Theorem creur 8879

Description: The real 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
creur	|- ( A e. CC -> E! x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )

Distinct variable group:   x, y, A

Proof of Theorem creur

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

Step	Hyp	Ref	Expression
1		cnre 7920	. 2 |- ( A e. CC -> E. z e. RR E. w e. RR A = ( z + ( _i x. w ) ) )
2		cru 8525	. . . . . . . . . . 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 ) ) )
3	2	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 ) ) )
4		eqcom 2173	. . . . . . . . . 10 |- ( ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) <-> ( x + ( _i x. y ) ) = ( z + ( _i x. w ) ) )
5		ancom 264	. . . . . . . . . 10 |- ( ( y = w /\ x = z ) <-> ( x = z /\ y = w ) )
6	3, 4, 5	3bitr4g 222	. . . . . . . . 9 |- ( ( ( z e. RR /\ w e. RR ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) <-> ( y = w /\ x = z ) ) )
7	6	anassrs 398	. . . . . . . 8 |- ( ( ( ( z e. RR /\ w e. RR ) /\ x e. RR ) /\ y e. RR ) -> ( ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) <-> ( y = w /\ x = z ) ) )
8	7	rexbidva 2468	. . . . . . 7 |- ( ( ( z e. RR /\ w e. RR ) /\ x e. RR ) -> ( E. y e. RR ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) <-> E. y e. RR ( y = w /\ x = z ) ) )
9		biidd 171	. . . . . . . . 9 |- ( y = w -> ( x = z <-> x = z ) )
10	9	ceqsrexv 2861	. . . . . . . 8 |- ( w e. RR -> ( E. y e. RR ( y = w /\ x = z ) <-> x = z ) )
11	10	ad2antlr 487	. . . . . . 7 |- ( ( ( z e. RR /\ w e. RR ) /\ x e. RR ) -> ( E. y e. RR ( y = w /\ x = z ) <-> x = z ) )
12	8, 11	bitrd 187	. . . . . 6 |- ( ( ( z e. RR /\ w e. RR ) /\ x e. RR ) -> ( E. y e. RR ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) <-> x = z ) )
13	12	ralrimiva 2544	. . . . 5 |- ( ( z e. RR /\ w e. RR ) -> A. x e. RR ( E. y e. RR ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) <-> x = z ) )
14		reu6i 2922	. . . . 5 |- ( ( z e. RR /\ A. x e. RR ( E. y e. RR ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) <-> x = z ) ) -> E! x e. RR E. y e. RR ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) )
15	13, 14	syldan 280	. . . 4 |- ( ( z e. RR /\ w e. RR ) -> E! x e. RR E. y e. RR ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) )
16		eqeq1 2178	. . . . . 6 |- ( A = ( z + ( _i x. w ) ) -> ( A = ( x + ( _i x. y ) ) <-> ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) ) )
17	16	rexbidv 2472	. . . . 5 |- ( A = ( z + ( _i x. w ) ) -> ( E. y e. RR A = ( x + ( _i x. y ) ) <-> E. y e. RR ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) ) )
18	17	reubidv 2654	. . . 4 |- ( A = ( z + ( _i x. w ) ) -> ( E! x e. RR E. y e. RR A = ( x + ( _i x. y ) ) <-> E! x e. RR E. y 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! x e. RR E. y e. RR A = ( x + ( _i x. y ) ) ) )
20	19	rexlimivv 2594	. 2 |- ( E. z e. RR E. w e. RR A = ( z + ( _i x. w ) ) -> E! x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
21	1, 20	syl 14	1 |- ( A e. CC -> E! x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1349   e. wcel 2142   A.wral 2449   E.wrex 2450   E!wreu 2451  (class class class)co 5857   CCcc 7776   RRcr 7777   _ici 7780   + caddc 7781   x. cmul 7783

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 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-sep 4108  ax-pow 4161  ax-pr 4195  ax-un 4419  ax-setind 4522  ax-cnex 7869  ax-resscn 7870  ax-1cn 7871  ax-1re 7872  ax-icn 7873  ax-addcl 7874  ax-addrcl 7875  ax-mulcl 7876  ax-mulrcl 7877  ax-addcom 7878  ax-mulcom 7879  ax-addass 7880  ax-mulass 7881  ax-distr 7882  ax-i2m1 7883  ax-0lt1 7884  ax-1rid 7885  ax-0id 7886  ax-rnegex 7887  ax-precex 7888  ax-cnre 7889  ax-pre-ltirr 7890  ax-pre-lttrn 7892  ax-pre-apti 7893  ax-pre-ltadd 7894  ax-pre-mulgt0 7895

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-fal 1355  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ne 2342  df-nel 2437  df-ral 2454  df-rex 2455  df-reu 2456  df-rab 2458  df-v 2733  df-sbc 2957  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-br 3991  df-opab 4052  df-id 4279  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-iota 5162  df-fun 5202  df-fv 5208  df-riota 5813  df-ov 5860  df-oprab 5861  df-mpo 5862  df-pnf 7960  df-mnf 7961  df-ltxr 7963  df-sub 8096  df-neg 8097  df-reap 8498

This theorem is referenced by: (None)