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Theorem dvdsnprmd 11200
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 10894 . . . 4  |-  ( A 
||  N  ->  ( A  e.  ZZ  /\  N  e.  ZZ ) )
3 divides 10891 . . . 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 8748 . . . . . . . . 9  |-  2  e.  ZZ
65a1i 9 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
2  e.  ZZ )
7 simplr 497 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
k  e.  ZZ )
8 dvdsnprmd.l . . . . . . . . . . . . 13  |-  ( ph  ->  A  <  N )
98adantr 270 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ZZ )  ->  A  < 
N )
109adantr 270 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  ->  A  <  N )
11 breq2 3841 . . . . . . . . . . . 12  |-  ( ( k  x.  A )  =  N  ->  ( A  <  ( k  x.  A )  <->  A  <  N ) )
1211adantl 271 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
( A  <  (
k  x.  A )  <-> 
A  <  N )
)
1310, 12mpbird 165 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  ->  A  <  ( k  x.  A ) )
14 dvdsnprmd.g . . . . . . . . . . . . . 14  |-  ( ph  ->  1  <  A )
15 zre 8724 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  ZZ  ->  A  e.  RR )
16153ad2ant1 964 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  ZZ  /\  1  <  A  /\  k  e.  ZZ )  ->  A  e.  RR )
17 zre 8724 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  ZZ  ->  k  e.  RR )
18173ad2ant3 966 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  ZZ  /\  1  <  A  /\  k  e.  ZZ )  ->  k  e.  RR )
19 0lt1 7589 . . . . . . . . . . . . . . . . . . . . 21  |-  0  <  1
20 0red 7468 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( A  e.  ZZ  ->  0  e.  RR )
21 1red 7482 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( A  e.  ZZ  ->  1  e.  RR )
22 lttr 7538 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  A  e.  RR )  ->  (
( 0  <  1  /\  1  <  A )  ->  0  <  A
) )
2320, 21, 15, 22syl3anc 1174 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A  e.  ZZ  ->  (
( 0  <  1  /\  1  <  A )  ->  0  <  A
) )
2419, 23mpani 421 . . . . . . . . . . . . . . . . . . . 20  |-  ( A  e.  ZZ  ->  (
1  <  A  ->  0  <  A ) )
2524imp 122 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  ZZ  /\  1  <  A )  -> 
0  <  A )
26253adant3 963 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  ZZ  /\  1  <  A  /\  k  e.  ZZ )  ->  0  <  A )
2716, 18, 263jca 1123 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  ZZ  /\  1  <  A  /\  k  e.  ZZ )  ->  ( A  e.  RR  /\  k  e.  RR  /\  0  < 
A ) )
28273exp 1142 . . . . . . . . . . . . . . . 16  |-  ( A  e.  ZZ  ->  (
1  <  A  ->  ( k  e.  ZZ  ->  ( A  e.  RR  /\  k  e.  RR  /\  0  <  A ) ) ) )
2928adantr 270 . . . . . . . . . . . . . . 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 122 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( A  e.  RR  /\  k  e.  RR  /\  0  < 
A ) )
3332adantr 270 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
( A  e.  RR  /\  k  e.  RR  /\  0  <  A ) )
34 ltmulgt12 8298 . . . . . . . . . . 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 165 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
1  <  k )
37 df-2 8452 . . . . . . . . . . 11  |-  2  =  ( 1  +  1 )
3837breq1i 3844 . . . . . . . . . 10  |-  ( 2  <_  k  <->  ( 1  +  1 )  <_ 
k )
39 1zzd 8747 . . . . . . . . . . . . . 14  |-  ( k  e.  ZZ  ->  1  e.  ZZ )
40 zltp1le 8774 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  ZZ  /\  k  e.  ZZ )  ->  ( 1  <  k  <->  ( 1  +  1 )  <_  k ) )
4139, 40mpancom 413 . . . . . . . . . . . . 13  |-  ( k  e.  ZZ  ->  (
1  <  k  <->  ( 1  +  1 )  <_ 
k ) )
4241bicomd 139 . . . . . . . . . . . 12  |-  ( k  e.  ZZ  ->  (
( 1  +  1 )  <_  k  <->  1  <  k ) )
4342adantl 271 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( ( 1  +  1 )  <_  k  <->  1  <  k ) )
4443adantr 270 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
( ( 1  +  1 )  <_  k  <->  1  <  k ) )
4538, 44syl5bb 190 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
( 2  <_  k  <->  1  <  k ) )
4636, 45mpbird 165 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
2  <_  k )
47 eluz2 8994 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  2
)  <->  ( 2  e.  ZZ  /\  k  e.  ZZ  /\  2  <_ 
k ) )
486, 7, 46, 47syl3anbrc 1127 . . . . . . 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 107 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  1  <  A )  ->  A  e.  ZZ )
51 1zzd 8747 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  ZZ  ->  1  e.  ZZ )
52 zltp1le 8774 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  e.  ZZ  /\  A  e.  ZZ )  ->  ( 1  <  A  <->  ( 1  +  1 )  <_  A ) )
5351, 52mpancom 413 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  ZZ  ->  (
1  <  A  <->  ( 1  +  1 )  <_  A ) )
5453biimpa 290 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  ZZ  /\  1  <  A )  -> 
( 1  +  1 )  <_  A )
5537breq1i 3844 . . . . . . . . . . . . . . . 16  |-  ( 2  <_  A  <->  ( 1  +  1 )  <_  A )
5654, 55sylibr 132 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  1  <  A )  -> 
2  <_  A )
5749, 50, 563jca 1123 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ZZ  /\  1  <  A )  -> 
( 2  e.  ZZ  /\  A  e.  ZZ  /\  2  <_  A ) )
5857ex 113 . . . . . . . . . . . . 13  |-  ( A  e.  ZZ  ->  (
1  <  A  ->  ( 2  e.  ZZ  /\  A  e.  ZZ  /\  2  <_  A ) ) )
5958adantr 270 . . . . . . . . . . . 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 8994 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  <->  ( 2  e.  ZZ  /\  A  e.  ZZ  /\  2  <_  A ) )
6361, 62sylibr 132 . . . . . . . . 9  |-  ( ph  ->  A  e.  ( ZZ>= ` 
2 ) )
6463adantr 270 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ZZ )  ->  A  e.  ( ZZ>= `  2 )
)
6564adantr 270 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  ->  A  e.  ( ZZ>= ` 
2 ) )
66 nprm 11198 . . . . . . 7  |-  ( ( k  e.  ( ZZ>= ` 
2 )  /\  A  e.  ( ZZ>= `  2 )
)  ->  -.  (
k  x.  A )  e.  Prime )
6748, 65, 66syl2anc 403 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  ->  -.  ( k  x.  A
)  e.  Prime )
68 eleq1 2150 . . . . . . . 8  |-  ( ( k  x.  A )  =  N  ->  (
( k  x.  A
)  e.  Prime  <->  N  e.  Prime ) )
6968notbid 627 . . . . . . 7  |-  ( ( k  x.  A )  =  N  ->  ( -.  ( k  x.  A
)  e.  Prime  <->  -.  N  e.  Prime ) )
7069adantl 271 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  -> 
( -.  ( k  x.  A )  e. 
Prime 
<->  -.  N  e.  Prime ) )
7167, 70mpbid 145 . . . . 5  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  x.  A )  =  N )  ->  -.  N  e.  Prime )
7271ex 113 . . . 4  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( ( k  x.  A )  =  N  ->  -.  N  e.  Prime ) )
7372rexlimdva 2489 . . 3  |-  ( ph  ->  ( E. k  e.  ZZ  ( k  x.  A )  =  N  ->  -.  N  e.  Prime ) )
744, 73sylbid 148 . 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 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   E.wrex 2360   class class class wbr 3837   ` cfv 5002  (class class class)co 5634   RRcr 7328   0cc0 7329   1c1 7330    + caddc 7332    x. cmul 7334    < clt 7501    <_ cle 7502   2c2 8444   ZZcz 8720   ZZ>=cuz 8988    || cdvds 10889   Primecprime 11182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443  ax-caucvg 7444
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-1o 6163  df-2o 6164  df-er 6272  df-en 6438  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-iseq 9818  df-seq3 9819  df-exp 9920  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-dvds 10890  df-prm 11183
This theorem is referenced by: (None)
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