Theorem resub 11431

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 8379	. . . 4 |- ( B e. CC -> -u B e. CC )
2		readd 11430	. . . 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 11429	. . . . 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 6034	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( Re ` A ) + ( Re ` -u B ) ) = ( ( Re ` A ) + -u ( Re ` B ) ) )
7	3, 6	eqtrd 2264	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A + -u B ) ) = ( ( Re ` A ) + -u ( Re ` B ) ) )
8		negsub 8427	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
9	8	fveq2d 5643	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A + -u B ) ) = ( Re ` ( A - B ) ) )
10		recl 11414	. . . 4 |- ( A e. CC -> ( Re ` A ) e. RR )
11	10	recnd 8208	. . 3 |- ( A e. CC -> ( Re ` A ) e. CC )
12		recl 11414	. . . 4 |- ( B e. CC -> ( Re ` B ) e. RR )
13	12	recnd 8208	. . 3 |- ( B e. CC -> ( Re ` B ) e. CC )
14		negsub 8427	. . 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 2272	1 |- ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A - B ) ) = ( ( Re ` A ) - ( Re ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6018   CCcc 8030   + caddc 8035   - cmin 8350   -ucneg 8351   Recre 11401

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-2 9202  df-cj 11403  df-re 11404  df-im 11405

This theorem is referenced by:  resubd  11522  recn2  11878