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Theorem divnumden 14293
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 457 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  A  e.  ZZ )
2 nnz 10907 . . . . 5  |-  ( B  e.  NN  ->  B  e.  ZZ )
32adantl 466 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  e.  ZZ )
4 nnne0 10589 . . . . . . . 8  |-  ( B  e.  NN  ->  B  =/=  0 )
54neneqd 2659 . . . . . . 7  |-  ( B  e.  NN  ->  -.  B  =  0 )
65adantl 466 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  -.  B  =  0 )
76intnand 916 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  -.  ( A  =  0  /\  B  =  0 ) )
8 gcdn0cl 14164 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
91, 3, 7, 8syl21anc 1227 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  NN )
10 gcddvds 14165 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
112, 10sylan2 474 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
12 gcddiv 14199 . . . 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 1231 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  gcd  B )  /  ( A  gcd  B ) )  =  ( ( A  /  ( A  gcd  B ) )  gcd  ( B  /  ( A  gcd  B ) ) ) )
149nncnd 10572 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  CC )
159nnne0d 10601 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  =/=  0 )
1614, 15dividd 10339 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  gcd  B )  /  ( A  gcd  B ) )  =  1 )
1713, 16eqtr3d 2500 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  / 
( A  gcd  B
) )  gcd  ( B  /  ( A  gcd  B ) ) )  =  1 )
18 zcn 10890 . . . 4  |-  ( A  e.  ZZ  ->  A  e.  CC )
1918adantr 465 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  A  e.  CC )
20 nncn 10564 . . . 4  |-  ( B  e.  NN  ->  B  e.  CC )
2120adantl 466 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  e.  CC )
224adantl 466 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  =/=  0 )
23 divcan7 10274 . . . 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 2465 . . 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 1237 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  B
)  =  ( ( A  /  ( A  gcd  B ) )  /  ( B  / 
( A  gcd  B
) ) ) )
26 znq 11211 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  B
)  e.  QQ )
2711simpld 459 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  ||  A )
28 gcdcl 14167 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  NN0 )
2928nn0zd 10988 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  ZZ )
302, 29sylan2 474 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  ZZ )
31 dvdsval2 14001 . . . . 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 1228 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  A  <->  ( A  /  ( A  gcd  B ) )  e.  ZZ ) )
3327, 32mpbid 210 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  ( A  gcd  B ) )  e.  ZZ )
3411simprd 463 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  ||  B )
35 simpr 461 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  e.  NN )
36 nndivdvds 14004 . . . . 5  |-  ( ( B  e.  NN  /\  ( A  gcd  B )  e.  NN )  -> 
( ( A  gcd  B )  ||  B  <->  ( B  /  ( A  gcd  B ) )  e.  NN ) )
3735, 9, 36syl2anc 661 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  B  <->  ( B  /  ( A  gcd  B ) )  e.  NN ) )
3834, 37mpbid 210 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( B  /  ( A  gcd  B ) )  e.  NN )
39 qnumdenbi 14289 . . 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 1228 . 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 921 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    / cdiv 10227   NNcn 10556   ZZcz 10885   QQcq 11207    || cdvds 13998    gcd cgcd 14156  numercnumer 14278  denomcdenom 14279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-dvds 13999  df-gcd 14157  df-numer 14280  df-denom 14281
This theorem is referenced by:  divdenle  14294  divnumden2  27769  qqhval2lem  28123
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