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Theorem divnumden 11885
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 108 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  A  e.  ZZ )
2 nnz 9085 . . . . 5  |-  ( B  e.  NN  ->  B  e.  ZZ )
32adantl 275 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  e.  ZZ )
4 nnne0 8760 . . . . . . . 8  |-  ( B  e.  NN  ->  B  =/=  0 )
54neneqd 2329 . . . . . . 7  |-  ( B  e.  NN  ->  -.  B  =  0 )
65adantl 275 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  -.  B  =  0 )
76intnand 916 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  -.  ( A  =  0  /\  B  =  0 ) )
8 gcdn0cl 11662 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
91, 3, 7, 8syl21anc 1215 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  NN )
10 gcddvds 11663 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
112, 10sylan2 284 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
12 gcddiv 11718 . . . 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 1219 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  gcd  B )  /  ( A  gcd  B ) )  =  ( ( A  /  ( A  gcd  B ) )  gcd  ( B  /  ( A  gcd  B ) ) ) )
149nncnd 8746 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  CC )
159nnap0d 8778 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
) #  0 )
1614, 15dividapd 8558 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  gcd  B )  /  ( A  gcd  B ) )  =  1 )
1713, 16eqtr3d 2174 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  / 
( A  gcd  B
) )  gcd  ( B  /  ( A  gcd  B ) ) )  =  1 )
18 zcn 9071 . . . 4  |-  ( A  e.  ZZ  ->  A  e.  CC )
1918adantr 274 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  A  e.  CC )
20 nncn 8740 . . . 4  |-  ( B  e.  NN  ->  B  e.  CC )
2120adantl 275 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  e.  CC )
22 simpr 109 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  e.  NN )
2322nnap0d 8778 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B #  0 )
24 divcanap7 8493 . . . 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
) )
2524eqcomd 2145 . . 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
) ) ) )
2619, 21, 23, 14, 15, 25syl122anc 1225 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  B
)  =  ( ( A  /  ( A  gcd  B ) )  /  ( B  / 
( A  gcd  B
) ) ) )
27 znq 9428 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  B
)  e.  QQ )
2811simpld 111 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  ||  A )
29 gcdcl 11666 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  NN0 )
3029nn0zd 9183 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  ZZ )
312, 30sylan2 284 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  ZZ )
329nnne0d 8777 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  =/=  0 )
33 dvdsval2 11507 . . . . 5  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  ( A  gcd  B )  =/=  0  /\  A  e.  ZZ )  ->  (
( A  gcd  B
)  ||  A  <->  ( A  /  ( A  gcd  B ) )  e.  ZZ ) )
3431, 32, 1, 33syl3anc 1216 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  A  <->  ( A  /  ( A  gcd  B ) )  e.  ZZ ) )
3528, 34mpbid 146 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  ( A  gcd  B ) )  e.  ZZ )
3611simprd 113 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  ||  B )
37 nndivdvds 11510 . . . . 5  |-  ( ( B  e.  NN  /\  ( A  gcd  B )  e.  NN )  -> 
( ( A  gcd  B )  ||  B  <->  ( B  /  ( A  gcd  B ) )  e.  NN ) )
3822, 9, 37syl2anc 408 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  B  <->  ( B  /  ( A  gcd  B ) )  e.  NN ) )
3936, 38mpbid 146 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( B  /  ( A  gcd  B ) )  e.  NN )
40 qnumdenbi 11881 . . 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 ) ) ) ) )
4127, 35, 39, 40syl3anc 1216 . 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 ) ) ) ) )
4217, 26, 41mpbi2and 927 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    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480    =/= wne 2308   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7630   0cc0 7632   1c1 7633   # cap 8355    / cdiv 8444   NNcn 8732   ZZcz 9066   QQcq 9423    || cdvds 11504    gcd cgcd 11646  numercnumer 11870  denomcdenom 11871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751  ax-caucvg 7752
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-n0 8990  df-z 9067  df-uz 9339  df-q 9424  df-rp 9454  df-fz 9803  df-fzo 9932  df-fl 10055  df-mod 10108  df-seqfrec 10231  df-exp 10305  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783  df-dvds 11505  df-gcd 11647  df-numer 11872  df-denom 11873
This theorem is referenced by:  divdenle  11886
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