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Theorem rediv 13018
Description: Real part of a division. Related to remul2 13017. (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 448 . . . . 5  |-  ( ( ( B  e.  RR  /\  B  =/=  0 )  /\  A  e.  CC ) 
<->  ( A  e.  CC  /\  ( B  e.  RR  /\  B  =/=  0 ) ) )
2 3anass 976 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  <->  ( A  e.  CC  /\  ( B  e.  RR  /\  B  =/=  0 ) ) )
31, 2bitr4i 252 . . . 4  |-  ( ( ( B  e.  RR  /\  B  =/=  0 )  /\  A  e.  CC ) 
<->  ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 ) )
4 rereccl 10221 . . . . 5  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( 1  /  B
)  e.  RR )
54anim1i 566 . . . 4  |-  ( ( ( B  e.  RR  /\  B  =/=  0 )  /\  A  e.  CC )  ->  ( ( 1  /  B )  e.  RR  /\  A  e.  CC ) )
63, 5sylbir 213 . . 3  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
( 1  /  B
)  e.  RR  /\  A  e.  CC )
)
7 remul2 13017 . . 3  |-  ( ( ( 1  /  B
)  e.  RR  /\  A  e.  CC )  ->  ( Re `  (
( 1  /  B
)  x.  A ) )  =  ( ( 1  /  B )  x.  ( Re `  A ) ) )
86, 7syl 17 . 2  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
Re `  ( (
1  /  B )  x.  A ) )  =  ( ( 1  /  B )  x.  ( Re `  A
) ) )
9 recn 9530 . . 3  |-  ( B  e.  RR  ->  B  e.  CC )
10 divrec2 10183 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  =  ( ( 1  /  B )  x.  A
) )
1110fveq2d 5807 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
Re `  ( A  /  B ) )  =  ( Re `  (
( 1  /  B
)  x.  A ) ) )
129, 11syl3an2 1262 . 2  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
Re `  ( A  /  B ) )  =  ( Re `  (
( 1  /  B
)  x.  A ) ) )
13 recl 12997 . . . . 5  |-  ( A  e.  CC  ->  (
Re `  A )  e.  RR )
1413recnd 9570 . . . 4  |-  ( A  e.  CC  ->  (
Re `  A )  e.  CC )
15143ad2ant1 1016 . . 3  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
Re `  A )  e.  CC )
1693ad2ant2 1017 . . 3  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  B  e.  CC )
17 simp3 997 . . 3  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  B  =/=  0 )
1815, 16, 17divrec2d 10283 . 2  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
( Re `  A
)  /  B )  =  ( ( 1  /  B )  x.  ( Re `  A
) ) )
198, 12, 183eqtr4d 2451 1  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
Re `  ( A  /  B ) )  =  ( ( Re `  A )  /  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840    =/= wne 2596   ` cfv 5523  (class class class)co 6232   CCcc 9438   RRcr 9439   0cc0 9440   1c1 9441    x. cmul 9445    / cdiv 10165   Recre 12984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-po 4741  df-so 4742  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-2 10553  df-cj 12986  df-re 12987  df-im 12988
This theorem is referenced by:  redivd  13116
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