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Theorem dvdsnprmd 11733
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 11425 . . . 4  |-  ( A 
||  N  ->  ( A  e.  ZZ  /\  N  e.  ZZ ) )
3 divides 11422 . . . 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 9050 . . . . . . . . 9  |-  2  e.  ZZ
65a1i 9 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
2  e.  ZZ )
7 simplr 504 . . . . . . . 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 3903 . . . . . . . . . . . 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 9026 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  ZZ  ->  A  e.  RR )
16153ad2ant1 987 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  ZZ  /\  1  <  A  /\  k  e.  ZZ )  ->  A  e.  RR )
17 zre 9026 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  ZZ  ->  k  e.  RR )
18173ad2ant3 989 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  ZZ  /\  1  <  A  /\  k  e.  ZZ )  ->  k  e.  RR )
19 0lt1 7857 . . . . . . . . . . . . . . . . . . . . 21  |-  0  <  1
20 0red 7735 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( A  e.  ZZ  ->  0  e.  RR )
21 1red 7749 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( A  e.  ZZ  ->  1  e.  RR )
22 lttr 7806 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  A  e.  RR )  ->  (
( 0  <  1  /\  1  <  A )  ->  0  <  A
) )
2320, 21, 15, 22syl3anc 1201 . . . . . . . . . . . . . . . . . . . . 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 986 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  ZZ  /\  1  <  A  /\  k  e.  ZZ )  ->  0  <  A )
2716, 18, 263jca 1146 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  ZZ  /\  1  <  A  /\  k  e.  ZZ )  ->  ( A  e.  RR  /\  k  e.  RR  /\  0  < 
A ) )
28273exp 1165 . . . . . . . . . . . . . . . 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 8591 . . . . . . . . . . 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 8747 . . . . . . . . . . 11  |-  2  =  ( 1  +  1 )
3837breq1i 3906 . . . . . . . . . 10  |-  ( 2  <_  k  <->  ( 1  +  1 )  <_ 
k )
39 1zzd 9049 . . . . . . . . . . . . . 14  |-  ( k  e.  ZZ  ->  1  e.  ZZ )
40 zltp1le 9076 . . . . . . . . . . . . . 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 9300 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  2
)  <->  ( 2  e.  ZZ  /\  k  e.  ZZ  /\  2  <_ 
k ) )
486, 7, 46, 47syl3anbrc 1150 . . . . . . 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 9049 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  ZZ  ->  1  e.  ZZ )
52 zltp1le 9076 . . . . . . . . . . . . . . . . . 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 3906 . . . . . . . . . . . . . . . 16  |-  ( 2  <_  A  <->  ( 1  +  1 )  <_  A )
5654, 55sylibr 133 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  1  <  A )  -> 
2  <_  A )
5749, 50, 563jca 1146 . . . . . . . . . . . . . 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 9300 . . . . . . . . . 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 11731 . . . . . . 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 2180 . . . . . . . 8  |-  ( ( k  x.  A )  =  N  ->  (
( k  x.  A
)  e.  Prime  <->  N  e.  Prime ) )
6968notbid 641 . . . . . . 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 2526 . . 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 947    = wceq 1316    e. wcel 1465   E.wrex 2394   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   RRcr 7587   0cc0 7588   1c1 7589    + caddc 7591    x. cmul 7593    < clt 7768    <_ cle 7769   2c2 8739   ZZcz 9022   ZZ>=cuz 9294    || cdvds 11420   Primecprime 11715
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-1o 6281  df-2o 6282  df-er 6397  df-en 6603  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-n0 8946  df-z 9023  df-uz 9295  df-q 9380  df-rp 9410  df-seqfrec 10187  df-exp 10261  df-cj 10582  df-re 10583  df-im 10584  df-rsqrt 10738  df-abs 10739  df-dvds 11421  df-prm 11716
This theorem is referenced by: (None)
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