Definition df-hfsum 9511

Description: Define the sum of two Hilbert space functionals. Definition of [Beran] p. 111. Note that unlike some authors, we define a functional as any function from H~ to CC, not just linear (or bounded linear) ones.

Assertion

Ref	Expression
df-hfsum	|- +fn = {<.<.f, g>., h>. | ((f:H~-->CC /\ g:H~-->CC) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) + (g` x)))})}

Distinct variable group:   f,g,h,x,y

Detailed syntax breakdown of Definition df-hfsum

Step	Hyp	Ref	Expression
1		chfs 8812	. 2 class +fn
2		chil 8790	. . . . . 6 class H~
3		cc 5251	. . . . . 6 class CC
4		vf	. . . . . . 7 set f
5	4	cv 957	. . . . . 6 class f
6	2, 3, 5	wf 3185	. . . . 5 wff f:H~-->CC
7		vg	. . . . . . 7 set g
8	7	cv 957	. . . . . 6 class g
9	2, 3, 8	wf 3185	. . . . 5 wff g:H~-->CC
10	6, 9	wa 223	. . . 4 wff (f:H~-->CC /\ g:H~-->CC)
11		vh	. . . . . 6 set h
12	11	cv 957	. . . . 5 class h
13		vx	. . . . . . . . 9 set x
14	13	cv 957	. . . . . . . 8 class x
15	14, 2	wcel 960	. . . . . . 7 wff x e. H~
16		vy	. . . . . . . . 9 set y
17	16	cv 957	. . . . . . . 8 class y
18	14, 5	cfv 3189	. . . . . . . . 9 class (f` x)
19	14, 8	cfv 3189	. . . . . . . . 9 class (g` x)
20		caddc 5256	. . . . . . . . 9 class +
21	18, 19, 20	co 3970	. . . . . . . 8 class ((f` x) + (g` x))
22	17, 21	wceq 958	. . . . . . 7 wff y = ((f` x) + (g` x))
23	15, 22	wa 223	. . . . . 6 wff (x e. H~ /\ y = ((f` x) + (g` x)))
24	23, 13, 16	copab 2672	. . . . 5 class {<.x, y>. | (x e. H~ /\ y = ((f` x) + (g` x)))}
25	12, 24	wceq 958	. . . 4 wff h = {<.x, y>. | (x e. H~ /\ y = ((f` x) + (g` x)))}
26	10, 25	wa 223	. . 3 wff ((f:H~-->CC /\ g:H~-->CC) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) + (g` x)))})
27	26, 4, 7, 11	copab2 3971	. 2 class {<.<.f, g>., h>. | ((f:H~-->CC /\ g:H~-->CC) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) + (g` x)))})}
28	1, 27	wceq 958	1 wff +fn = {<.<.f, g>., h>. | ((f:H~-->CC /\ g:H~-->CC) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) + (g` x)))})}

This definition is referenced by:  hfsmvalt 9516

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