Theorem creur 8854

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 7895	. 2 |- ( A e. CC -> E. z e. RR E. w e. RR A = ( z + ( _i x. w ) ) )
2		cru 8500	. . . . . . . . . . 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 2167	. . . . . . . . . 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 2463	. . . . . . 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 2856	. . . . . . . 8 |- ( w e. RR -> ( E. y e. RR ( y = w /\ x = z ) <-> x = z ) )
11	10	ad2antlr 481	. . . . . . 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 2539	. . . . 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 2917	. . . . 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 2172	. . . . . 6 |- ( A = ( z + ( _i x. w ) ) -> ( A = ( x + ( _i x. y ) ) <-> ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) ) )
17	16	rexbidv 2467	. . . . 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 2649	. . . 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 2589	. 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 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   E!wreu 2446  (class class class)co 5842   CCcc 7751   RRcr 7752   _ici 7755   + caddc 7756   x. cmul 7758

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-ltxr 7938  df-sub 8071  df-neg 8072  df-reap 8473

This theorem is referenced by: (None)