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Theorem prime 9143
Description: Two ways to express " A is a prime number (or 1)." (Contributed by NM, 4-May-2005.)
Assertion
Ref Expression
prime  |-  ( A  e.  NN  ->  ( A. x  e.  NN  ( ( A  /  x )  e.  NN  ->  ( x  =  1  \/  x  =  A ) )  <->  A. x  e.  NN  ( ( 1  <  x  /\  x  <_  A  /\  ( A  /  x )  e.  NN )  ->  x  =  A ) ) )
Distinct variable group:    x, A

Proof of Theorem prime
StepHypRef Expression
1 nnz 9066 . . . . . . 7  |-  ( x  e.  NN  ->  x  e.  ZZ )
2 1z 9073 . . . . . . . 8  |-  1  e.  ZZ
3 zdceq 9119 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  1  e.  ZZ )  -> DECID  x  =  1 )
42, 3mpan2 421 . . . . . . 7  |-  ( x  e.  ZZ  -> DECID  x  =  1
)
5 dfordc 877 . . . . . . . 8  |-  (DECID  x  =  1  ->  ( (
x  =  1  \/  x  =  A )  <-> 
( -.  x  =  1  ->  x  =  A ) ) )
6 df-ne 2307 . . . . . . . . 9  |-  ( x  =/=  1  <->  -.  x  =  1 )
76imbi1i 237 . . . . . . . 8  |-  ( ( x  =/=  1  ->  x  =  A )  <->  ( -.  x  =  1  ->  x  =  A ) )
85, 7syl6bbr 197 . . . . . . 7  |-  (DECID  x  =  1  ->  ( (
x  =  1  \/  x  =  A )  <-> 
( x  =/=  1  ->  x  =  A ) ) )
91, 4, 83syl 17 . . . . . 6  |-  ( x  e.  NN  ->  (
( x  =  1  \/  x  =  A )  <->  ( x  =/=  1  ->  x  =  A ) ) )
109imbi2d 229 . . . . 5  |-  ( x  e.  NN  ->  (
( ( A  /  x )  e.  NN  ->  ( x  =  1  \/  x  =  A ) )  <->  ( ( A  /  x )  e.  NN  ->  ( x  =/=  1  ->  x  =  A ) ) ) )
11 impexp 261 . . . . . 6  |-  ( ( ( x  =/=  1  /\  ( A  /  x
)  e.  NN )  ->  x  =  A )  <->  ( x  =/=  1  ->  ( ( A  /  x )  e.  NN  ->  x  =  A ) ) )
12 bi2.04 247 . . . . . 6  |-  ( ( x  =/=  1  -> 
( ( A  /  x )  e.  NN  ->  x  =  A ) )  <->  ( ( A  /  x )  e.  NN  ->  ( x  =/=  1  ->  x  =  A ) ) )
1311, 12bitri 183 . . . . 5  |-  ( ( ( x  =/=  1  /\  ( A  /  x
)  e.  NN )  ->  x  =  A )  <->  ( ( A  /  x )  e.  NN  ->  ( x  =/=  1  ->  x  =  A ) ) )
1410, 13syl6bbr 197 . . . 4  |-  ( x  e.  NN  ->  (
( ( A  /  x )  e.  NN  ->  ( x  =  1  \/  x  =  A ) )  <->  ( (
x  =/=  1  /\  ( A  /  x
)  e.  NN )  ->  x  =  A ) ) )
1514adantl 275 . . 3  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( ( A  /  x )  e.  NN  ->  ( x  =  1  \/  x  =  A ) )  <->  ( (
x  =/=  1  /\  ( A  /  x
)  e.  NN )  ->  x  =  A ) ) )
16 nngt1ne1 8748 . . . . . . 7  |-  ( x  e.  NN  ->  (
1  <  x  <->  x  =/=  1 ) )
1716adantl 275 . . . . . 6  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( 1  <  x  <->  x  =/=  1 ) )
1817anbi1d 460 . . . . 5  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( 1  < 
x  /\  ( A  /  x )  e.  NN ) 
<->  ( x  =/=  1  /\  ( A  /  x
)  e.  NN ) ) )
19 nnz 9066 . . . . . . . . 9  |-  ( ( A  /  x )  e.  NN  ->  ( A  /  x )  e.  ZZ )
20 nnre 8720 . . . . . . . . . . . . 13  |-  ( x  e.  NN  ->  x  e.  RR )
21 gtndiv 9139 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  A  e.  NN  /\  A  <  x )  ->  -.  ( A  /  x
)  e.  ZZ )
22213expia 1183 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  A  e.  NN )  ->  ( A  <  x  ->  -.  ( A  /  x )  e.  ZZ ) )
2320, 22sylan 281 . . . . . . . . . . . 12  |-  ( ( x  e.  NN  /\  A  e.  NN )  ->  ( A  <  x  ->  -.  ( A  /  x )  e.  ZZ ) )
2423con2d 613 . . . . . . . . . . 11  |-  ( ( x  e.  NN  /\  A  e.  NN )  ->  ( ( A  /  x )  e.  ZZ  ->  -.  A  <  x
) )
25 nnre 8720 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  e.  RR )
26 lenlt 7833 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  A  e.  RR )  ->  ( x  <_  A  <->  -.  A  <  x ) )
2720, 25, 26syl2an 287 . . . . . . . . . . 11  |-  ( ( x  e.  NN  /\  A  e.  NN )  ->  ( x  <_  A  <->  -.  A  <  x ) )
2824, 27sylibrd 168 . . . . . . . . . 10  |-  ( ( x  e.  NN  /\  A  e.  NN )  ->  ( ( A  /  x )  e.  ZZ  ->  x  <_  A )
)
2928ancoms 266 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( A  /  x )  e.  ZZ  ->  x  <_  A )
)
3019, 29syl5 32 . . . . . . . 8  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( A  /  x )  e.  NN  ->  x  <_  A )
)
3130pm4.71rd 391 . . . . . . 7  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( A  /  x )  e.  NN  <->  ( x  <_  A  /\  ( A  /  x
)  e.  NN ) ) )
3231anbi2d 459 . . . . . 6  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( 1  < 
x  /\  ( A  /  x )  e.  NN ) 
<->  ( 1  <  x  /\  ( x  <_  A  /\  ( A  /  x
)  e.  NN ) ) ) )
33 3anass 966 . . . . . 6  |-  ( ( 1  <  x  /\  x  <_  A  /\  ( A  /  x )  e.  NN )  <->  ( 1  <  x  /\  (
x  <_  A  /\  ( A  /  x
)  e.  NN ) ) )
3432, 33syl6bbr 197 . . . . 5  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( 1  < 
x  /\  ( A  /  x )  e.  NN ) 
<->  ( 1  <  x  /\  x  <_  A  /\  ( A  /  x
)  e.  NN ) ) )
3518, 34bitr3d 189 . . . 4  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( x  =/=  1  /\  ( A  /  x )  e.  NN )  <->  ( 1  <  x  /\  x  <_  A  /\  ( A  /  x )  e.  NN ) ) )
3635imbi1d 230 . . 3  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( ( x  =/=  1  /\  ( A  /  x )  e.  NN )  ->  x  =  A )  <->  ( (
1  <  x  /\  x  <_  A  /\  ( A  /  x )  e.  NN )  ->  x  =  A ) ) )
3715, 36bitrd 187 . 2  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( ( A  /  x )  e.  NN  ->  ( x  =  1  \/  x  =  A ) )  <->  ( (
1  <  x  /\  x  <_  A  /\  ( A  /  x )  e.  NN )  ->  x  =  A ) ) )
3837ralbidva 2431 1  |-  ( A  e.  NN  ->  ( A. x  e.  NN  ( ( A  /  x )  e.  NN  ->  ( x  =  1  \/  x  =  A ) )  <->  A. x  e.  NN  ( ( 1  <  x  /\  x  <_  A  /\  ( A  /  x )  e.  NN )  ->  x  =  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697  DECID wdc 819    /\ w3a 962    = wceq 1331    e. wcel 1480    =/= wne 2306   A.wral 2414   class class class wbr 3924  (class class class)co 5767   RRcr 7612   1c1 7614    < clt 7793    <_ cle 7794    / cdiv 8425   NNcn 8713   ZZcz 9047
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-id 4210  df-po 4213  df-iso 4214  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-n0 8971  df-z 9048
This theorem is referenced by: (None)
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