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Theorem dvdsbnd 11645
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 9174 . . . 4 |- ( ( A e. ZZ /\ A =/= 0 ) -> A e. CC )
32abscld 10953 . . 3 |- ( ( A e. ZZ /\ A =/= 0 ) -> ( abs ` A ) e. RR )
4 arch 8974 . . 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 8733 . . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> n e. RR )
9 eluzelz 9335 . . . . . . . . . 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 9173 . . . . . . . 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 9338 . . . . . . . . 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 8185 . . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> ( abs ` A ) < m )
16 zabscl 10858 . . . . . . . . 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 9100 . . . . . . . 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 11543 . . . . . . . 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 7619   0cc0 7620   < clt 7800   <_ cle 7801   NNcn 8720   ZZcz 9054   ZZ>=cuz 9326   abscabs 10769   || cdvds 11493
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 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740
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 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-dvds 11494
This theorem is referenced by:  gcdsupex  11646  gcdsupcl  11647
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