Theorem creui 9033

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 8068	. 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 2207	. . . . . . . . . 10 |- ( ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) <-> ( x + ( _i x. y ) ) = ( z + ( _i x. w ) ) )
4		cru 8675	. . . . . . . . . . 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 2503	. . . . . . 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 2903	. . . . . . . 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 2579	. . . . 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 2964	. . . . 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 2212	. . . . . 6 |- ( A = ( z + ( _i x. w ) ) -> ( A = ( x + ( _i x. y ) ) <-> ( z + ( _i x. w ) ) = ( x + ( _i x. y ) ) ) )
17	16	rexbidv 2507	. . . . 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 2690	. . . 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 2629	. 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 2176   A.wral 2484   E.wrex 2485   E!wreu 2486  (class class class)co 5944   CCcc 7923   RRcr 7924   _ici 7927   + caddc 7928   x. cmul 7930

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-sub 8245  df-neg 8246  df-reap 8648

This theorem is referenced by: (None)