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Theorem imdiv 11623
Description: Imaginary part of a division. Related to immul2 11622. (Contributed by Mario Carneiro, 20-Jun-2015.)
Assertion
Ref Expression
imdiv  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
Im `  ( A  /  B ) )  =  ( ( Im `  A )  /  B
) )

Proof of Theorem imdiv
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 9478 . . . . 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 immul2 11622 . . 3  |-  ( ( ( 1  /  B
)  e.  RR  /\  A  e.  CC )  ->  ( Im `  (
( 1  /  B
)  x.  A ) )  =  ( ( 1  /  B )  x.  ( Im `  A ) ) )
86, 7syl 15 . 2  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
Im `  ( (
1  /  B )  x.  A ) )  =  ( ( 1  /  B )  x.  ( Im `  A
) ) )
9 recn 8827 . . 3  |-  ( B  e.  RR  ->  B  e.  CC )
10 divrec2 9441 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  =  ( ( 1  /  B )  x.  A
) )
1110fveq2d 5529 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
Im `  ( A  /  B ) )  =  ( Im `  (
( 1  /  B
)  x.  A ) ) )
129, 11syl3an2 1216 . 2  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
Im `  ( A  /  B ) )  =  ( Im `  (
( 1  /  B
)  x.  A ) ) )
13 imcl 11596 . . . . 5  |-  ( A  e.  CC  ->  (
Im `  A )  e.  RR )
1413recnd 8861 . . . 4  |-  ( A  e.  CC  ->  (
Im `  A )  e.  CC )
15143ad2ant1 976 . . 3  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
Im `  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 9540 . 2  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
( Im `  A
)  /  B )  =  ( ( 1  /  B )  x.  ( Im `  A
) ) )
198, 12, 183eqtr4d 2325 1  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
Im `  ( A  /  B ) )  =  ( ( Im `  A )  /  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    / cdiv 9423   Imcim 11583
This theorem is referenced by:  imdivd  11715
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-cj 11584  df-re 11585  df-im 11586
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