Theorem creui 8104

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 7177	. 2 |- ( A e. CC -> E. z e. RR E. w e. RR A = ( z + ( _i x. w ) ) )
2		simpr 108	. . . . 5 |- ( ( z e. RR /\ w e. RR ) -> w e. RR )
3		eqcom 2084	. . . . . . . . . 10 |- ( ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) <-> ( x + ( _i x. y ) ) = ( z + ( _i x. w ) ) )
4		cru 7769	. . . . . . . . . . 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 264	. . . . . . . . . 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 190	. . . . . . . . 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 536	. . . . . . . 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 2366	. . . . . . 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 170	. . . . . . . . 9 |- ( x = z -> ( y = w <-> y = w ) )
10	9	ceqsrexv 2726	. . . . . . . 8 |- ( z e. RR -> ( E. x e. RR ( x = z /\ y = w ) <-> y = w ) )
11	10	ad2antrr 472	. . . . . . 7 |- ( ( ( z e. RR /\ w e. RR ) /\ y e. RR ) -> ( E. x e. RR ( x = z /\ y = w ) <-> y = w ) )
12	8, 11	bitrd 186	. . . . . 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 2435	. . . . 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 2784	. . . . 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 403	. . . 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 2088	. . . . . 6 |- ( A = ( z + ( _i x. w ) ) -> ( A = ( x + ( _i x. y ) ) <-> ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) ) )
17	16	rexbidv 2370	. . . . 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 2538	. . . 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 155	. . 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 2483	. 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 102   <-> wb 103   = wceq 1285   e. wcel 1434   A.wral 2349   E.wrex 2350   E!wreu 2351  (class class class)co 5543   CCcc 7041   RRcr 7042   _ici 7045   + caddc 7046   x. cmul 7048

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-ltxr 7220  df-sub 7348  df-neg 7349  df-reap 7742

This theorem is referenced by: (None)