Definition df-hfsum 10934

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 10234	. 2 class +fn
2		chil 10212	. . . . . 6 class ~H
3		cc 6180	. . . . . 6 class CC
4		vf	. . . . . . 7 set f
5	4	cv 1135	. . . . . 6 class f
6	2, 3, 5	wf 3805	. . . . 5 wff f:~H-->CC
7		vg	. . . . . . 7 set g
8	7	cv 1135	. . . . . 6 class g
9	2, 3, 8	wf 3805	. . . . 5 wff g:~H-->CC
10	6, 9	wa 239	. . . 4 wff (f:~H-->CC /\ g:~H-->CC)
11		vh	. . . . . 6 set h
12	11	cv 1135	. . . . 5 class h
13		vx	. . . . . . . . 9 set x
14	13	cv 1135	. . . . . . . 8 class x
15	14, 2	wcel 1138	. . . . . . 7 wff x e. ~H
16		vy	. . . . . . . . 9 set y
17	16	cv 1135	. . . . . . . 8 class y
18	14, 5	cfv 3809	. . . . . . . . 9 class (f` x)
19	14, 8	cfv 3809	. . . . . . . . 9 class (g` x)
20		caddc 6185	. . . . . . . . 9 class +
21	18, 19, 20	co 4695	. . . . . . . 8 class ((f` x) + (g` x))
22	17, 21	wceq 1136	. . . . . . 7 wff y = ((f` x) + (g` x))
23	15, 22	wa 239	. . . . . 6 wff (x e. ~H /\ y = ((f` x) + (g` x)))
24	23, 13, 16	copab 3213	. . . . 5 class {<.x, y>. | (x e. ~H /\ y = ((f` x) + (g` x)))}
25	12, 24	wceq 1136	. . . 4 wff h = {<.x, y>. | (x e. ~H /\ y = ((f` x) + (g` x)))}
26	10, 25	wa 239	. . 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 4696	. 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 1136	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:  hfsmval 10939

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