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Theorem dvdsnprmd 10901
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 10595 . . . 4  |-  ( A 
||  N  ->  ( A  e.  ZZ  /\  N  e.  ZZ ) )
3 divides 10592 . . . 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 8688 . . . . . . . . 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 3818 . . . . . . . . . . . 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 8664 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  ZZ  ->  A  e.  RR )
16153ad2ant1 962 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  ZZ  /\  1  <  A  /\  k  e.  ZZ )  ->  A  e.  RR )
17 zre 8664 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  ZZ  ->  k  e.  RR )
18173ad2ant3 964 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  ZZ  /\  1  <  A  /\  k  e.  ZZ )  ->  k  e.  RR )
19 0lt1 7531 . . . . . . . . . . . . . . . . . . . . 21  |-  0  <  1
20 0red 7410 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( A  e.  ZZ  ->  0  e.  RR )
21 1red 7424 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( A  e.  ZZ  ->  1  e.  RR )
22 lttr 7480 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  A  e.  RR )  ->  (
( 0  <  1  /\  1  <  A )  ->  0  <  A
) )
2320, 21, 15, 22syl3anc 1172 . . . . . . . . . . . . . . . . . . . . 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 961 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  ZZ  /\  1  <  A  /\  k  e.  ZZ )  ->  0  <  A )
2716, 18, 263jca 1121 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  ZZ  /\  1  <  A  /\  k  e.  ZZ )  ->  ( A  e.  RR  /\  k  e.  RR  /\  0  < 
A ) )
28273exp 1140 . . . . . . . . . . . . . . . 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 8238 . . . . . . . . . . 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 8393 . . . . . . . . . . 11  |-  2  =  ( 1  +  1 )
3837breq1i 3821 . . . . . . . . . 10  |-  ( 2  <_  k  <->  ( 1  +  1 )  <_ 
k )
39 1zzd 8687 . . . . . . . . . . . . . 14  |-  ( k  e.  ZZ  ->  1  e.  ZZ )
40 zltp1le 8714 . . . . . . . . . . . . . 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 8934 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  2
)  <->  ( 2  e.  ZZ  /\  k  e.  ZZ  /\  2  <_ 
k ) )
486, 7, 46, 47syl3anbrc 1125 . . . . . . 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 8687 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  ZZ  ->  1  e.  ZZ )
52 zltp1le 8714 . . . . . . . . . . . . . . . . . 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 3821 . . . . . . . . . . . . . . . 16  |-  ( 2  <_  A  <->  ( 1  +  1 )  <_  A )
5654, 55sylibr 132 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  1  <  A )  -> 
2  <_  A )
5749, 50, 563jca 1121 . . . . . . . . . . . . . 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 8934 . . . . . . . . . 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 10899 . . . . . . 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 2147 . . . . . . . 8  |-  ( ( k  x.  A )  =  N  ->  (
( k  x.  A
)  e.  Prime  <->  N  e.  Prime ) )
6968notbid 625 . . . . . . 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 2485 . . 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 922    = wceq 1287    e. wcel 1436   E.wrex 2356   class class class wbr 3814   ` cfv 4972  (class class class)co 5594   RRcr 7270   0cc0 7271   1c1 7272    + caddc 7274    x. cmul 7276    < clt 7443    <_ cle 7444   2c2 8384   ZZcz 8660   ZZ>=cuz 8928    || cdvds 10590   Primecprime 10883
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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-mulrcl 7365  ax-addcom 7366  ax-mulcom 7367  ax-addass 7368  ax-mulass 7369  ax-distr 7370  ax-i2m1 7371  ax-0lt1 7372  ax-1rid 7373  ax-0id 7374  ax-rnegex 7375  ax-precex 7376  ax-cnre 7377  ax-pre-ltirr 7378  ax-pre-ltwlin 7379  ax-pre-lttrn 7380  ax-pre-apti 7381  ax-pre-ltadd 7382  ax-pre-mulgt0 7383  ax-pre-mulext 7384  ax-arch 7385  ax-caucvg 7386
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-if 3377  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-id 4087  df-po 4090  df-iso 4091  df-iord 4160  df-on 4162  df-ilim 4163  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-recs 6005  df-frec 6091  df-1o 6116  df-2o 6117  df-er 6225  df-en 6391  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449  df-sub 7576  df-neg 7577  df-reap 7970  df-ap 7977  df-div 8056  df-inn 8335  df-2 8393  df-3 8394  df-4 8395  df-n0 8584  df-z 8661  df-uz 8929  df-q 9014  df-rp 9044  df-iseq 9755  df-iexp 9806  df-cj 10117  df-re 10118  df-im 10119  df-rsqrt 10272  df-abs 10273  df-dvds 10591  df-prm 10884
This theorem is referenced by: (None)
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