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Theorem dvdsnprmd 11795
Description: If a number is divisible by an integer greater than 1 and less then the number, the number is not prime. (Contributed by AV, 24-Jul-2021.)
Hypotheses
Ref Expression
dvdsnprmd.g  |-  ( ph  ->  1  <  A )
dvdsnprmd.l  |-  ( ph  ->  A  <  N )
dvdsnprmd.d  |-  ( ph  ->  A  ||  N )
Assertion
Ref Expression
dvdsnprmd  |-  ( ph  ->  -.  N  e.  Prime )

Proof of Theorem dvdsnprmd
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 dvdsnprmd.d . 2  |-  ( ph  ->  A  ||  N )
2 dvdszrcl 11487 . . . 4  |-  ( A 
||  N  ->  ( A  e.  ZZ  /\  N  e.  ZZ ) )
3 divides 11484 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  ||  N  <->  E. k  e.  ZZ  (
k  x.  A )  =  N ) )
41, 2, 33syl 17 . . 3  |-  ( ph  ->  ( A  ||  N  <->  E. k  e.  ZZ  (
k  x.  A )  =  N ) )
5 2z 9075 . . . . . . . . 9  |-  2  e.  ZZ
65a1i 9 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
2  e.  ZZ )
7 simplr 519 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
k  e.  ZZ )
8 dvdsnprmd.l . . . . . . . . . . . . 13  |-  ( ph  ->  A  <  N )
98adantr 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ZZ )  ->  A  < 
N )
109adantr 274 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  ->  A  <  N )
11 breq2 3928 . . . . . . . . . . . 12  |-  ( ( k  x.  A )  =  N  ->  ( A  <  ( k  x.  A )  <->  A  <  N ) )
1211adantl 275 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
( A  <  (
k  x.  A )  <-> 
A  <  N )
)
1310, 12mpbird 166 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  ->  A  <  ( k  x.  A ) )
14 dvdsnprmd.g . . . . . . . . . . . . . 14  |-  ( ph  ->  1  <  A )
15 zre 9051 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  ZZ  ->  A  e.  RR )
16153ad2ant1 1002 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  ZZ  /\  1  <  A  /\  k  e.  ZZ )  ->  A  e.  RR )
17 zre 9051 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  ZZ  ->  k  e.  RR )
18173ad2ant3 1004 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  ZZ  /\  1  <  A  /\  k  e.  ZZ )  ->  k  e.  RR )
19 0lt1 7882 . . . . . . . . . . . . . . . . . . . . 21  |-  0  <  1
20 0red 7760 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( A  e.  ZZ  ->  0  e.  RR )
21 1red 7774 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( A  e.  ZZ  ->  1  e.  RR )
22 lttr 7831 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  A  e.  RR )  ->  (
( 0  <  1  /\  1  <  A )  ->  0  <  A
) )
2320, 21, 15, 22syl3anc 1216 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A  e.  ZZ  ->  (
( 0  <  1  /\  1  <  A )  ->  0  <  A
) )
2419, 23mpani 426 . . . . . . . . . . . . . . . . . . . 20  |-  ( A  e.  ZZ  ->  (
1  <  A  ->  0  <  A ) )
2524imp 123 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  ZZ  /\  1  <  A )  -> 
0  <  A )
26253adant3 1001 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  ZZ  /\  1  <  A  /\  k  e.  ZZ )  ->  0  <  A )
2716, 18, 263jca 1161 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  ZZ  /\  1  <  A  /\  k  e.  ZZ )  ->  ( A  e.  RR  /\  k  e.  RR  /\  0  < 
A ) )
28273exp 1180 . . . . . . . . . . . . . . . 16  |-  ( A  e.  ZZ  ->  (
1  <  A  ->  ( k  e.  ZZ  ->  ( A  e.  RR  /\  k  e.  RR  /\  0  <  A ) ) ) )
2928adantr 274 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ )  ->  ( 1  <  A  ->  ( k  e.  ZZ  ->  ( A  e.  RR  /\  k  e.  RR  /\  0  <  A ) ) ) )
301, 2, 293syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1  <  A  ->  ( k  e.  ZZ  ->  ( A  e.  RR  /\  k  e.  RR  /\  0  <  A ) ) ) )
3114, 30mpd 13 . . . . . . . . . . . . 13  |-  ( ph  ->  ( k  e.  ZZ  ->  ( A  e.  RR  /\  k  e.  RR  /\  0  <  A ) ) )
3231imp 123 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( A  e.  RR  /\  k  e.  RR  /\  0  < 
A ) )
3332adantr 274 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
( A  e.  RR  /\  k  e.  RR  /\  0  <  A ) )
34 ltmulgt12 8616 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  k  e.  RR  /\  0  <  A )  ->  (
1  <  k  <->  A  <  ( k  x.  A ) ) )
3533, 34syl 14 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
( 1  <  k  <->  A  <  ( k  x.  A ) ) )
3613, 35mpbird 166 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
1  <  k )
37 df-2 8772 . . . . . . . . . . 11  |-  2  =  ( 1  +  1 )
3837breq1i 3931 . . . . . . . . . 10  |-  ( 2  <_  k  <->  ( 1  +  1 )  <_ 
k )
39 1zzd 9074 . . . . . . . . . . . . . 14  |-  ( k  e.  ZZ  ->  1  e.  ZZ )
40 zltp1le 9101 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  ZZ  /\  k  e.  ZZ )  ->  ( 1  <  k  <->  ( 1  +  1 )  <_  k ) )
4139, 40mpancom 418 . . . . . . . . . . . . 13  |-  ( k  e.  ZZ  ->  (
1  <  k  <->  ( 1  +  1 )  <_ 
k ) )
4241bicomd 140 . . . . . . . . . . . 12  |-  ( k  e.  ZZ  ->  (
( 1  +  1 )  <_  k  <->  1  <  k ) )
4342adantl 275 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( ( 1  +  1 )  <_  k  <->  1  <  k ) )
4443adantr 274 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
( ( 1  +  1 )  <_  k  <->  1  <  k ) )
4538, 44syl5bb 191 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
( 2  <_  k  <->  1  <  k ) )
4636, 45mpbird 166 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
2  <_  k )
47 eluz2 9325 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  2
)  <->  ( 2  e.  ZZ  /\  k  e.  ZZ  /\  2  <_ 
k ) )
486, 7, 46, 47syl3anbrc 1165 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
k  e.  ( ZZ>= ` 
2 ) )
495a1i 9 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  1  <  A )  -> 
2  e.  ZZ )
50 simpl 108 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  1  <  A )  ->  A  e.  ZZ )
51 1zzd 9074 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  ZZ  ->  1  e.  ZZ )
52 zltp1le 9101 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  e.  ZZ  /\  A  e.  ZZ )  ->  ( 1  <  A  <->  ( 1  +  1 )  <_  A ) )
5351, 52mpancom 418 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  ZZ  ->  (
1  <  A  <->  ( 1  +  1 )  <_  A ) )
5453biimpa 294 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  ZZ  /\  1  <  A )  -> 
( 1  +  1 )  <_  A )
5537breq1i 3931 . . . . . . . . . . . . . . . 16  |-  ( 2  <_  A  <->  ( 1  +  1 )  <_  A )
5654, 55sylibr 133 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  1  <  A )  -> 
2  <_  A )
5749, 50, 563jca 1161 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ZZ  /\  1  <  A )  -> 
( 2  e.  ZZ  /\  A  e.  ZZ  /\  2  <_  A ) )
5857ex 114 . . . . . . . . . . . . 13  |-  ( A  e.  ZZ  ->  (
1  <  A  ->  ( 2  e.  ZZ  /\  A  e.  ZZ  /\  2  <_  A ) ) )
5958adantr 274 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ )  ->  ( 1  <  A  ->  ( 2  e.  ZZ  /\  A  e.  ZZ  /\  2  <_  A ) ) )
601, 2, 593syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  <  A  ->  ( 2  e.  ZZ  /\  A  e.  ZZ  /\  2  <_  A ) ) )
6114, 60mpd 13 . . . . . . . . . 10  |-  ( ph  ->  ( 2  e.  ZZ  /\  A  e.  ZZ  /\  2  <_  A ) )
62 eluz2 9325 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  <->  ( 2  e.  ZZ  /\  A  e.  ZZ  /\  2  <_  A ) )
6361, 62sylibr 133 . . . . . . . . 9  |-  ( ph  ->  A  e.  ( ZZ>= ` 
2 ) )
6463adantr 274 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ZZ )  ->  A  e.  ( ZZ>= `  2 )
)
6564adantr 274 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  ->  A  e.  ( ZZ>= ` 
2 ) )
66 nprm 11793 . . . . . . 7  |-  ( ( k  e.  ( ZZ>= ` 
2 )  /\  A  e.  ( ZZ>= `  2 )
)  ->  -.  (
k  x.  A )  e.  Prime )
6748, 65, 66syl2anc 408 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  ->  -.  ( k  x.  A
)  e.  Prime )
68 eleq1 2200 . . . . . . . 8  |-  ( ( k  x.  A )  =  N  ->  (
( k  x.  A
)  e.  Prime  <->  N  e.  Prime ) )
6968notbid 656 . . . . . . 7  |-  ( ( k  x.  A )  =  N  ->  ( -.  ( k  x.  A
)  e.  Prime  <->  -.  N  e.  Prime ) )
7069adantl 275 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
( -.  ( k  x.  A )  e. 
Prime 
<->  -.  N  e.  Prime ) )
7167, 70mpbid 146 . . . . 5  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  ->  -.  N  e.  Prime )
7271ex 114 . . . 4  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( ( k  x.  A )  =  N  ->  -.  N  e.  Prime ) )
7372rexlimdva 2547 . . 3  |-  ( ph  ->  ( E. k  e.  ZZ  ( k  x.  A )  =  N  ->  -.  N  e.  Prime ) )
744, 73sylbid 149 . 2  |-  ( ph  ->  ( A  ||  N  ->  -.  N  e.  Prime ) )
751, 74mpd 13 1  |-  ( ph  ->  -.  N  e.  Prime )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   E.wrex 2415   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   RRcr 7612   0cc0 7613   1c1 7614    + caddc 7616    x. cmul 7618    < clt 7793    <_ cle 7794   2c2 8764   ZZcz 9047   ZZ>=cuz 9319    || cdvds 11482   Primecprime 11777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-1o 6306  df-2o 6307  df-er 6422  df-en 6628  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-seqfrec 10212  df-exp 10286  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-dvds 11483  df-prm 11778
This theorem is referenced by: (None)
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