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Theorem dvdsbnd 12196
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 9478 . . . 4 |- ( ( A e. ZZ /\ A =/= 0 ) -> A e. CC )
32abscld 11411 . . 3 |- ( ( A e. ZZ /\ A =/= 0 ) -> ( abs ` A ) e. RR )
4 arch 9274 . . 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 534 . . . . . . . . 9 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> n e. NN )
87nnred 9031 . . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> n e. RR )
9 eluzelz 9639 . . . . . . . . . 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 9477 . . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> m e. RR )
12 simplr 528 . . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> ( abs ` A ) < n )
13 eluzle 9642 . . . . . . . . 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 8478 . . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> ( abs ` A ) < m )
16 zabscl 11316 . . . . . . . . 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 9400 . . . . . . . 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 528 . . . . . . . 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 12075 . . . . . . . 8 |- ( ( m e. ZZ /\ A e. ZZ /\ A =/= 0 ) -> ( m || A -> m <_ ( abs ` A ) ) )
2524con3d 632 . . . . . . 7 |- ( ( m e. ZZ /\ A e. ZZ /\ A =/= 0 ) -> ( -. m <_ ( abs ` A ) -> -. m || A ) )
2610, 21, 23, 25syl3anc 1249 . . . . . 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 2578 . . . 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 2607 . 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 980   e. wcel 2175   =/= wne 2375   A.wral 2483   E.wrex 2484   class class class wbr 4043   ` cfv 5268   RRcr 7906   0cc0 7907   < clt 8089   <_ cle 8090   NNcn 9018   ZZcz 9354   ZZ>=cuz 9630   abscabs 11227   || cdvds 12017
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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-iinf 4634  ax-cnex 7998  ax-resscn 7999  ax-1cn 8000  ax-1re 8001  ax-icn 8002  ax-addcl 8003  ax-addrcl 8004  ax-mulcl 8005  ax-mulrcl 8006  ax-addcom 8007  ax-mulcom 8008  ax-addass 8009  ax-mulass 8010  ax-distr 8011  ax-i2m1 8012  ax-0lt1 8013  ax-1rid 8014  ax-0id 8015  ax-rnegex 8016  ax-precex 8017  ax-cnre 8018  ax-pre-ltirr 8019  ax-pre-ltwlin 8020  ax-pre-lttrn 8021  ax-pre-apti 8022  ax-pre-ltadd 8023  ax-pre-mulgt0 8024  ax-pre-mulext 8025  ax-arch 8026  ax-caucvg 8027
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4338  df-po 4341  df-iso 4342  df-iord 4411  df-on 4413  df-ilim 4414  df-suc 4416  df-iom 4637  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-riota 5889  df-ov 5937  df-oprab 5938  df-mpo 5939  df-1st 6216  df-2nd 6217  df-recs 6381  df-frec 6467  df-pnf 8091  df-mnf 8092  df-xr 8093  df-ltxr 8094  df-le 8095  df-sub 8227  df-neg 8228  df-reap 8630  df-ap 8637  df-div 8728  df-inn 9019  df-2 9077  df-3 9078  df-4 9079  df-n0 9278  df-z 9355  df-uz 9631  df-q 9723  df-rp 9758  df-seqfrec 10574  df-exp 10665  df-cj 11072  df-re 11073  df-im 11074  df-rsqrt 11228  df-abs 11229  df-dvds 12018
This theorem is referenced by:  gcdsupex  12197  gcdsupcl  12198
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