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Theorem rediv 11612
Description: Real part of a division. Related to remul2 11611. (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 437 . . . . 5  |-  ( ( ( B  e.  RR  /\  B  =/=  0 )  /\  A  e.  CC ) 
<->  ( A  e.  CC  /\  ( B  e.  RR  /\  B  =/=  0 ) ) )
2 3anass 938 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  <->  ( A  e.  CC  /\  ( B  e.  RR  /\  B  =/=  0 ) ) )
31, 2bitr4i 243 . . . 4  |-  ( ( ( B  e.  RR  /\  B  =/=  0 )  /\  A  e.  CC ) 
<->  ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 ) )
4 rereccl 9474 . . . . 5  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( 1  /  B
)  e.  RR )
54anim1i 551 . . . 4  |-  ( ( ( B  e.  RR  /\  B  =/=  0 )  /\  A  e.  CC )  ->  ( ( 1  /  B )  e.  RR  /\  A  e.  CC ) )
63, 5sylbir 204 . . 3  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
( 1  /  B
)  e.  RR  /\  A  e.  CC )
)
7 remul2 11611 . . 3  |-  ( ( ( 1  /  B
)  e.  RR  /\  A  e.  CC )  ->  ( Re `  (
( 1  /  B
)  x.  A ) )  =  ( ( 1  /  B )  x.  ( Re `  A ) ) )
86, 7syl 15 . 2  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
Re `  ( (
1  /  B )  x.  A ) )  =  ( ( 1  /  B )  x.  ( Re `  A
) ) )
9 recn 8823 . . 3  |-  ( B  e.  RR  ->  B  e.  CC )
10 divrec2 9437 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  =  ( ( 1  /  B )  x.  A
) )
1110fveq2d 5490 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
Re `  ( A  /  B ) )  =  ( Re `  (
( 1  /  B
)  x.  A ) ) )
129, 11syl3an2 1216 . 2  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
Re `  ( A  /  B ) )  =  ( Re `  (
( 1  /  B
)  x.  A ) ) )
13 recl 11591 . . . . 5  |-  ( A  e.  CC  ->  (
Re `  A )  e.  RR )
1413recnd 8857 . . . 4  |-  ( A  e.  CC  ->  (
Re `  A )  e.  CC )
15143ad2ant1 976 . . 3  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
Re `  A )  e.  CC )
1693ad2ant2 977 . . 3  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  B  e.  CC )
17 simp3 957 . . 3  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  B  =/=  0 )
1815, 16, 17divrec2d 9536 . 2  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
( Re `  A
)  /  B )  =  ( ( 1  /  B )  x.  ( Re `  A
) ) )
198, 12, 183eqtr4d 2326 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 358    /\ w3a 934    = wceq 1623    e. wcel 1685    =/= wne 2447   ` cfv 5221  (class class class)co 5820   CCcc 8731   RRcr 8732   0cc0 8733   1c1 8734    x. cmul 8738    / cdiv 9419   Recre 11578
This theorem is referenced by:  redivd  11710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-iota 6253  df-riota 6300  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-2 9800  df-cj 11580  df-re 11581  df-im 11582
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