Theorem reneg 11557

Description: Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Assertion

Ref	Expression
reneg	|- ( A e. CC -> ( Re ` -u A ) = -u ( Re ` A ) )

Proof of Theorem reneg

Step	Hyp	Ref	Expression
1		recl 11542	. . . . . 6 |- ( A e. CC -> ( Re ` A ) e. RR )
2	1	recnd 8304	. . . . 5 |- ( A e. CC -> ( Re ` A ) e. CC )
3		ax-icn 8224	. . . . . 6 |- _i e. CC
4		imcl 11543	. . . . . . 7 |- ( A e. CC -> ( Im ` A ) e. RR )
5	4	recnd 8304	. . . . . 6 |- ( A e. CC -> ( Im ` A ) e. CC )
6		mulcl 8256	. . . . . 6 |- ( ( _i e. CC /\ ( Im ` A ) e. CC ) -> ( _i x. ( Im ` A ) ) e. CC )
7	3, 5, 6	sylancr 414	. . . . 5 |- ( A e. CC -> ( _i x. ( Im ` A ) ) e. CC )
8	2, 7	negdid 8599	. . . 4 |- ( A e. CC -> -u ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) = ( -u ( Re ` A ) + -u ( _i x. ( Im ` A ) ) ) )
9		replim 11548	. . . . 5 |- ( A e. CC -> A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) )
10	9	negeqd 8470	. . . 4 |- ( A e. CC -> -u A = -u ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) )
11		mulneg2 8671	. . . . . 6 |- ( ( _i e. CC /\ ( Im ` A ) e. CC ) -> ( _i x. -u ( Im ` A ) ) = -u ( _i x. ( Im ` A ) ) )
12	3, 5, 11	sylancr 414	. . . . 5 |- ( A e. CC -> ( _i x. -u ( Im ` A ) ) = -u ( _i x. ( Im ` A ) ) )
13	12	oveq2d 6068	. . . 4 |- ( A e. CC -> ( -u ( Re ` A ) + ( _i x. -u ( Im ` A ) ) ) = ( -u ( Re ` A ) + -u ( _i x. ( Im ` A ) ) ) )
14	8, 10, 13	3eqtr4d 2277	. . 3 |- ( A e. CC -> -u A = ( -u ( Re ` A ) + ( _i x. -u ( Im ` A ) ) ) )
15	14	fveq2d 5676	. 2 |- ( A e. CC -> ( Re ` -u A ) = ( Re ` ( -u ( Re ` A ) + ( _i x. -u ( Im ` A ) ) ) ) )
16	1	renegcld 8655	. . 3 |- ( A e. CC -> -u ( Re ` A ) e. RR )
17	4	renegcld 8655	. . 3 |- ( A e. CC -> -u ( Im ` A ) e. RR )
18		crre 11546	. . 3 |- ( ( -u ( Re ` A ) e. RR /\ -u ( Im ` A ) e. RR ) -> ( Re ` ( -u ( Re ` A ) + ( _i x. -u ( Im ` A ) ) ) ) = -u ( Re ` A ) )
19	16, 17, 18	syl2anc 411	. 2 |- ( A e. CC -> ( Re ` ( -u ( Re ` A ) + ( _i x. -u ( Im ` A ) ) ) ) = -u ( Re ` A ) )
20	15, 19	eqtrd 2267	1 |- ( A e. CC -> ( Re ` -u A ) = -u ( Re ` A ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   ` cfv 5354  (class class class)co 6052   CCcc 8127   RRcr 8128   _ici 8131   + caddc 8132   x. cmul 8134   -ucneg 8447   Recre 11529   Imcim 11530

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-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247

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-rmo 2530  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-mpt 4175  df-id 4416  df-po 4419  df-iso 4420  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-2 9298  df-cj 11531  df-re 11532  df-im 11533

This theorem is referenced by:  resub  11559  cjneg  11579  renegi  11613  renegd  11643