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Theorem divnumden 12835
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 443 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  A  e.  ZZ )
2 nnz 10061 . . . . 5  |-  ( B  e.  NN  ->  B  e.  ZZ )
32adantl 452 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  e.  ZZ )
4 nnne0 9794 . . . . . . . 8  |-  ( B  e.  NN  ->  B  =/=  0 )
54neneqd 2475 . . . . . . 7  |-  ( B  e.  NN  ->  -.  B  =  0 )
65adantl 452 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  -.  B  =  0 )
76intnand 882 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  -.  ( A  =  0  /\  B  =  0 ) )
8 gcdn0cl 12709 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
91, 3, 7, 8syl21anc 1181 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  NN )
10 gcddvds 12710 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
112, 10sylan2 460 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
12 gcddiv 12744 . . . 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 1185 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  gcd  B )  /  ( A  gcd  B ) )  =  ( ( A  /  ( A  gcd  B ) )  gcd  ( B  /  ( A  gcd  B ) ) ) )
149nncnd 9778 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  CC )
15 nnne0 9794 . . . . 5  |-  ( ( A  gcd  B )  e.  NN  ->  ( A  gcd  B )  =/=  0 )
169, 15syl 15 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  =/=  0 )
1714, 16dividd 9550 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  gcd  B )  /  ( A  gcd  B ) )  =  1 )
1813, 17eqtr3d 2330 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  / 
( A  gcd  B
) )  gcd  ( B  /  ( A  gcd  B ) ) )  =  1 )
19 zcn 10045 . . . 4  |-  ( A  e.  ZZ  ->  A  e.  CC )
2019adantr 451 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  A  e.  CC )
21 nncn 9770 . . . 4  |-  ( B  e.  NN  ->  B  e.  CC )
2221adantl 452 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  e.  CC )
234adantl 452 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  =/=  0 )
24 divcan7 9485 . . . 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 2301 . . 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
) ) ) )
2620, 22, 23, 14, 16, 25syl122anc 1191 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  B
)  =  ( ( A  /  ( A  gcd  B ) )  /  ( B  / 
( A  gcd  B
) ) ) )
27 znq 10336 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  B
)  e.  QQ )
2811simpld 445 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  ||  A )
29 gcdcl 12712 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  NN0 )
3029nn0zd 10131 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  ZZ )
312, 30sylan2 460 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  ZZ )
32 dvdsval2 12550 . . . . 5  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  ( A  gcd  B )  =/=  0  /\  A  e.  ZZ )  ->  (
( A  gcd  B
)  ||  A  <->  ( A  /  ( A  gcd  B ) )  e.  ZZ ) )
3331, 16, 1, 32syl3anc 1182 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  A  <->  ( A  /  ( A  gcd  B ) )  e.  ZZ ) )
3428, 33mpbid 201 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  ( A  gcd  B ) )  e.  ZZ )
3511simprd 449 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  gcd  B
)  ||  B )
36 simpr 447 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  B  e.  NN )
37 nndivdvds 12553 . . . . 5  |-  ( ( B  e.  NN  /\  ( A  gcd  B )  e.  NN )  -> 
( ( A  gcd  B )  ||  B  <->  ( B  /  ( A  gcd  B ) )  e.  NN ) )
3836, 9, 37syl2anc 642 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  B  <->  ( B  /  ( A  gcd  B ) )  e.  NN ) )
3935, 38mpbid 201 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( B  /  ( A  gcd  B ) )  e.  NN )
40 qnumdenbi 12831 . . 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, 34, 39, 40syl3anc 1182 . 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 ) ) ) ) )
4218, 26, 41mpbi2and 887 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    / cdiv 9439   NNcn 9762   ZZcz 10040   QQcq 10332    || cdivides 12547    gcd cgcd 12701  numercnumer 12820  denomcdenom 12821
This theorem is referenced by:  divdenle  12836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-numer 12822  df-denom 12823
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