Theorem creui 9068

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 8103	. 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 2209	. . . . . . . . . 10 |- ( ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) <-> ( x + ( _i x. y ) ) = ( z + ( _i x. w ) ) )
4		cru 8710	. . . . . . . . . . 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 2505	. . . . . . 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 2910	. . . . . . . 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 2581	. . . . 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 2971	. . . . 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 2214	. . . . . 6 |- ( A = ( z + ( _i x. w ) ) -> ( A = ( x + ( _i x. y ) ) <-> ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) ) )
17	16	rexbidv 2509	. . . . 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 2693	. . . 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 2631	. 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 1373   e. wcel 2178   A.wral 2486   E.wrex 2487   E!wreu 2488  (class class class)co 5967   CCcc 7958   RRcr 7959   _ici 7962   + caddc 7963   x. cmul 7965

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-sub 8280  df-neg 8281  df-reap 8683

This theorem is referenced by: (None)