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Theorem imdiv 11719
Description: Imaginary part of a division. Related to immul2 11718. (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 9568 . . . . 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 11718 . . 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 8917 . . 3  |-  ( B  e.  RR  ->  B  e.  CC )
10 divrec2 9531 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  =  ( ( 1  /  B )  x.  A
) )
1110fveq2d 5612 . . 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 11692 . . . . 5  |-  ( A  e.  CC  ->  (
Im `  A )  e.  RR )
1413recnd 8951 . . . 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 9630 . 2  |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 )  ->  (
( Im `  A
)  /  B )  =  ( ( 1  /  B )  x.  ( Im `  A
) ) )
198, 12, 183eqtr4d 2400 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 1642    e. wcel 1710    =/= wne 2521   ` cfv 5337  (class class class)co 5945   CCcc 8825   RRcr 8826   0cc0 8827   1c1 8828    x. cmul 8832    / cdiv 9513   Imcim 11679
This theorem is referenced by:  imdivd  11811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-po 4396  df-so 4397  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-riota 6391  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-2 9894  df-cj 11680  df-re 11681  df-im 11682
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