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Theorem rediv 11928
Description: Real part of a division. Related to remul2 11927. (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
StepHypRef Expression
1 ancom 438 . . . . 5  |-  ( ( ( B  e.  RR  /\  B  =/=  0 )  /\  A  e.  CC ) 
<->  ( A  e.  CC  /\  ( B  e.  RR  /\  B  =/=  0 ) ) )
2 3anass 940 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  <->  ( A  e.  CC  /\  ( B  e.  RR  /\  B  =/=  0 ) ) )
31, 2bitr4i 244 . . . 4  |-  ( ( ( B  e.  RR  /\  B  =/=  0 )  /\  A  e.  CC ) 
<->  ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 ) )
4 rereccl 9724 . . . . 5  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( 1  /  B
)  e.  RR )
54anim1i 552 . . . 4  |-  ( ( ( B  e.  RR  /\  B  =/=  0 )  /\  A  e.  CC )  ->  ( ( 1  /  B )  e.  RR  /\  A  e.  CC ) )
63, 5sylbir 205 . . 3  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
( 1  /  B
)  e.  RR  /\  A  e.  CC )
)
7 remul2 11927 . . 3  |-  ( ( ( 1  /  B
)  e.  RR  /\  A  e.  CC )  ->  ( Re `  (
( 1  /  B
)  x.  A ) )  =  ( ( 1  /  B )  x.  ( Re `  A ) ) )
86, 7syl 16 . 2  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
Re `  ( (
1  /  B )  x.  A ) )  =  ( ( 1  /  B )  x.  ( Re `  A
) ) )
9 recn 9072 . . 3  |-  ( B  e.  RR  ->  B  e.  CC )
10 divrec2 9687 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  =  ( ( 1  /  B )  x.  A
) )
1110fveq2d 5724 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
Re `  ( A  /  B ) )  =  ( Re `  (
( 1  /  B
)  x.  A ) ) )
129, 11syl3an2 1218 . 2  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
Re `  ( A  /  B ) )  =  ( Re `  (
( 1  /  B
)  x.  A ) ) )
13 recl 11907 . . . . 5  |-  ( A  e.  CC  ->  (
Re `  A )  e.  RR )
1413recnd 9106 . . . 4  |-  ( A  e.  CC  ->  (
Re `  A )  e.  CC )
15143ad2ant1 978 . . 3  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
Re `  A )  e.  CC )
1693ad2ant2 979 . . 3  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  B  e.  CC )
17 simp3 959 . . 3  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  B  =/=  0 )
1815, 16, 17divrec2d 9786 . 2  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
( Re `  A
)  /  B )  =  ( ( 1  /  B )  x.  ( Re `  A
) ) )
198, 12, 183eqtr4d 2477 1  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
Re `  ( A  /  B ) )  =  ( ( Re `  A )  /  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    x. cmul 8987    / cdiv 9669   Recre 11894
This theorem is referenced by:  redivd  12026
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-2 10050  df-cj 11896  df-re 11897  df-im 11898
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