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Theorem cjdiv 11971
Description: Complex conjugate distributes over division. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
Assertion
Ref Expression
cjdiv  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
* `  ( A  /  B ) )  =  ( ( * `  A )  /  (
* `  B )
) )

Proof of Theorem cjdiv
StepHypRef Expression
1 divcl 9686 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  e.  CC )
2 cjcl 11912 . . . 4  |-  ( ( A  /  B )  e.  CC  ->  (
* `  ( A  /  B ) )  e.  CC )
31, 2syl 16 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
* `  ( A  /  B ) )  e.  CC )
4 simp2 959 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  B  e.  CC )
5 cjcl 11912 . . . 4  |-  ( B  e.  CC  ->  (
* `  B )  e.  CC )
64, 5syl 16 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
* `  B )  e.  CC )
7 simp3 960 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  B  =/=  0 )
8 cjne0 11970 . . . . 5  |-  ( B  e.  CC  ->  ( B  =/=  0  <->  ( * `  B )  =/=  0
) )
94, 8syl 16 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  =/=  0  <->  ( * `  B )  =/=  0
) )
107, 9mpbid 203 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
* `  B )  =/=  0 )
113, 6, 10divcan4d 9798 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( ( * `  ( A  /  B
) )  x.  (
* `  B )
)  /  ( * `
 B ) )  =  ( * `  ( A  /  B
) ) )
12 cjmul 11949 . . . . 5  |-  ( ( ( A  /  B
)  e.  CC  /\  B  e.  CC )  ->  ( * `  (
( A  /  B
)  x.  B ) )  =  ( ( * `  ( A  /  B ) )  x.  ( * `  B ) ) )
131, 4, 12syl2anc 644 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
* `  ( ( A  /  B )  x.  B ) )  =  ( ( * `  ( A  /  B
) )  x.  (
* `  B )
) )
14 divcan1 9689 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  /  B
)  x.  B )  =  A )
1514fveq2d 5734 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
* `  ( ( A  /  B )  x.  B ) )  =  ( * `  A
) )
1613, 15eqtr3d 2472 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( * `  ( A  /  B ) )  x.  ( * `  B ) )  =  ( * `  A
) )
1716oveq1d 6098 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( ( * `  ( A  /  B
) )  x.  (
* `  B )
)  /  ( * `
 B ) )  =  ( ( * `
 A )  / 
( * `  B
) ) )
1811, 17eqtr3d 2472 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
* `  ( A  /  B ) )  =  ( ( * `  A )  /  (
* `  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992    x. cmul 8997    / cdiv 9679   *ccj 11903
This theorem is referenced by:  cjdivi  11998  cjdivd  12030  dipcj  22215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-2 10060  df-cj 11906  df-re 11907  df-im 11908
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