Theorem resub 11381

Description: Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)

Assertion

Ref	Expression
resub	|- ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A - B ) ) = ( ( Re ` A ) - ( Re ` B ) ) )

Proof of Theorem resub

Step	Hyp	Ref	Expression
1		negcl 8346	. . . 4 |- ( B e. CC -> -u B e. CC )
2		readd 11380	. . . 4 |- ( ( A e. CC /\ -u B e. CC ) -> ( Re ` ( A + -u B ) ) = ( ( Re ` A ) + ( Re ` -u B ) ) )
3	1, 2	sylan2 286	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A + -u B ) ) = ( ( Re ` A ) + ( Re ` -u B ) ) )
4		reneg 11379	. . . . 5 |- ( B e. CC -> ( Re ` -u B ) = -u ( Re ` B ) )
5	4	adantl 277	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( Re ` -u B ) = -u ( Re ` B ) )
6	5	oveq2d 6017	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( Re ` A ) + ( Re ` -u B ) ) = ( ( Re ` A ) + -u ( Re ` B ) ) )
7	3, 6	eqtrd 2262	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A + -u B ) ) = ( ( Re ` A ) + -u ( Re ` B ) ) )
8		negsub 8394	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
9	8	fveq2d 5631	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A + -u B ) ) = ( Re ` ( A - B ) ) )
10		recl 11364	. . . 4 |- ( A e. CC -> ( Re ` A ) e. RR )
11	10	recnd 8175	. . 3 |- ( A e. CC -> ( Re ` A ) e. CC )
12		recl 11364	. . . 4 |- ( B e. CC -> ( Re ` B ) e. RR )
13	12	recnd 8175	. . 3 |- ( B e. CC -> ( Re ` B ) e. CC )
14		negsub 8394	. . 3 |- ( ( ( Re ` A ) e. CC /\ ( Re ` B ) e. CC ) -> ( ( Re ` A ) + -u ( Re ` B ) ) = ( ( Re ` A ) - ( Re ` B ) ) )
15	11, 13, 14	syl2an 289	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( Re ` A ) + -u ( Re ` B ) ) = ( ( Re ` A ) - ( Re ` B ) ) )
16	7, 9, 15	3eqtr3d 2270	1 |- ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A - B ) ) = ( ( Re ` A ) - ( Re ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   ` cfv 5318  (class class class)co 6001   CCcc 7997   + caddc 8002   - cmin 8317   -ucneg 8318   Recre 11351

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-2 9169  df-cj 11353  df-re 11354  df-im 11355

This theorem is referenced by:  resubd  11472  recn2  11828