Definition df-unop 9709

Description: Define the set of unitary operators on Hilbert space.

Assertion

Ref	Expression
df-unop	⊢ UniOp = {t∣(t: ℋ –onto→ ℋ ⋀ ∀x ∈ ℋ ∀y ∈ ℋ ((t ‘x) ·ih (t ‘y)) = (x ·ih y))}

Distinct variable group:   x,t,y

Detailed syntax breakdown of Definition df-unop

Step	Hyp	Ref	Expression
1		cuo 8757	. 2 class UniOp
2		chil 8727	. . . . 5 class ℋ
3		vt	. . . . . 6 set t
4	3	cv 953	. . . . 5 class t
5	2, 2, 4	wfo 3175	. . . 4 wff t: ℋ –onto→ ℋ
6		vx	. . . . . . . . . 10 set x
7	6	cv 953	. . . . . . . . 9 class x
8	7, 4	cfv 3177	. . . . . . . 8 class (t ‘x)
9		vy	. . . . . . . . . 10 set y
10	9	cv 953	. . . . . . . . 9 class y
11	10, 4	cfv 3177	. . . . . . . 8 class (t ‘y)
12		csp 8732	. . . . . . . 8 class ·ih
13	8, 11, 12	co 3954	. . . . . . 7 class ((t ‘x) ·ih (t ‘y))
14	7, 10, 12	co 3954	. . . . . . 7 class (x ·ih y)
15	13, 14	wceq 954	. . . . . 6 wff ((t ‘x) ·ih (t ‘y)) = (x ·ih y)
16	15, 9, 2	wral 1642	. . . . 5 wff ∀y ∈ ℋ ((t ‘x) ·ih (t ‘y)) = (x ·ih y)
17	16, 6, 2	wral 1642	. . . 4 wff ∀x ∈ ℋ ∀y ∈ ℋ ((t ‘x) ·ih (t ‘y)) = (x ·ih y)
18	5, 17	wa 223	. . 3 wff (t: ℋ –onto→ ℋ ⋀ ∀x ∈ ℋ ∀y ∈ ℋ ((t ‘x) ·ih (t ‘y)) = (x ·ih y))
19	18, 3	cab 1461	. 2 class {t∣(t: ℋ –onto→ ℋ ⋀ ∀x ∈ ℋ ∀y ∈ ℋ ((t ‘x) ·ih (t ‘y)) = (x ·ih y))}
20	1, 19	wceq 954	1 wff UniOp = {t∣(t: ℋ –onto→ ℋ ⋀ ∀x ∈ ℋ ∀y ∈ ℋ ((t ‘x) ·ih (t ‘y)) = (x ·ih y))}

This definition is referenced by:  elunopt 9739

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