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Theorem prime 9480
Description: Two ways to express " A is a prime number (or 1)." See also isprm 11885. (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 bi2.04 348 . . . 4  |-  ( ( x  =/=  1  -> 
( ( A  /  x )  e.  NN  ->  x  =  A ) )  <->  ( ( A  /  x )  e.  NN  ->  ( x  =/=  1  ->  x  =  A ) ) )
2 impexp 427 . . . 4  |-  ( ( ( x  =/=  1  /\  ( A  /  x
)  e.  NN )  ->  x  =  A )  <->  ( x  =/=  1  ->  ( ( A  /  x )  e.  NN  ->  x  =  A ) ) )
3 neor 2268 . . . . 5  |-  ( ( x  =  1  \/  x  =  A )  <-> 
( x  =/=  1  ->  x  =  A ) )
43imbi2i 301 . . . 4  |-  ( ( ( A  /  x
)  e.  NN  ->  ( x  =  1  \/  x  =  A ) )  <->  ( ( A  /  x )  e.  NN  ->  ( x  =/=  1  ->  x  =  A ) ) )
51, 2, 43bitr4ri 267 . . 3  |-  ( ( ( A  /  x
)  e.  NN  ->  ( x  =  1  \/  x  =  A ) )  <->  ( ( x  =/=  1  /\  ( A  /  x )  e.  NN )  ->  x  =  A ) )
6 nngt1ne1 9164 . . . . . . 7  |-  ( x  e.  NN  ->  (
1  <  x  <->  x  =/=  1 ) )
76adantl 446 . . . . . 6  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( 1  <  x  <->  x  =/=  1 ) )
87anbi1d 676 . . . . 5  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( 1  < 
x  /\  ( A  /  x )  e.  NN ) 
<->  ( x  =/=  1  /\  ( A  /  x
)  e.  NN ) ) )
9 nnz 9433 . . . . . . . . 9  |-  ( ( A  /  x )  e.  NN  ->  ( A  /  x )  e.  ZZ )
10 nnre 9144 . . . . . . . . . . . . 13  |-  ( x  e.  NN  ->  x  e.  RR )
11 gtndiv 9477 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  A  e.  NN  /\  A  <  x )  ->  -.  ( A  /  x
)  e.  ZZ )
12113expia 1112 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  A  e.  NN )  ->  ( A  <  x  ->  -.  ( A  /  x )  e.  ZZ ) )
1310, 12sylan 451 . . . . . . . . . . . 12  |-  ( ( x  e.  NN  /\  A  e.  NN )  ->  ( A  <  x  ->  -.  ( A  /  x )  e.  ZZ ) )
1413con2d 105 . . . . . . . . . . 11  |-  ( ( x  e.  NN  /\  A  e.  NN )  ->  ( ( A  /  x )  e.  ZZ  ->  -.  A  <  x
) )
15 nnre 9144 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  e.  RR )
16 lenlt 8330 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  A  e.  RR )  ->  ( x  <_  A  <->  -.  A  <  x ) )
1710, 15, 16syl2an 457 . . . . . . . . . . 11  |-  ( ( x  e.  NN  /\  A  e.  NN )  ->  ( x  <_  A  <->  -.  A  <  x ) )
1814, 17sylibrd 223 . . . . . . . . . 10  |-  ( ( x  e.  NN  /\  A  e.  NN )  ->  ( ( A  /  x )  e.  ZZ  ->  x  <_  A )
)
1918ancoms 433 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( A  /  x )  e.  ZZ  ->  x  <_  A )
)
209, 19syl5 28 . . . . . . . 8  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( A  /  x )  e.  NN  ->  x  <_  A )
)
2120pm4.71rd 609 . . . . . . 7  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( A  /  x )  e.  NN  <->  ( x  <_  A  /\  ( A  /  x
)  e.  NN ) ) )
2221anbi2d 675 . . . . . 6  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( 1  < 
x  /\  ( A  /  x )  e.  NN ) 
<->  ( 1  <  x  /\  ( x  <_  A  /\  ( A  /  x
)  e.  NN ) ) ) )
23 3anass 900 . . . . . 6  |-  ( ( 1  <  x  /\  x  <_  A  /\  ( A  /  x )  e.  NN )  <->  ( 1  <  x  /\  (
x  <_  A  /\  ( A  /  x
)  e.  NN ) ) )
2422, 23syl6bbr 252 . . . . 5  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( 1  < 
x  /\  ( A  /  x )  e.  NN ) 
<->  ( 1  <  x  /\  x  <_  A  /\  ( A  /  x
)  e.  NN ) ) )
258, 24bitr3d 244 . . . 4  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( x  =/=  1  /\  ( A  /  x )  e.  NN )  <->  ( 1  <  x  /\  x  <_  A  /\  ( A  /  x )  e.  NN ) ) )
2625imbi1d 306 . . 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 ) ) )
275, 26syl5bb 246 . 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 ) ) )
2827ralbidva 2293 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    <-> wb 174    \/ wo 355    /\ wa 356    /\ w3a 896    = wceq 1524    e. wcel 1526    =/= wne 2187   A.wral 2279   class class class wbr 3596  (class class class)co 5370   RRcr 8167   1c1 8169    <_ cle 8294    < clt 8298    / cdiv 8828   NNcn 9137   ZZcz 9412
This theorem is referenced by:  infpnlem1  12082
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1446  ax-6 1447  ax-7 1448  ax-gen 1449  ax-8 1528  ax-11 1529  ax-13 1530  ax-14 1531  ax-17 1533  ax-12o 1567  ax-10 1581  ax-9 1587  ax-4 1594  ax-16 1780  ax-ext 2051  ax-sep 3711  ax-nul 3719  ax-pow 3755  ax-pr 3779  ax-un 4071  ax-resscn 8224  ax-1cn 8225  ax-icn 8226  ax-addcl 8227  ax-addrcl 8228  ax-mulcl 8229  ax-mulrcl 8230  ax-mulcom 8231  ax-addass 8232  ax-mulass 8233  ax-distr 8234  ax-i2m1 8235  ax-1ne0 8236  ax-1rid 8237  ax-rnegex 8238  ax-rrecex 8239  ax-cnre 8240  ax-pre-lttri 8241  ax-pre-lttrn 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 897  df-3an 898  df-tru 1259  df-ex 1451  df-sb 1741  df-eu 1963  df-mo 1964  df-clab 2057  df-cleq 2062  df-clel 2065  df-ne 2189  df-nel 2190  df-ral 2283  df-rex 2284  df-reu 2285  df-rab 2286  df-v 2482  df-sbc 2656  df-csb 2738  df-dif 2801  df-un 2803  df-in 2805  df-ss 2809  df-pss 2811  df-nul 3078  df-if 3187  df-pw 3248  df-sn 3266  df-pr 3267  df-tp 3268  df-op 3269  df-uni 3435  df-iun 3512  df-br 3597  df-opab 3651  df-mpt 3652  df-tr 3684  df-eprel 3866  df-id 3870  df-po 3875  df-so 3876  df-fr 3913  df-we 3915  df-ord 3956  df-on 3957  df-lim 3958  df-suc 3959  df-om 4234  df-xp 4280  df-rel 4281  df-cnv 4282  df-co 4283  df-dm 4284  df-rn 4285  df-res 4286  df-ima 4287  df-fun 4288  df-fn 4289  df-f 4290  df-f1 4291  df-fo 4292  df-f1o 4293  df-fv 4294  df-ov 5373  df-oprab 5374  df-mpt2 5375  df-iota 5780  df-recs 5853  df-rdg 5888  df-er 6125  df-en 6312  df-dom 6313  df-sdom 6314  df-riota 6478  df-pnf 8299  df-mnf 8300  df-xr 8301  df-ltxr 8302  df-le 8303  df-sub 8464  df-neg 8465  df-div 8829  df-n 9138  df-n0 9354  df-z 9413
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