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Theorem facndiv 11301
Description: No natural number (greater than one) divides the factorial plus one of an equal or larger number. (Contributed by NM, 3-May-2005.)
Assertion
Ref Expression
facndiv  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( 1  < 
N  /\  N  <_  M ) )  ->  -.  ( ( ( ! `
 M )  +  1 )  /  N
)  e.  ZZ )

Proof of Theorem facndiv
StepHypRef Expression
1 nnre 9753 . . . 4  |-  ( N  e.  NN  ->  N  e.  RR )
2 recnz 10087 . . . 4  |-  ( ( N  e.  RR  /\  1  <  N )  ->  -.  ( 1  /  N
)  e.  ZZ )
31, 2sylan 457 . . 3  |-  ( ( N  e.  NN  /\  1  <  N )  ->  -.  ( 1  /  N
)  e.  ZZ )
43ad2ant2lr 728 . 2  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( 1  < 
N  /\  N  <_  M ) )  ->  -.  ( 1  /  N
)  e.  ZZ )
5 facdiv 11300 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN  /\  N  <_  M )  ->  (
( ! `  M
)  /  N )  e.  NN )
653expa 1151 . . . . . 6  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  N  <_  M
)  ->  ( ( ! `  M )  /  N )  e.  NN )
76nnzd 10116 . . . . 5  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  N  <_  M
)  ->  ( ( ! `  M )  /  N )  e.  ZZ )
87adantrl 696 . . . 4  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( 1  < 
N  /\  N  <_  M ) )  ->  (
( ! `  M
)  /  N )  e.  ZZ )
9 zsubcl 10061 . . . . 5  |-  ( ( ( ( ( ! `
 M )  +  1 )  /  N
)  e.  ZZ  /\  ( ( ! `  M )  /  N
)  e.  ZZ )  ->  ( ( ( ( ! `  M
)  +  1 )  /  N )  -  ( ( ! `  M )  /  N
) )  e.  ZZ )
109ex 423 . . . 4  |-  ( ( ( ( ! `  M )  +  1 )  /  N )  e.  ZZ  ->  (
( ( ! `  M )  /  N
)  e.  ZZ  ->  ( ( ( ( ! `
 M )  +  1 )  /  N
)  -  ( ( ! `  M )  /  N ) )  e.  ZZ ) )
118, 10syl5com 26 . . 3  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( 1  < 
N  /\  N  <_  M ) )  ->  (
( ( ( ! `
 M )  +  1 )  /  N
)  e.  ZZ  ->  ( ( ( ( ! `
 M )  +  1 )  /  N
)  -  ( ( ! `  M )  /  N ) )  e.  ZZ ) )
12 faccl 11298 . . . . . . . . 9  |-  ( M  e.  NN0  ->  ( ! `
 M )  e.  NN )
1312nncnd 9762 . . . . . . . 8  |-  ( M  e.  NN0  ->  ( ! `
 M )  e.  CC )
14 peano2cn 8984 . . . . . . . 8  |-  ( ( ! `  M )  e.  CC  ->  (
( ! `  M
)  +  1 )  e.  CC )
1513, 14syl 15 . . . . . . 7  |-  ( M  e.  NN0  ->  ( ( ! `  M )  +  1 )  e.  CC )
1615ad2antrr 706 . . . . . 6  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( 1  < 
N  /\  N  <_  M ) )  ->  (
( ! `  M
)  +  1 )  e.  CC )
1713ad2antrr 706 . . . . . 6  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( 1  < 
N  /\  N  <_  M ) )  ->  ( ! `  M )  e.  CC )
18 nncn 9754 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  CC )
19 nnne0 9778 . . . . . . . 8  |-  ( N  e.  NN  ->  N  =/=  0 )
2018, 19jca 518 . . . . . . 7  |-  ( N  e.  NN  ->  ( N  e.  CC  /\  N  =/=  0 ) )
2120ad2antlr 707 . . . . . 6  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( 1  < 
N  /\  N  <_  M ) )  ->  ( N  e.  CC  /\  N  =/=  0 ) )
22 divsubdir 9456 . . . . . 6  |-  ( ( ( ( ! `  M )  +  1 )  e.  CC  /\  ( ! `  M )  e.  CC  /\  ( N  e.  CC  /\  N  =/=  0 ) )  -> 
( ( ( ( ! `  M )  +  1 )  -  ( ! `  M ) )  /  N )  =  ( ( ( ( ! `  M
)  +  1 )  /  N )  -  ( ( ! `  M )  /  N
) ) )
2316, 17, 21, 22syl3anc 1182 . . . . 5  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( 1  < 
N  /\  N  <_  M ) )  ->  (
( ( ( ! `
 M )  +  1 )  -  ( ! `  M )
)  /  N )  =  ( ( ( ( ! `  M
)  +  1 )  /  N )  -  ( ( ! `  M )  /  N
) ) )
24 ax-1cn 8795 . . . . . . . 8  |-  1  e.  CC
25 pncan2 9058 . . . . . . . 8  |-  ( ( ( ! `  M
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( ! `
 M )  +  1 )  -  ( ! `  M )
)  =  1 )
2613, 24, 25sylancl 643 . . . . . . 7  |-  ( M  e.  NN0  ->  ( ( ( ! `  M
)  +  1 )  -  ( ! `  M ) )  =  1 )
2726oveq1d 5873 . . . . . 6  |-  ( M  e.  NN0  ->  ( ( ( ( ! `  M )  +  1 )  -  ( ! `
 M ) )  /  N )  =  ( 1  /  N
) )
2827ad2antrr 706 . . . . 5  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( 1  < 
N  /\  N  <_  M ) )  ->  (
( ( ( ! `
 M )  +  1 )  -  ( ! `  M )
)  /  N )  =  ( 1  /  N ) )
2923, 28eqtr3d 2317 . . . 4  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( 1  < 
N  /\  N  <_  M ) )  ->  (
( ( ( ! `
 M )  +  1 )  /  N
)  -  ( ( ! `  M )  /  N ) )  =  ( 1  /  N ) )
3029eleq1d 2349 . . 3  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( 1  < 
N  /\  N  <_  M ) )  ->  (
( ( ( ( ! `  M )  +  1 )  /  N )  -  (
( ! `  M
)  /  N ) )  e.  ZZ  <->  ( 1  /  N )  e.  ZZ ) )
3111, 30sylibd 205 . 2  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( 1  < 
N  /\  N  <_  M ) )  ->  (
( ( ( ! `
 M )  +  1 )  /  N
)  e.  ZZ  ->  ( 1  /  N )  e.  ZZ ) )
324, 31mtod 168 1  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( 1  < 
N  /\  N  <_  M ) )  ->  -.  ( ( ( ! `
 M )  +  1 )  /  N
)  e.  ZZ )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024   !cfa 11288
This theorem is referenced by:  infpnlem1  12957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-fac 11289
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