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Theorem divnumden 13140
Description: Calculate the reduced form of a quotient using  gcd. (Contributed by Stefan O'Rear, 13-Sep-2014.)
Assertion
Ref Expression
divnumden  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( (numer `  ( A  /  B ) )  =  ( A  / 
( A  gcd  B
) )  /\  (denom `  ( A  /  B
) )  =  ( B  /  ( A  gcd  B ) ) ) )

Proof of Theorem divnumden
StepHypRef Expression
1 simpl 444 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  A  e.  ZZ )
2 nnz 10303 . . . . 5  |-  ( B  e.  NN  ->  B  e.  ZZ )
32adantl 453 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  e.  ZZ )
4 nnne0 10032 . . . . . . . 8  |-  ( B  e.  NN  ->  B  =/=  0 )
54neneqd 2617 . . . . . . 7  |-  ( B  e.  NN  ->  -.  B  =  0 )
65adantl 453 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  -.  B  =  0 )
76intnand 883 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  -.  ( A  =  0  /\  B  =  0 ) )
8 gcdn0cl 13014 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
91, 3, 7, 8syl21anc 1183 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  NN )
10 gcddvds 13015 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
112, 10sylan2 461 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
12 gcddiv 13049 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( A  gcd  B )  e.  NN )  /\  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )  ->  ( ( A  gcd  B )  / 
( A  gcd  B
) )  =  ( ( A  /  ( A  gcd  B ) )  gcd  ( B  / 
( A  gcd  B
) ) ) )
131, 3, 9, 11, 12syl31anc 1187 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  gcd  B )  /  ( A  gcd  B ) )  =  ( ( A  /  ( A  gcd  B ) )  gcd  ( B  /  ( A  gcd  B ) ) ) )
149nncnd 10016 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  CC )
159nnne0d 10044 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  =/=  0 )
1614, 15dividd 9788 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  gcd  B )  /  ( A  gcd  B ) )  =  1 )
1713, 16eqtr3d 2470 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  / 
( A  gcd  B
) )  gcd  ( B  /  ( A  gcd  B ) ) )  =  1 )
18 zcn 10287 . . . 4  |-  ( A  e.  ZZ  ->  A  e.  CC )
1918adantr 452 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  A  e.  CC )
20 nncn 10008 . . . 4  |-  ( B  e.  NN  ->  B  e.  CC )
2120adantl 453 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  e.  CC )
224adantl 453 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  =/=  0 )
23 divcan7 9723 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( ( A  gcd  B )  e.  CC  /\  ( A  gcd  B )  =/=  0 ) )  -> 
( ( A  / 
( A  gcd  B
) )  /  ( B  /  ( A  gcd  B ) ) )  =  ( A  /  B
) )
2423eqcomd 2441 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( ( A  gcd  B )  e.  CC  /\  ( A  gcd  B )  =/=  0 ) )  -> 
( A  /  B
)  =  ( ( A  /  ( A  gcd  B ) )  /  ( B  / 
( A  gcd  B
) ) ) )
2519, 21, 22, 14, 15, 24syl122anc 1193 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  B
)  =  ( ( A  /  ( A  gcd  B ) )  /  ( B  / 
( A  gcd  B
) ) ) )
26 znq 10578 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  B
)  e.  QQ )
2711simpld 446 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  ||  A )
28 gcdcl 13017 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  NN0 )
2928nn0zd 10373 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  ZZ )
302, 29sylan2 461 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  ZZ )
31 dvdsval2 12855 . . . . 5  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  ( A  gcd  B )  =/=  0  /\  A  e.  ZZ )  ->  (
( A  gcd  B
)  ||  A  <->  ( A  /  ( A  gcd  B ) )  e.  ZZ ) )
3230, 15, 1, 31syl3anc 1184 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  A  <->  ( A  /  ( A  gcd  B ) )  e.  ZZ ) )
3327, 32mpbid 202 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  ( A  gcd  B ) )  e.  ZZ )
3411simprd 450 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  ||  B )
35 simpr 448 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  e.  NN )
36 nndivdvds 12858 . . . . 5  |-  ( ( B  e.  NN  /\  ( A  gcd  B )  e.  NN )  -> 
( ( A  gcd  B )  ||  B  <->  ( B  /  ( A  gcd  B ) )  e.  NN ) )
3735, 9, 36syl2anc 643 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  B  <->  ( B  /  ( A  gcd  B ) )  e.  NN ) )
3834, 37mpbid 202 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( B  /  ( A  gcd  B ) )  e.  NN )
39 qnumdenbi 13136 . . 3  |-  ( ( ( A  /  B
)  e.  QQ  /\  ( A  /  ( A  gcd  B ) )  e.  ZZ  /\  ( B  /  ( A  gcd  B ) )  e.  NN )  ->  ( ( ( ( A  /  ( A  gcd  B ) )  gcd  ( B  / 
( A  gcd  B
) ) )  =  1  /\  ( A  /  B )  =  ( ( A  / 
( A  gcd  B
) )  /  ( B  /  ( A  gcd  B ) ) ) )  <-> 
( (numer `  ( A  /  B ) )  =  ( A  / 
( A  gcd  B
) )  /\  (denom `  ( A  /  B
) )  =  ( B  /  ( A  gcd  B ) ) ) ) )
4026, 33, 38, 39syl3anc 1184 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( ( ( A  /  ( A  gcd  B ) )  gcd  ( B  / 
( A  gcd  B
) ) )  =  1  /\  ( A  /  B )  =  ( ( A  / 
( A  gcd  B
) )  /  ( B  /  ( A  gcd  B ) ) ) )  <-> 
( (numer `  ( A  /  B ) )  =  ( A  / 
( A  gcd  B
) )  /\  (denom `  ( A  /  B
) )  =  ( B  /  ( A  gcd  B ) ) ) ) )
4117, 25, 40mpbi2and 888 1  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( (numer `  ( A  /  B ) )  =  ( A  / 
( A  gcd  B
) )  /\  (denom `  ( A  /  B
) )  =  ( B  /  ( A  gcd  B ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    / cdiv 9677   NNcn 10000   ZZcz 10282   QQcq 10574    || cdivides 12852    gcd cgcd 13006  numercnumer 13125  denomcdenom 13126
This theorem is referenced by:  divdenle  13141  divnumden2  24161  qqhval2lem  24365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007  df-numer 13127  df-denom 13128
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