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Theorem nprm 12772
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 10238 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  ZZ )
21adantr 451 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  A  e.  ZZ )
32zred 10117 . . 3  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  A  e.  RR )
4 eluz2b2 10290 . . . . . 6  |-  ( B  e.  ( ZZ>= `  2
)  <->  ( B  e.  NN  /\  1  < 
B ) )
54simprbi 450 . . . . 5  |-  ( B  e.  ( ZZ>= `  2
)  ->  1  <  B )
65adantl 452 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  1  <  B )
7 eluzelz 10238 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  2
)  ->  B  e.  ZZ )
87adantl 452 . . . . . 6  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  B  e.  ZZ )
98zred 10117 . . . . 5  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  B  e.  RR )
10 eluz2b2 10290 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  <->  ( A  e.  NN  /\  1  < 
A ) )
1110simplbi 446 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  NN )
1211adantr 451 . . . . . 6  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  A  e.  NN )
1312nngt0d 9789 . . . . 5  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  0  <  A )
14 ltmulgt11 9616 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  A )  ->  (
1  <  B  <->  A  <  ( A  x.  B ) ) )
153, 9, 13, 14syl3anc 1182 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  ( 1  <  B  <->  A  <  ( A  x.  B ) ) )
166, 15mpbid 201 . . 3  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  A  <  ( A  x.  B ) )
173, 16ltned 8955 . 2  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  A  =/=  ( A  x.  B
) )
18 dvdsmul1 12550 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  A  ||  ( A  x.  B ) )
191, 7, 18syl2an 463 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  A  ||  ( A  x.  B )
)
20 isprm4 12768 . . . . . . 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 450 . . . . . 6  |-  ( ( A  x.  B )  e.  Prime  ->  A. x  e.  ( ZZ>= `  2 )
( x  ||  ( A  x.  B )  ->  x  =  ( A  x.  B ) ) )
22 breq1 4026 . . . . . . . 8  |-  ( x  =  A  ->  (
x  ||  ( A  x.  B )  <->  A  ||  ( A  x.  B )
) )
23 eqeq1 2289 . . . . . . . 8  |-  ( x  =  A  ->  (
x  =  ( A  x.  B )  <->  A  =  ( A  x.  B
) ) )
2422, 23imbi12d 311 . . . . . . 7  |-  ( x  =  A  ->  (
( x  ||  ( A  x.  B )  ->  x  =  ( A  x.  B ) )  <-> 
( A  ||  ( A  x.  B )  ->  A  =  ( A  x.  B ) ) ) )
2524rspcv 2880 . . . . . 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 28 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A  x.  B )  e.  Prime  ->  ( A  ||  ( A  x.  B
)  ->  A  =  ( A  x.  B
) ) ) )
2726adantr 451 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  ( ( A  x.  B )  e.  Prime  ->  ( A  ||  ( A  x.  B
)  ->  A  =  ( A  x.  B
) ) ) )
2819, 27mpid 37 . . 3  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  ( ( A  x.  B )  e.  Prime  ->  A  =  ( A  x.  B
) ) )
2928necon3ad 2482 . 2  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  B  e.  ( ZZ>= `  2 )
)  ->  ( A  =/=  ( A  x.  B
)  ->  -.  ( A  x.  B )  e.  Prime ) )
3017, 29mpd 14 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    < clt 8867   NNcn 9746   2c2 9795   ZZcz 10024   ZZ>=cuz 10230    || cdivides 12531   Primecprime 12758
This theorem is referenced by:  nprmi  12773  sqnprm  12777  mersenne  20466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-dvds 12532  df-prm 12759
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