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Theorem nprm 13093
Description: A product of two integers greater than one is composite. (Contributed by Mario Carneiro, 20-Jun-2015.)
Assertion
Ref Expression
nprm  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  -.  ( A  x.  B )  e.  Prime )

Proof of Theorem nprm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eluzelz 10496 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  ZZ )
21adantr 452 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  A  e.  ZZ )
32zred 10375 . . 3  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  A  e.  RR )
4 eluz2b2 10548 . . . . . 6  |-  ( B  e.  ( ZZ>= `  2
)  <->  ( B  e.  NN  /\  1  < 
B ) )
54simprbi 451 . . . . 5  |-  ( B  e.  ( ZZ>= `  2
)  ->  1  <  B )
65adantl 453 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  1  <  B )
7 eluzelz 10496 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  2
)  ->  B  e.  ZZ )
87adantl 453 . . . . . 6  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  B  e.  ZZ )
98zred 10375 . . . . 5  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  B  e.  RR )
10 eluz2b2 10548 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  <->  ( A  e.  NN  /\  1  < 
A ) )
1110simplbi 447 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  NN )
1211adantr 452 . . . . . 6  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  A  e.  NN )
1312nngt0d 10043 . . . . 5  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  0  <  A )
14 ltmulgt11 9870 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  A )  ->  (
1  <  B  <->  A  <  ( A  x.  B ) ) )
153, 9, 13, 14syl3anc 1184 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  ( 1  <  B  <->  A  <  ( A  x.  B ) ) )
166, 15mpbid 202 . . 3  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  A  <  ( A  x.  B ) )
173, 16ltned 9209 . 2  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  A  =/=  ( A  x.  B
) )
18 dvdsmul1 12871 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  A  ||  ( A  x.  B ) )
191, 7, 18syl2an 464 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  A  ||  ( A  x.  B )
)
20 isprm4 13089 . . . . . . 7  |-  ( ( A  x.  B )  e.  Prime  <->  ( ( A  x.  B )  e.  ( ZZ>= `  2 )  /\  A. x  e.  (
ZZ>= `  2 ) ( x  ||  ( A  x.  B )  ->  x  =  ( A  x.  B ) ) ) )
2120simprbi 451 . . . . . 6  |-  ( ( A  x.  B )  e.  Prime  ->  A. x  e.  ( ZZ>= `  2 )
( x  ||  ( A  x.  B )  ->  x  =  ( A  x.  B ) ) )
22 breq1 4215 . . . . . . . 8  |-  ( x  =  A  ->  (
x  ||  ( A  x.  B )  <->  A  ||  ( A  x.  B )
) )
23 eqeq1 2442 . . . . . . . 8  |-  ( x  =  A  ->  (
x  =  ( A  x.  B )  <->  A  =  ( A  x.  B
) ) )
2422, 23imbi12d 312 . . . . . . 7  |-  ( x  =  A  ->  (
( x  ||  ( A  x.  B )  ->  x  =  ( A  x.  B ) )  <-> 
( A  ||  ( A  x.  B )  ->  A  =  ( A  x.  B ) ) ) )
2524rspcv 3048 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A. x  e.  ( ZZ>= ` 
2 ) ( x 
||  ( A  x.  B )  ->  x  =  ( A  x.  B ) )  -> 
( A  ||  ( A  x.  B )  ->  A  =  ( A  x.  B ) ) ) )
2621, 25syl5 30 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A  x.  B )  e.  Prime  ->  ( A  ||  ( A  x.  B
)  ->  A  =  ( A  x.  B
) ) ) )
2726adantr 452 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  ( ( A  x.  B )  e.  Prime  ->  ( A  ||  ( A  x.  B
)  ->  A  =  ( A  x.  B
) ) ) )
2819, 27mpid 39 . . 3  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  ( ( A  x.  B )  e.  Prime  ->  A  =  ( A  x.  B
) ) )
2928necon3ad 2637 . 2  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  ( A  =/=  ( A  x.  B
)  ->  -.  ( A  x.  B )  e.  Prime ) )
3017, 29mpd 15 1  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  -.  ( A  x.  B )  e.  Prime )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   RRcr 8989   0cc0 8990   1c1 8991    x. cmul 8995    < clt 9120   NNcn 10000   2c2 10049   ZZcz 10282   ZZ>=cuz 10488    || cdivides 12852   Primecprime 13079
This theorem is referenced by:  nprmi  13094  sqnprm  13098  mersenne  21011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-dvds 12853  df-prm 13080
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