Theorem redivap 11661

Description: Real part of a division. Related to remul2 11660. (Contributed by Jim Kingdon, 14-Jun-2020.)

Assertion

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

Proof of Theorem redivap

Step	Hyp	Ref	Expression
1		ancom 266	. . . . 5 |- ( ( ( B e. RR /\ B # 0 ) /\ A e. CC ) <-> ( A e. CC /\ ( B e. RR /\ B # 0 ) ) )
2		3anass 1013	. . . . 5 |- ( ( A e. CC /\ B e. RR /\ B # 0 ) <-> ( A e. CC /\ ( B e. RR /\ B # 0 ) ) )
3	1, 2	bitr4i 187	. . . 4 |- ( ( ( B e. RR /\ B # 0 ) /\ A e. CC ) <-> ( A e. CC /\ B e. RR /\ B # 0 ) )
4		rerecclap 9068	. . . . 5 |- ( ( B e. RR /\ B # 0 ) -> ( 1 / B ) e. RR )
5	4	anim1i 340	. . . 4 |- ( ( ( B e. RR /\ B # 0 ) /\ A e. CC ) -> ( ( 1 / B ) e. RR /\ A e. CC ) )
6	3, 5	sylbir 135	. . 3 |- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( ( 1 / B ) e. RR /\ A e. CC ) )
7		remul2 11660	. . 3 |- ( ( ( 1 / B ) e. RR /\ A e. CC ) -> ( Re ` ( ( 1 / B ) x. A ) ) = ( ( 1 / B ) x. ( Re ` A ) ) )
8	6, 7	syl 14	. 2 |- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( Re ` ( ( 1 / B ) x. A ) ) = ( ( 1 / B ) x. ( Re ` A ) ) )
9		recn 8318	. . 3 |- ( B e. RR -> B e. CC )
10		divrecap2 9027	. . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) = ( ( 1 / B ) x. A ) )
11	10	fveq2d 5702	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( Re ` ( A / B ) ) = ( Re ` ( ( 1 / B ) x. A ) ) )
12	9, 11	syl3an2 1312	. 2 |- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( Re ` ( A / B ) ) = ( Re ` ( ( 1 / B ) x. A ) ) )
13		recl 11640	. . . . 5 |- ( A e. CC -> ( Re ` A ) e. RR )
14	13	recnd 8360	. . . 4 |- ( A e. CC -> ( Re ` A ) e. CC )
15	14	3ad2ant1 1049	. . 3 |- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( Re ` A ) e. CC )
16	9	3ad2ant2 1050	. . 3 |- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> B e. CC )
17		simp3 1030	. . 3 |- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> B # 0 )
18	15, 16, 17	divrecap2d 9132	. 2 |- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( ( Re ` A ) / B ) = ( ( 1 / B ) x. ( Re ` A ) ) )
19	8, 12, 18	3eqtr4d 2281	1 |- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( Re ` ( A / B ) ) = ( ( Re ` A ) / B ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   CCcc 8183   RRcr 8184   0cc0 8185   1c1 8186   x. cmul 8190   # cap 8917   / cdiv 9010   Recre 11627

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-po 4443  df-iso 4444  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-2 9372  df-cj 11629  df-re 11630  df-im 11631

This theorem is used by:  redivapd  11762