Theorem creur 9235

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 8272	. 2 |- ( A e. CC -> E. z e. RR E. w e. RR A = ( z + ( _i x. w ) ) )
2		cru 8878	. . . . . . . . . . 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 2236	. . . . . . . . . 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 400	. . . . . . . 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 2541	. . . . . . 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 2949	. . . . . . . 8 |- ( w e. RR -> ( E. y e. RR ( y = w /\ x = z ) <-> x = z ) )
11	10	ad2antlr 489	. . . . . . 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 2617	. . . . 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 3010	. . . . 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 2241	. . . . . 6 |- ( A = ( z + ( _i x. w ) ) -> ( A = ( x + ( _i x. y ) ) <-> ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) ) )
17	16	rexbidv 2545	. . . . 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 2731	. . . 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 2668	. 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 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   E!wreu 2524  (class class class)co 6052   CCcc 8127   RRcr 8128   _ici 8131   + caddc 8132   x. cmul 8134

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-ltxr 8315  df-sub 8448  df-neg 8449  df-reap 8851

This theorem is referenced by: (None)