Theorem creui 9130

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 8165	. 2 |- ( A e. CC -> E. z e. RR E. w e. RR A = ( z + ( _i x. w ) ) )
2		simpr 110	. . . . 5 |- ( ( z e. RR /\ w e. RR ) -> w e. RR )
3		eqcom 2231	. . . . . . . . . 10 |- ( ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) <-> ( x + ( _i x. y ) ) = ( z + ( _i x. w ) ) )
4		cru 8772	. . . . . . . . . . 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 268	. . . . . . . . . 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	bitrid 192	. . . . . . . . 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 571	. . . . . . . 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 2527	. . . . . . 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 172	. . . . . . . . 9 |- ( x = z -> ( y = w <-> y = w ) )
10	9	ceqsrexv 2934	. . . . . . . 8 |- ( z e. RR -> ( E. x e. RR ( x = z /\ y = w ) <-> y = w ) )
11	10	ad2antrr 488	. . . . . . 7 |- ( ( ( z e. RR /\ w e. RR ) /\ y e. RR ) -> ( E. x e. RR ( x = z /\ y = w ) <-> y = w ) )
12	8, 11	bitrd 188	. . . . . 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 2603	. . . . 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 2995	. . . . 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 411	. . . 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 2236	. . . . . 6 |- ( A = ( z + ( _i x. w ) ) -> ( A = ( x + ( _i x. y ) ) <-> ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) ) )
17	16	rexbidv 2531	. . . . 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 2716	. . . 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 157	. . 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 2654	. 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 104   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   E!wreu 2510  (class class class)co 6013   CCcc 8020   RRcr 8021   _ici 8024   + caddc 8025   x. cmul 8027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-sub 8342  df-neg 8343  df-reap 8745

This theorem is referenced by: (None)