Theorem hoaddcom 9615

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 8792	. . . . 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 3800	. . . . 5 |- (x e. H~ -> (S` x) e. H~)
4		hoeq.2	. . . . . 6 |- T:H~-->H~
5	4	ffvelrni 3800	. . . . 5 |- (x e. H~ -> (T` x) e. H~)
6	1, 3, 5	sylanc 471	. . . 4 |- (x e. H~ -> ((S` x) +h (T` x)) = ((T` x) +h (S` x)))
7		hosvaltOLD 9434	. . . . 5 |- (((S:H~-->H~ /\ T:H~-->H~) /\ x e. H~) -> ((S +op T)` x) = ((S` x) +h (T` x)))
8	2, 4, 7	mpanl12 706	. . . 4 |- (x e. H~ -> ((S +op T)` x) = ((S` x) +h (T` x)))
9		hosvaltOLD 9434	. . . . 5 |- (((T:H~-->H~ /\ S:H~-->H~) /\ x e. H~) -> ((T +op S)` x) = ((T` x) +h (S` x)))
10	4, 2, 9	mpanl12 706	. . . 4 |- (x e. H~ -> ((T +op S)` x) = ((T` x) +h (S` x)))
11	6, 8, 10	3eqtr4d 1509	. . 3 |- (x e. H~ -> ((S +op T)` x) = ((T +op S)` x))
12	11	rgen 1690	. 2 |- A.x e. H~ ((S +op T)` x) = ((T +op S)` x)
13	2, 4	hoaddcl 9611	. . 3 |- (S +op T):H~-->H~
14	4, 2	hoaddcl 9611	. . 3 |- (T +op S):H~-->H~
15	13, 14	hoeq 9604	. 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 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:  hoaddcomt 9617  hoadd12 9620  hoadd23 9621  hoaddsub 9664  hosd1 9665  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-hvcom 8792

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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