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Theorem dvdsbnd 11656
Description: There is an upper bound to the divisors of a nonzero integer. (Contributed by Jim Kingdon, 11-Dec-2021.)
Assertion
Ref Expression
dvdsbnd  |-  ( ( A  e.  ZZ  /\  A  =/=  0 )  ->  E. n  e.  NN  A. m  e.  ( ZZ>= `  n )  -.  m  ||  A )
Distinct variable group:    A, m, n

Proof of Theorem dvdsbnd
StepHypRef Expression
1 simpl 108 . . . . 5  |-  ( ( A  e.  ZZ  /\  A  =/=  0 )  ->  A  e.  ZZ )
21zcnd 9186 . . . 4  |-  ( ( A  e.  ZZ  /\  A  =/=  0 )  ->  A  e.  CC )
32abscld 10965 . . 3  |-  ( ( A  e.  ZZ  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  RR )
4 arch 8986 . . 3  |-  ( ( abs `  A )  e.  RR  ->  E. n  e.  NN  ( abs `  A
)  <  n )
53, 4syl 14 . 2  |-  ( ( A  e.  ZZ  /\  A  =/=  0 )  ->  E. n  e.  NN  ( abs `  A )  <  n )
63ad3antrrr 483 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  n  e.  NN )  /\  ( abs `  A )  < 
n )  /\  m  e.  ( ZZ>= `  n )
)  ->  ( abs `  A )  e.  RR )
7 simpllr 523 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  n  e.  NN )  /\  ( abs `  A )  < 
n )  /\  m  e.  ( ZZ>= `  n )
)  ->  n  e.  NN )
87nnred 8745 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  n  e.  NN )  /\  ( abs `  A )  < 
n )  /\  m  e.  ( ZZ>= `  n )
)  ->  n  e.  RR )
9 eluzelz 9347 . . . . . . . . . 10  |-  ( m  e.  ( ZZ>= `  n
)  ->  m  e.  ZZ )
109adantl 275 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  n  e.  NN )  /\  ( abs `  A )  < 
n )  /\  m  e.  ( ZZ>= `  n )
)  ->  m  e.  ZZ )
1110zred 9185 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  n  e.  NN )  /\  ( abs `  A )  < 
n )  /\  m  e.  ( ZZ>= `  n )
)  ->  m  e.  RR )
12 simplr 519 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  n  e.  NN )  /\  ( abs `  A )  < 
n )  /\  m  e.  ( ZZ>= `  n )
)  ->  ( abs `  A )  <  n
)
13 eluzle 9350 . . . . . . . . 9  |-  ( m  e.  ( ZZ>= `  n
)  ->  n  <_  m )
1413adantl 275 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  n  e.  NN )  /\  ( abs `  A )  < 
n )  /\  m  e.  ( ZZ>= `  n )
)  ->  n  <_  m )
156, 8, 11, 12, 14ltletrd 8197 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  n  e.  NN )  /\  ( abs `  A )  < 
n )  /\  m  e.  ( ZZ>= `  n )
)  ->  ( abs `  A )  <  m
)
16 zabscl 10870 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  ( abs `  A )  e.  ZZ )
1716ad4antr 485 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  n  e.  NN )  /\  ( abs `  A )  < 
n )  /\  m  e.  ( ZZ>= `  n )
)  ->  ( abs `  A )  e.  ZZ )
18 zltnle 9112 . . . . . . . 8  |-  ( ( ( abs `  A
)  e.  ZZ  /\  m  e.  ZZ )  ->  ( ( abs `  A
)  <  m  <->  -.  m  <_  ( abs `  A
) ) )
1917, 10, 18syl2anc 408 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  n  e.  NN )  /\  ( abs `  A )  < 
n )  /\  m  e.  ( ZZ>= `  n )
)  ->  ( ( abs `  A )  < 
m  <->  -.  m  <_  ( abs `  A ) ) )
2015, 19mpbid 146 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  n  e.  NN )  /\  ( abs `  A )  < 
n )  /\  m  e.  ( ZZ>= `  n )
)  ->  -.  m  <_  ( abs `  A
) )
211ad3antrrr 483 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  n  e.  NN )  /\  ( abs `  A )  < 
n )  /\  m  e.  ( ZZ>= `  n )
)  ->  A  e.  ZZ )
22 simplr 519 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  n  e.  NN )  ->  A  =/=  0
)
2322ad2antrr 479 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  n  e.  NN )  /\  ( abs `  A )  < 
n )  /\  m  e.  ( ZZ>= `  n )
)  ->  A  =/=  0 )
24 dvdsleabs 11554 . . . . . . . 8  |-  ( ( m  e.  ZZ  /\  A  e.  ZZ  /\  A  =/=  0 )  ->  (
m  ||  A  ->  m  <_  ( abs `  A
) ) )
2524con3d 620 . . . . . . 7  |-  ( ( m  e.  ZZ  /\  A  e.  ZZ  /\  A  =/=  0 )  ->  ( -.  m  <_  ( abs `  A )  ->  -.  m  ||  A ) )
2610, 21, 23, 25syl3anc 1216 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  n  e.  NN )  /\  ( abs `  A )  < 
n )  /\  m  e.  ( ZZ>= `  n )
)  ->  ( -.  m  <_  ( abs `  A
)  ->  -.  m  ||  A ) )
2720, 26mpd 13 . . . . 5  |-  ( ( ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  n  e.  NN )  /\  ( abs `  A )  < 
n )  /\  m  e.  ( ZZ>= `  n )
)  ->  -.  m  ||  A )
2827ralrimiva 2505 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  n  e.  NN )  /\  ( abs `  A )  < 
n )  ->  A. m  e.  ( ZZ>= `  n )  -.  m  ||  A )
2928ex 114 . . 3  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  n  e.  NN )  ->  ( ( abs `  A )  <  n  ->  A. m  e.  (
ZZ>= `  n )  -.  m  ||  A ) )
3029reximdva 2534 . 2  |-  ( ( A  e.  ZZ  /\  A  =/=  0 )  -> 
( E. n  e.  NN  ( abs `  A
)  <  n  ->  E. n  e.  NN  A. m  e.  ( ZZ>= `  n )  -.  m  ||  A ) )
315, 30mpd 13 1  |-  ( ( A  e.  ZZ  /\  A  =/=  0 )  ->  E. n  e.  NN  A. m  e.  ( ZZ>= `  n )  -.  m  ||  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    e. wcel 1480    =/= wne 2308   A.wral 2416   E.wrex 2417   class class class wbr 3929   ` cfv 5123   RRcr 7631   0cc0 7632    < clt 7812    <_ cle 7813   NNcn 8732   ZZcz 9066   ZZ>=cuz 9338   abscabs 10781    || cdvds 11504
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-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-seqfrec 10231  df-exp 10305  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783  df-dvds 11505
This theorem is referenced by:  gcdsupex  11657  gcdsupcl  11658
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