Definition df-re 8001

Description: Define a function whose value is the real part of a complex number. See reval 8005 for its value, recli 8015 for its closure, and replim 8011 for its use in decomposing a complex number.

Assertion

Ref	Expression
df-re	|- Re = {<.x, y>. | (x e. CC /\ y = U.{z e. RR | E.w e. RR x = (z + (_i x. w))})}

Distinct variable group:   x,y,z,w

Detailed syntax breakdown of Definition df-re

Step	Hyp	Ref	Expression
1		cre 7997	. 2 class Re
2		vx	. . . . . 6 set x
3	2	cv 1297	. . . . 5 class x
4		cc 6384	. . . . 5 class CC
5	3, 4	wcel 1300	. . . 4 wff x e. CC
6		vy	. . . . . 6 set y
7	6	cv 1297	. . . . 5 class y
8		vz	. . . . . . . . . . 11 set z
9	8	cv 1297	. . . . . . . . . 10 class z
10		ci 6388	. . . . . . . . . . 11 class _i
11		vw	. . . . . . . . . . . 12 set w
12	11	cv 1297	. . . . . . . . . . 11 class w
13		cmul 6391	. . . . . . . . . . 11 class x.
14	10, 12, 13	co 4884	. . . . . . . . . 10 class (_i x. w)
15		caddc 6389	. . . . . . . . . 10 class +
16	9, 14, 15	co 4884	. . . . . . . . 9 class (z + (_i x. w))
17	3, 16	wceq 1298	. . . . . . . 8 wff x = (z + (_i x. w))
18		cr 6385	. . . . . . . 8 class RR
19	17, 11, 18	wrex 2106	. . . . . . 7 wff E.w e. RR x = (z + (_i x. w))
20	19, 8, 18	crab 2108	. . . . . 6 class {z e. RR | E.w e. RR x = (z + (_i x. w))}
21	20	cuni 3177	. . . . 5 class U.{z e. RR | E.w e. RR x = (z + (_i x. w))}
22	7, 21	wceq 1298	. . . 4 wff y = U.{z e. RR | E.w e. RR x = (z + (_i x. w))}
23	5, 22	wa 240	. . 3 wff (x e. CC /\ y = U.{z e. RR | E.w e. RR x = (z + (_i x. w))})
24	23, 2, 6	copab 3395	. 2 class {<.x, y>. | (x e. CC /\ y = U.{z e. RR | E.w e. RR x = (z + (_i x. w))})}
25	1, 24	wceq 1298	1 wff Re = {<.x, y>. | (x e. CC /\ y = U.{z e. RR | E.w e. RR x = (z + (_i x. w))})}

This definition is referenced by:  reval 8005  ref 8009

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