Theorem replim 11544

Description: Reconstruct a complex number from its real and imaginary parts. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)

Assertion

Ref	Expression
replim	|- ( A e. CC -> A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) )

Proof of Theorem replim

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnre 8270	. 2 |- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
2		crre 11542	. . . . . 6 |- ( ( x e. RR /\ y e. RR ) -> ( Re ` ( x + ( _i x. y ) ) ) = x )
3		crim 11543	. . . . . . 7 |- ( ( x e. RR /\ y e. RR ) -> ( Im ` ( x + ( _i x. y ) ) ) = y )
4	3	oveq2d 6066	. . . . . 6 |- ( ( x e. RR /\ y e. RR ) -> ( _i x. ( Im ` ( x + ( _i x. y ) ) ) ) = ( _i x. y ) )
5	2, 4	oveq12d 6068	. . . . 5 |- ( ( x e. RR /\ y e. RR ) -> ( ( Re ` ( x + ( _i x. y ) ) ) + ( _i x. ( Im ` ( x + ( _i x. y ) ) ) ) ) = ( x + ( _i x. y ) ) )
6	5	eqcomd 2238	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( x + ( _i x. y ) ) = ( ( Re ` ( x + ( _i x. y ) ) ) + ( _i x. ( Im ` ( x + ( _i x. y ) ) ) ) ) )
7		id 19	. . . . 5 |- ( A = ( x + ( _i x. y ) ) -> A = ( x + ( _i x. y ) ) )
8		fveq2 5670	. . . . . 6 |- ( A = ( x + ( _i x. y ) ) -> ( Re ` A ) = ( Re ` ( x + ( _i x. y ) ) ) )
9		fveq2 5670	. . . . . . 7 |- ( A = ( x + ( _i x. y ) ) -> ( Im ` A ) = ( Im ` ( x + ( _i x. y ) ) ) )
10	9	oveq2d 6066	. . . . . 6 |- ( A = ( x + ( _i x. y ) ) -> ( _i x. ( Im ` A ) ) = ( _i x. ( Im ` ( x + ( _i x. y ) ) ) ) )
11	8, 10	oveq12d 6068	. . . . 5 |- ( A = ( x + ( _i x. y ) ) -> ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) = ( ( Re ` ( x + ( _i x. y ) ) ) + ( _i x. ( Im ` ( x + ( _i x. y ) ) ) ) ) )
12	7, 11	eqeq12d 2247	. . . 4 |- ( A = ( x + ( _i x. y ) ) -> ( A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) <-> ( x + ( _i x. y ) ) = ( ( Re ` ( x + ( _i x. y ) ) ) + ( _i x. ( Im ` ( x + ( _i x. y ) ) ) ) ) ) )
13	6, 12	syl5ibrcom 157	. . 3 |- ( ( x e. RR /\ y e. RR ) -> ( A = ( x + ( _i x. y ) ) -> A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) ) )
14	13	rexlimivv 2666	. 2 |- ( E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) -> A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) )
15	1, 14	syl 14	1 |- ( A e. CC -> A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   E.wrex 2521   ` cfv 5352  (class class class)co 6050   CCcc 8125   RRcr 8126   _ici 8129   + caddc 8130   x. cmul 8132   Recre 11525   Imcim 11526

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-2 9296  df-cj 11527  df-re 11528  df-im 11529

This theorem is referenced by:  remim  11545  reim0b  11547  rereb  11548  mulreap  11549  cjreb  11551  reneg  11553  readd  11554  remullem  11556  imneg  11561  imadd  11562  cjcj  11568  imval2  11579  cnrecnv  11595  replimi  11599  replimd  11626  cnreim  11663  abs00ap  11747  recan  11794  efeul  12420  absef  12456  absefib  12457  efieq1re  12458