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Theorem dvdsbnd 12677
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 109 . . . . 5 |- ( ( A e. ZZ /\ A =/= 0 ) -> A e. ZZ )
21zcnd 9719 . . . 4 |- ( ( A e. ZZ /\ A =/= 0 ) -> A e. CC )
32abscld 11891 . . 3 |- ( ( A e. ZZ /\ A =/= 0 ) -> ( abs ` A ) e. RR )
4 arch 9510 . . 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 492 . . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> ( abs ` A ) e. RR )
7 simpllr 536 . . . . . . . . 9 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> n e. NN )
87nnred 9267 . . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> n e. RR )
9 eluzelz 9881 . . . . . . . . . 10 |- ( m e. ( ZZ>= ` n ) -> m e. ZZ )
109adantl 277 . . . . . . . . 9 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> m e. ZZ )
1110zred 9718 . . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> m e. RR )
12 simplr 529 . . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> ( abs ` A ) < n )
13 eluzle 9884 . . . . . . . . 9 |- ( m e. ( ZZ>= ` n ) -> n <_ m )
1413adantl 277 . . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> n <_ m )
156, 8, 11, 12, 14ltletrd 8714 . . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> ( abs ` A ) < m )
16 zabscl 11796 . . . . . . . . 9 |- ( A e. ZZ -> ( abs ` A ) e. ZZ )
1716ad4antr 494 . . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> ( abs ` A ) e. ZZ )
18 zltnle 9640 . . . . . . . 8 |- ( ( ( abs ` A ) e. ZZ /\ m e. ZZ ) -> ( ( abs ` A ) < m <-> -. m <_ ( abs ` A ) ) )
1917, 10, 18syl2anc 411 . . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> ( ( abs ` A ) < m <-> -. m <_ ( abs ` A ) ) )
2015, 19mpbid 147 . . . . . 6 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> -. m <_ ( abs ` A ) )
211ad3antrrr 492 . . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> A e. ZZ )
22 simplr 529 . . . . . . . 8 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) -> A =/= 0 )
2322ad2antrr 488 . . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> A =/= 0 )
24 dvdsleabs 12556 . . . . . . . 8 |- ( ( m e. ZZ /\ A e. ZZ /\ A =/= 0 ) -> ( m || A -> m <_ ( abs ` A ) ) )
2524con3d 636 . . . . . . 7 |- ( ( m e. ZZ /\ A e. ZZ /\ A =/= 0 ) -> ( -. m <_ ( abs ` A ) -> -. m || A ) )
2610, 21, 23, 25syl3anc 1274 . . . . . 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 2617 . . . 4 |- ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) -> A. m e. ( ZZ>= ` n ) -. m || A )
2928ex 115 . . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) -> ( ( abs ` A ) < n -> A. m e. ( ZZ>= ` n ) -. m || A ) )
3029reximdva 2646 . 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 104   <-> wb 105   /\ w3a 1005   e. wcel 2205   =/= wne 2414   A.wral 2522   E.wrex 2523   class class class wbr 4114   ` cfv 5357   RRcr 8142   0cc0 8143   < clt 8324   <_ cle 8325   NNcn 9254   ZZcz 9594   ZZ>=cuz 9871   abscabs 11707   || cdvds 12498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263
This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499
This theorem is referenced by:  gcdsupex  12678  gcdsupcl  12679
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