Theorem creur 9279

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 8312	. 2 |- ( A e. CC -> E. z e. RR E. w e. RR A = ( z + ( _i x. w ) ) )
2		cru 8920	. . . . . . . . . . 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 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 ) ) )
4		eqcom 2240	. . . . . . . . . 10 |- ( ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) <-> ( x + ( _i x. y ) ) = ( z + ( _i x. w ) ) )
5		ancom 266	. . . . . . . . . 10 |- ( ( y = w /\ x = z ) <-> ( x = z /\ y = w ) )
6	3, 4, 5	3bitr4g 223	. . . . . . . . 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 404	. . . . . . . 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 2547	. . . . . . 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 172	. . . . . . . . 9 |- ( y = w -> ( x = z <-> x = z ) )
10	9	ceqsrexv 2956	. . . . . . . 8 |- ( w e. RR -> ( E. y e. RR ( y = w /\ x = z ) <-> x = z ) )
11	10	ad2antlr 493	. . . . . . 7 |- ( ( ( z e. RR /\ w e. RR ) /\ x e. RR ) -> ( E. y e. RR ( y = w /\ x = z ) <-> x = z ) )
12	8, 11	bitrd 188	. . . . . 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 2623	. . . . 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 3017	. . . . 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 282	. . . 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 2245	. . . . . 6 |- ( A = ( z + ( _i x. w ) ) -> ( A = ( x + ( _i x. y ) ) <-> ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) ) )
17	16	rexbidv 2551	. . . . 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 2737	. . . 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 157	. . 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 2674	. 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 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   E!wreu 2530  (class class class)co 6075   CCcc 8167   RRcr 8168   _ici 8171   + caddc 8172   x. cmul 8174

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-sub 8489  df-neg 8490  df-reap 8893

This theorem is referenced by: (None)