Definition df-lnfn 9774

Description: Define the set of linear functionals on Hilbert space.

Assertion

Ref	Expression
df-lnfn	|- LinFn = {t | (t:H~-->CC /\ A.x e. CC A.y e. H~ A.z e. H~ (t` ((x .h y) +h z)) = ((x x. (t` y)) + (t` z)))}

Distinct variable group:   x,t,y,z

Detailed syntax breakdown of Definition df-lnfn

Step	Hyp	Ref	Expression
1		clf 8823	. 2 class LinFn
2		chil 8788	. . . . 5 class H~
3		cc 5232	. . . . 5 class CC
4		vt	. . . . . 6 set t
5	4	cv 955	. . . . 5 class t
6	2, 3, 5	wf 3178	. . . 4 wff t:H~-->CC
7		vx	. . . . . . . . . . . 12 set x
8	7	cv 955	. . . . . . . . . . 11 class x
9		vy	. . . . . . . . . . . 12 set y
10	9	cv 955	. . . . . . . . . . 11 class y
11		csm 8790	. . . . . . . . . . 11 class .h
12	8, 10, 11	co 3963	. . . . . . . . . 10 class (x .h y)
13		vz	. . . . . . . . . . 11 set z
14	13	cv 955	. . . . . . . . . 10 class z
15		cva 8789	. . . . . . . . . 10 class +h
16	12, 14, 15	co 3963	. . . . . . . . 9 class ((x .h y) +h z)
17	16, 5	cfv 3182	. . . . . . . 8 class (t` ((x .h y) +h z))
18	10, 5	cfv 3182	. . . . . . . . . 10 class (t` y)
19		cmul 5239	. . . . . . . . . 10 class x.
20	8, 18, 19	co 3963	. . . . . . . . 9 class (x x. (t` y))
21	14, 5	cfv 3182	. . . . . . . . 9 class (t` z)
22		caddc 5237	. . . . . . . . 9 class +
23	20, 21, 22	co 3963	. . . . . . . 8 class ((x x. (t` y)) + (t` z))
24	17, 23	wceq 956	. . . . . . 7 wff (t` ((x .h y) +h z)) = ((x x. (t` y)) + (t` z))
25	24, 13, 2	wral 1645	. . . . . 6 wff A.z e. H~ (t` ((x .h y) +h z)) = ((x x. (t` y)) + (t` z))
26	25, 9, 2	wral 1645	. . . . 5 wff A.y e. H~ A.z e. H~ (t` ((x .h y) +h z)) = ((x x. (t` y)) + (t` z))
27	26, 7, 3	wral 1645	. . . 4 wff A.x e. CC A.y e. H~ A.z e. H~ (t` ((x .h y) +h z)) = ((x x. (t` y)) + (t` z))
28	6, 27	wa 223	. . 3 wff (t:H~-->CC /\ A.x e. CC A.y e. H~ A.z e. H~ (t` ((x .h y) +h z)) = ((x x. (t` y)) + (t` z)))
29	28, 4	cab 1463	. 2 class {t | (t:H~-->CC /\ A.x e. CC A.y e. H~ A.z e. H~ (t` ((x .h y) +h z)) = ((x x. (t` y)) + (t` z)))}
30	1, 29	wceq 956	1 wff LinFn = {t | (t:H~-->CC /\ A.x e. CC A.y e. H~ A.z e. H~ (t` ((x .h y) +h z)) = ((x x. (t` y)) + (t` z)))}

This definition is referenced by:  ellnfnt 9810

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