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Theorem cjdiv 12775
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 10115 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  e.  CC )
2 cjcl 12716 . . . 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 989 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  B  e.  CC )
5 cjcl 12716 . . . 4  |-  ( B  e.  CC  ->  (
* `  B )  e.  CC )
64, 5syl 16 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
* `  B )  e.  CC )
7 simp3 990 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  B  =/=  0 )
8 cjne0 12774 . . . . 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 210 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
* `  B )  =/=  0 )
113, 6, 10divcan4d 10228 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( ( * `  ( A  /  B
) )  x.  (
* `  B )
)  /  ( * `
 B ) )  =  ( * `  ( A  /  B
) ) )
12 cjmul 12753 . . . . 5  |-  ( ( ( A  /  B
)  e.  CC  /\  B  e.  CC )  ->  ( * `  (
( A  /  B
)  x.  B ) )  =  ( ( * `  ( A  /  B ) )  x.  ( * `  B ) ) )
131, 4, 12syl2anc 661 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
* `  ( ( A  /  B )  x.  B ) )  =  ( ( * `  ( A  /  B
) )  x.  (
* `  B )
) )
14 divcan1 10118 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  /  B
)  x.  B )  =  A )
1514fveq2d 5806 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
* `  ( ( A  /  B )  x.  B ) )  =  ( * `  A
) )
1613, 15eqtr3d 2497 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( * `  ( A  /  B ) )  x.  ( * `  B ) )  =  ( * `  A
) )
1716oveq1d 6218 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( ( * `  ( A  /  B
) )  x.  (
* `  B )
)  /  ( * `
 B ) )  =  ( ( * `
 A )  / 
( * `  B
) ) )
1811, 17eqtr3d 2497 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
* `  ( A  /  B ) )  =  ( ( * `  A )  /  (
* `  B )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   ` cfv 5529  (class class class)co 6203   CCcc 9395   0cc0 9397    x. cmul 9402    / cdiv 10108   *ccj 12707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-2 10495  df-cj 12710  df-re 12711  df-im 12712
This theorem is referenced by:  cjdivi  12802  cjdivd  12834  dipcj  24291
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