hoadd12 9698  hoadd23 9699  hoaddsub 9742  hosd1 9743  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-hvcom 8866

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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