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Theorem redivap 10803
Description: Real part of a division. Related to remul2 10802. (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
StepHypRef Expression
1 ancom 264 . . . . 5  |-  ( ( ( B  e.  RR  /\  B #  0 )  /\  A  e.  CC )  <->  ( A  e.  CC  /\  ( B  e.  RR  /\  B #  0 ) ) )
2 3anass 971 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B #  0 )  <->  ( A  e.  CC  /\  ( B  e.  RR  /\  B #  0 ) ) )
31, 2bitr4i 186 . . . 4  |-  ( ( ( B  e.  RR  /\  B #  0 )  /\  A  e.  CC )  <->  ( A  e.  CC  /\  B  e.  RR  /\  B #  0 ) )
4 rerecclap 8618 . . . . 5  |-  ( ( B  e.  RR  /\  B #  0 )  ->  (
1  /  B )  e.  RR )
54anim1i 338 . . . 4  |-  ( ( ( B  e.  RR  /\  B #  0 )  /\  A  e.  CC )  ->  ( ( 1  /  B )  e.  RR  /\  A  e.  CC ) )
63, 5sylbir 134 . . 3  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B #  0 )  ->  (
( 1  /  B
)  e.  RR  /\  A  e.  CC )
)
7 remul2 10802 . . 3  |-  ( ( ( 1  /  B
)  e.  RR  /\  A  e.  CC )  ->  ( Re `  (
( 1  /  B
)  x.  A ) )  =  ( ( 1  /  B )  x.  ( Re `  A ) ) )
86, 7syl 14 . 2  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B #  0 )  ->  (
Re `  ( (
1  /  B )  x.  A ) )  =  ( ( 1  /  B )  x.  ( Re `  A
) ) )
9 recn 7878 . . 3  |-  ( B  e.  RR  ->  B  e.  CC )
10 divrecap2 8577 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  /  B )  =  ( ( 1  /  B )  x.  A
) )
1110fveq2d 5485 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  (
Re `  ( A  /  B ) )  =  ( Re `  (
( 1  /  B
)  x.  A ) ) )
129, 11syl3an2 1261 . 2  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B #  0 )  ->  (
Re `  ( A  /  B ) )  =  ( Re `  (
( 1  /  B
)  x.  A ) ) )
13 recl 10782 . . . . 5  |-  ( A  e.  CC  ->  (
Re `  A )  e.  RR )
1413recnd 7919 . . . 4  |-  ( A  e.  CC  ->  (
Re `  A )  e.  CC )
15143ad2ant1 1007 . . 3  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B #  0 )  ->  (
Re `  A )  e.  CC )
1693ad2ant2 1008 . . 3  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B #  0 )  ->  B  e.  CC )
17 simp3 988 . . 3  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B #  0 )  ->  B #  0 )
1815, 16, 17divrecap2d 8682 . 2  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B #  0 )  ->  (
( Re `  A
)  /  B )  =  ( ( 1  /  B )  x.  ( Re `  A
) ) )
198, 12, 183eqtr4d 2207 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 103    /\ w3a 967    = wceq 1342    e. wcel 2135   class class class wbr 3977   ` cfv 5183  (class class class)co 5837   CCcc 7743   RRcr 7744   0cc0 7745   1c1 7746    x. cmul 7750   # cap 8471    / cdiv 8560   Recre 10769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182  ax-un 4406  ax-setind 4509  ax-cnex 7836  ax-resscn 7837  ax-1cn 7838  ax-1re 7839  ax-icn 7840  ax-addcl 7841  ax-addrcl 7842  ax-mulcl 7843  ax-mulrcl 7844  ax-addcom 7845  ax-mulcom 7846  ax-addass 7847  ax-mulass 7848  ax-distr 7849  ax-i2m1 7850  ax-0lt1 7851  ax-1rid 7852  ax-0id 7853  ax-rnegex 7854  ax-precex 7855  ax-cnre 7856  ax-pre-ltirr 7857  ax-pre-ltwlin 7858  ax-pre-lttrn 7859  ax-pre-apti 7860  ax-pre-ltadd 7861  ax-pre-mulgt0 7862  ax-pre-mulext 7863
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2724  df-sbc 2948  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-br 3978  df-opab 4039  df-mpt 4040  df-id 4266  df-po 4269  df-iso 4270  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-rn 4610  df-res 4611  df-ima 4612  df-iota 5148  df-fun 5185  df-fn 5186  df-f 5187  df-fv 5191  df-riota 5793  df-ov 5840  df-oprab 5841  df-mpo 5842  df-pnf 7927  df-mnf 7928  df-xr 7929  df-ltxr 7930  df-le 7931  df-sub 8063  df-neg 8064  df-reap 8465  df-ap 8472  df-div 8561  df-2 8908  df-cj 10771  df-re 10772  df-im 10773
This theorem is referenced by:  redivapd  10903
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