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Theorem dvdsbnd 11440
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 9026 . . . 4 |- ( ( A e. ZZ /\ A =/= 0 ) -> A e. CC )
32abscld 10793 . . 3 |- ( ( A e. ZZ /\ A =/= 0 ) -> ( abs ` A ) e. RR )
4 arch 8826 . . 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 479 . . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> ( abs ` A ) e. RR )
7 simpllr 504 . . . . . . . . 9 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> n e. NN )
87nnred 8591 . . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> n e. RR )
9 eluzelz 9185 . . . . . . . . . 10 |- ( m e. ( ZZ>= ` n ) -> m e. ZZ )
109adantl 273 . . . . . . . . 9 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> m e. ZZ )
1110zred 9025 . . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> m e. RR )
12 simplr 500 . . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> ( abs ` A ) < n )
13 eluzle 9188 . . . . . . . . 9 |- ( m e. ( ZZ>= ` n ) -> n <_ m )
1413adantl 273 . . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> n <_ m )
156, 8, 11, 12, 14ltletrd 8052 . . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> ( abs ` A ) < m )
16 zabscl 10698 . . . . . . . . 9 |- ( A e. ZZ -> ( abs ` A ) e. ZZ )
1716ad4antr 481 . . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> ( abs ` A ) e. ZZ )
18 zltnle 8952 . . . . . . . 8 |- ( ( ( abs ` A ) e. ZZ /\ m e. ZZ ) -> ( ( abs ` A ) < m <-> -. m <_ ( abs ` A ) ) )
1917, 10, 18syl2anc 406 . . . . . . 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 479 . . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> A e. ZZ )
22 simplr 500 . . . . . . . 8 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) -> A =/= 0 )
2322ad2antrr 475 . . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> A =/= 0 )
24 dvdsleabs 11338 . . . . . . . 8 |- ( ( m e. ZZ /\ A e. ZZ /\ A =/= 0 ) -> ( m || A -> m <_ ( abs ` A ) ) )
2524con3d 601 . . . . . . 7 |- ( ( m e. ZZ /\ A e. ZZ /\ A =/= 0 ) -> ( -. m <_ ( abs ` A ) -> -. m || A ) )
2610, 21, 23, 25syl3anc 1184 . . . . . 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 2464 . . . 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 2493 . 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 930   e. wcel 1448   =/= wne 2267   A.wral 2375   E.wrex 2376   class class class wbr 3875   ` cfv 5059   RRcr 7499   0cc0 7500   < clt 7672   <_ cle 7673   NNcn 8578   ZZcz 8906   ZZ>=cuz 9176   abscabs 10609   || cdvds 11288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-q 9262  df-rp 9292  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611  df-dvds 11289
This theorem is referenced by:  gcdsupex  11441  gcdsupcl  11442
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