Theorem rediv 13871

Description: Real part of a division. Related to remul2 13870. (Contributed by David A. Wheeler, 10-Jun-2015.)

Assertion

Ref	Expression
rediv	|- ( ( A e. CC /\ B e. RR /\ B =/= 0 ) -> ( Re ` ( A / B ) ) = ( ( Re ` A ) / B ) )

Proof of Theorem rediv

Step	Hyp	Ref	Expression
1		ancom 466	. . . . 5 |- ( ( ( B e. RR /\ B =/= 0 ) /\ A e. CC ) <-> ( A e. CC /\ ( B e. RR /\ B =/= 0 ) ) )
2		3anass 1042	. . . . 5 |- ( ( A e. CC /\ B e. RR /\ B =/= 0 ) <-> ( A e. CC /\ ( B e. RR /\ B =/= 0 ) ) )
3	1, 2	bitr4i 267	. . . 4 |- ( ( ( B e. RR /\ B =/= 0 ) /\ A e. CC ) <-> ( A e. CC /\ B e. RR /\ B =/= 0 ) )
4		rereccl 10743	. . . . 5 |- ( ( B e. RR /\ B =/= 0 ) -> ( 1 / B ) e. RR )
5	4	anim1i 592	. . . 4 |- ( ( ( B e. RR /\ B =/= 0 ) /\ A e. CC ) -> ( ( 1 / B ) e. RR /\ A e. CC ) )
6	3, 5	sylbir 225	. . 3 |- ( ( A e. CC /\ B e. RR /\ B =/= 0 ) -> ( ( 1 / B ) e. RR /\ A e. CC ) )
7		remul2 13870	. . 3 |- ( ( ( 1 / B ) e. RR /\ A e. CC ) -> ( Re ` ( ( 1 / B ) x. A ) ) = ( ( 1 / B ) x. ( Re ` A ) ) )
8	6, 7	syl 17	. 2 |- ( ( A e. CC /\ B e. RR /\ B =/= 0 ) -> ( Re ` ( ( 1 / B ) x. A ) ) = ( ( 1 / B ) x. ( Re ` A ) ) )
9		recn 10026	. . 3 |- ( B e. RR -> B e. CC )
10		divrec2 10702	. . . 4 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( A / B ) = ( ( 1 / B ) x. A ) )
11	10	fveq2d 6195	. . 3 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( Re ` ( A / B ) ) = ( Re ` ( ( 1 / B ) x. A ) ) )
12	9, 11	syl3an2 1360	. 2 |- ( ( A e. CC /\ B e. RR /\ B =/= 0 ) -> ( Re ` ( A / B ) ) = ( Re ` ( ( 1 / B ) x. A ) ) )
13		recl 13850	. . . . 5 |- ( A e. CC -> ( Re ` A ) e. RR )
14	13	recnd 10068	. . . 4 |- ( A e. CC -> ( Re ` A ) e. CC )
15	14	3ad2ant1 1082	. . 3 |- ( ( A e. CC /\ B e. RR /\ B =/= 0 ) -> ( Re ` A ) e. CC )
16	9	3ad2ant2 1083	. . 3 |- ( ( A e. CC /\ B e. RR /\ B =/= 0 ) -> B e. CC )
17		simp3 1063	. . 3 |- ( ( A e. CC /\ B e. RR /\ B =/= 0 ) -> B =/= 0 )
18	15, 16, 17	divrec2d 10805	. 2 |- ( ( A e. CC /\ B e. RR /\ B =/= 0 ) -> ( ( Re ` A ) / B ) = ( ( 1 / B ) x. ( Re ` A ) ) )
19	8, 12, 18	3eqtr4d 2666	1 |- ( ( A e. CC /\ B e. RR /\ B =/= 0 ) -> ( Re ` ( A / B ) ) = ( ( Re ` A ) / B ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   / cdiv 10684   Recre 13837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-2 11079  df-cj 13839  df-re 13840  df-im 13841

This theorem is referenced by:  redivd  13969