Theorem creui 8721

Description: The imaginary part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
creui	|- ( A e. CC -> E! y e. RR E. x e. RR A = ( x + ( _i x. y ) ) )

Distinct variable group:   x, y, A

Proof of Theorem creui

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnre 7765	. 2 |- ( A e. CC -> E. z e. RR E. w e. RR A = ( z + ( _i x. w ) ) )
2		simpr 109	. . . . 5 |- ( ( z e. RR /\ w e. RR ) -> w e. RR )
3		eqcom 2141	. . . . . . . . . 10 |- ( ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) <-> ( x + ( _i x. y ) ) = ( z + ( _i x. w ) ) )
4		cru 8367	. . . . . . . . . . 11 |- ( ( ( x e. RR /\ y e. RR ) /\ ( z e. RR /\ w e. RR ) ) -> ( ( x + ( _i x. y ) ) = ( z + ( _i x. w ) ) <-> ( x = z /\ y = w ) ) )
5	4	ancoms 266	. . . . . . . . . 10 |- ( ( ( z e. RR /\ w e. RR ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( x + ( _i x. y ) ) = ( z + ( _i x. w ) ) <-> ( x = z /\ y = w ) ) )
6	3, 5	syl5bb 191	. . . . . . . . 9 |- ( ( ( z e. RR /\ w e. RR ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) <-> ( x = z /\ y = w ) ) )
7	6	anass1rs 560	. . . . . . . 8 |- ( ( ( ( z e. RR /\ w e. RR ) /\ y e. RR ) /\ x e. RR ) -> ( ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) <-> ( x = z /\ y = w ) ) )
8	7	rexbidva 2434	. . . . . . 7 |- ( ( ( z e. RR /\ w e. RR ) /\ y e. RR ) -> ( E. x e. RR ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) <-> E. x e. RR ( x = z /\ y = w ) ) )
9		biidd 171	. . . . . . . . 9 |- ( x = z -> ( y = w <-> y = w ) )
10	9	ceqsrexv 2815	. . . . . . . 8 |- ( z e. RR -> ( E. x e. RR ( x = z /\ y = w ) <-> y = w ) )
11	10	ad2antrr 479	. . . . . . 7 |- ( ( ( z e. RR /\ w e. RR ) /\ y e. RR ) -> ( E. x e. RR ( x = z /\ y = w ) <-> y = w ) )
12	8, 11	bitrd 187	. . . . . 6 |- ( ( ( z e. RR /\ w e. RR ) /\ y e. RR ) -> ( E. x e. RR ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) <-> y = w ) )
13	12	ralrimiva 2505	. . . . 5 |- ( ( z e. RR /\ w e. RR ) -> A. y e. RR ( E. x e. RR ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) <-> y = w ) )
14		reu6i 2875	. . . . 5 |- ( ( w e. RR /\ A. y e. RR ( E. x e. RR ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) <-> y = w ) ) -> E! y e. RR E. x e. RR ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) )
15	2, 13, 14	syl2anc 408	. . . 4 |- ( ( z e. RR /\ w e. RR ) -> E! y e. RR E. x e. RR ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) )
16		eqeq1 2146	. . . . . 6 |- ( A = ( z + ( _i x. w ) ) -> ( A = ( x + ( _i x. y ) ) <-> ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) ) )
17	16	rexbidv 2438	. . . . 5 |- ( A = ( z + ( _i x. w ) ) -> ( E. x e. RR A = ( x + ( _i x. y ) ) <-> E. x e. RR ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) ) )
18	17	reubidv 2614	. . . 4 |- ( A = ( z + ( _i x. w ) ) -> ( E! y e. RR E. x e. RR A = ( x + ( _i x. y ) ) <-> E! y e. RR E. x e. RR ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) ) )
19	15, 18	syl5ibrcom 156	. . 3 |- ( ( z e. RR /\ w e. RR ) -> ( A = ( z + ( _i x. w ) ) -> E! y e. RR E. x e. RR A = ( x + ( _i x. y ) ) ) )
20	19	rexlimivv 2555	. 2 |- ( E. z e. RR E. w e. RR A = ( z + ( _i x. w ) ) -> E! y e. RR E. x e. RR A = ( x + ( _i x. y ) ) )
21	1, 20	syl 14	1 |- ( A e. CC -> E! y e. RR E. x e. RR A = ( x + ( _i x. y ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   A.wral 2416   E.wrex 2417   E!wreu 2418  (class class class)co 5774   CCcc 7621   RRcr 7622   _ici 7625   + caddc 7626   x. cmul 7628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7714  ax-resscn 7715  ax-1cn 7716  ax-1re 7717  ax-icn 7718  ax-addcl 7719  ax-addrcl 7720  ax-mulcl 7721  ax-mulrcl 7722  ax-addcom 7723  ax-mulcom 7724  ax-addass 7725  ax-mulass 7726  ax-distr 7727  ax-i2m1 7728  ax-0lt1 7729  ax-1rid 7730  ax-0id 7731  ax-rnegex 7732  ax-precex 7733  ax-cnre 7734  ax-pre-ltirr 7735  ax-pre-lttrn 7737  ax-pre-apti 7738  ax-pre-ltadd 7739  ax-pre-mulgt0 7740

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7805  df-mnf 7806  df-ltxr 7808  df-sub 7938  df-neg 7939  df-reap 8340

This theorem is referenced by: (None)