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Theorem dvdsnprmd 12046
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 11722 . . . 4  |-  ( A 
||  N  ->  ( A  e.  ZZ  /\  N  e.  ZZ ) )
3 divides 11719 . . . 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 9211 . . . . . . . . 9  |-  2  e.  ZZ
65a1i 9 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
2  e.  ZZ )
7 simplr 520 . . . . . . . 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 3981 . . . . . . . . . . . 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 9187 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  ZZ  ->  A  e.  RR )
16153ad2ant1 1007 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  ZZ  /\  1  <  A  /\  k  e.  ZZ )  ->  A  e.  RR )
17 zre 9187 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  ZZ  ->  k  e.  RR )
18173ad2ant3 1009 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  ZZ  /\  1  <  A  /\  k  e.  ZZ )  ->  k  e.  RR )
19 0lt1 8017 . . . . . . . . . . . . . . . . . . . . 21  |-  0  <  1
20 0red 7892 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( A  e.  ZZ  ->  0  e.  RR )
21 1red 7906 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( A  e.  ZZ  ->  1  e.  RR )
22 lttr 7964 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  A  e.  RR )  ->  (
( 0  <  1  /\  1  <  A )  ->  0  <  A
) )
2320, 21, 15, 22syl3anc 1227 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A  e.  ZZ  ->  (
( 0  <  1  /\  1  <  A )  ->  0  <  A
) )
2419, 23mpani 427 . . . . . . . . . . . . . . . . . . . 20  |-  ( A  e.  ZZ  ->  (
1  <  A  ->  0  <  A ) )
2524imp 123 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  ZZ  /\  1  <  A )  -> 
0  <  A )
26253adant3 1006 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  ZZ  /\  1  <  A  /\  k  e.  ZZ )  ->  0  <  A )
2716, 18, 263jca 1166 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  ZZ  /\  1  <  A  /\  k  e.  ZZ )  ->  ( A  e.  RR  /\  k  e.  RR  /\  0  < 
A ) )
28273exp 1191 . . . . . . . . . . . . . . . 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 8752 . . . . . . . . . . 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 8908 . . . . . . . . . . 11  |-  2  =  ( 1  +  1 )
3837breq1i 3984 . . . . . . . . . 10  |-  ( 2  <_  k  <->  ( 1  +  1 )  <_ 
k )
39 1zzd 9210 . . . . . . . . . . . . . 14  |-  ( k  e.  ZZ  ->  1  e.  ZZ )
40 zltp1le 9237 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  ZZ  /\  k  e.  ZZ )  ->  ( 1  <  k  <->  ( 1  +  1 )  <_  k ) )
4139, 40mpancom 419 . . . . . . . . . . . . 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 9464 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  2
)  <->  ( 2  e.  ZZ  /\  k  e.  ZZ  /\  2  <_ 
k ) )
486, 7, 46, 47syl3anbrc 1170 . . . . . . 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 9210 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  ZZ  ->  1  e.  ZZ )
52 zltp1le 9237 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  e.  ZZ  /\  A  e.  ZZ )  ->  ( 1  <  A  <->  ( 1  +  1 )  <_  A ) )
5351, 52mpancom 419 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  ZZ  ->  (
1  <  A  <->  ( 1  +  1 )  <_  A ) )
5453biimpa 294 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  ZZ  /\  1  <  A )  -> 
( 1  +  1 )  <_  A )
5537breq1i 3984 . . . . . . . . . . . . . . . 16  |-  ( 2  <_  A  <->  ( 1  +  1 )  <_  A )
5654, 55sylibr 133 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  1  <  A )  -> 
2  <_  A )
5749, 50, 563jca 1166 . . . . . . . . . . . . . 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 9464 . . . . . . . . . 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 12044 . . . . . . 7  |-  ( ( k  e.  ( ZZ>= ` 
2 )  /\  A  e.  ( ZZ>= `  2 )
)  ->  -.  (
k  x.  A )  e.  Prime )
6748, 65, 66syl2anc 409 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  ->  -.  ( k  x.  A
)  e.  Prime )
68 eleq1 2227 . . . . . . . 8  |-  ( ( k  x.  A )  =  N  ->  (
( k  x.  A
)  e.  Prime  <->  N  e.  Prime ) )
6968notbid 657 . . . . . . 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 2581 . . 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 967    = wceq 1342    e. wcel 2135   E.wrex 2443   class class class wbr 3977   ` cfv 5183  (class class class)co 5837   RRcr 7744   0cc0 7745   1c1 7746    + caddc 7748    x. cmul 7750    < clt 7925    <_ cle 7926   2c2 8900   ZZcz 9183   ZZ>=cuz 9458    || cdvds 11717   Primecprime 12028
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4092  ax-sep 4095  ax-nul 4103  ax-pow 4148  ax-pr 4182  ax-un 4406  ax-setind 4509  ax-iinf 4560  ax-cnex 7836  ax-resscn 7837  ax-1cn 7838  ax-1re 7839  ax-icn 7840  ax-addcl 7841  ax-addrcl 7842  ax-mulcl 7843  ax-mulrcl 7844  ax-addcom 7845  ax-mulcom 7846  ax-addass 7847  ax-mulass 7848  ax-distr 7849  ax-i2m1 7850  ax-0lt1 7851  ax-1rid 7852  ax-0id 7853  ax-rnegex 7854  ax-precex 7855  ax-cnre 7856  ax-pre-ltirr 7857  ax-pre-ltwlin 7858  ax-pre-lttrn 7859  ax-pre-apti 7860  ax-pre-ltadd 7861  ax-pre-mulgt0 7862  ax-pre-mulext 7863  ax-arch 7864  ax-caucvg 7865
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2724  df-sbc 2948  df-csb 3042  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-nul 3406  df-if 3517  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-int 3820  df-iun 3863  df-br 3978  df-opab 4039  df-mpt 4040  df-tr 4076  df-id 4266  df-po 4269  df-iso 4270  df-iord 4339  df-on 4341  df-ilim 4342  df-suc 4344  df-iom 4563  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-rn 4610  df-res 4611  df-ima 4612  df-iota 5148  df-fun 5185  df-fn 5186  df-f 5187  df-f1 5188  df-fo 5189  df-f1o 5190  df-fv 5191  df-riota 5793  df-ov 5840  df-oprab 5841  df-mpo 5842  df-1st 6101  df-2nd 6102  df-recs 6265  df-frec 6351  df-1o 6376  df-2o 6377  df-er 6493  df-en 6699  df-pnf 7927  df-mnf 7928  df-xr 7929  df-ltxr 7930  df-le 7931  df-sub 8063  df-neg 8064  df-reap 8465  df-ap 8472  df-div 8561  df-inn 8850  df-2 8908  df-3 8909  df-4 8910  df-n0 9107  df-z 9184  df-uz 9459  df-q 9550  df-rp 9582  df-seqfrec 10372  df-exp 10446  df-cj 10774  df-re 10775  df-im 10776  df-rsqrt 10930  df-abs 10931  df-dvds 11718  df-prm 12029
This theorem is referenced by: (None)
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