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Theorem dvdsbnd 11992
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 9407 . . . 4 |- ( ( A e. ZZ /\ A =/= 0 ) -> A e. CC )
32abscld 11225 . . 3 |- ( ( A e. ZZ /\ A =/= 0 ) -> ( abs ` A ) e. RR )
4 arch 9204 . . 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 8963 . . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> n e. RR )
9 eluzelz 9568 . . . . . . . . . 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 9406 . . . . . . . 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 9571 . . . . . . . . 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 8411 . . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ A =/= 0 ) /\ n e. NN ) /\ ( abs ` A ) < n ) /\ m e. ( ZZ>= ` n ) ) -> ( abs ` A ) < m )
16 zabscl 11130 . . . . . . . . 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 9330 . . . . . . . 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 11886 . . . . . . . 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 2563 . . . 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 2592 . 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 2160   =/= wne 2360   A.wral 2468   E.wrex 2469   class class class wbr 4018   ` cfv 5235   RRcr 7841   0cc0 7842   < clt 8023   <_ cle 8024   NNcn 8950   ZZcz 9284   ZZ>=cuz 9559   abscabs 11041   || cdvds 11829
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960  ax-arch 7961  ax-caucvg 7962
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-frec 6417  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-inn 8951  df-2 9009  df-3 9010  df-4 9011  df-n0 9208  df-z 9285  df-uz 9560  df-q 9652  df-rp 9686  df-seqfrec 10479  df-exp 10554  df-cj 10886  df-re 10887  df-im 10888  df-rsqrt 11042  df-abs 11043  df-dvds 11830
This theorem is referenced by:  gcdsupex  11993  gcdsupcl  11994
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