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Theorem prime 10139
Description: Two ways to express " A is a prime number (or 1)." See also isprm 12807. (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 bi2.04 350 . . . 4  |-  ( ( x  =/=  1  -> 
( ( A  /  x )  e.  NN  ->  x  =  A ) )  <->  ( ( A  /  x )  e.  NN  ->  ( x  =/=  1  ->  x  =  A ) ) )
2 impexp 433 . . . 4  |-  ( ( ( x  =/=  1  /\  ( A  /  x
)  e.  NN )  ->  x  =  A )  <->  ( x  =/=  1  ->  ( ( A  /  x )  e.  NN  ->  x  =  A ) ) )
3 neor 2563 . . . . 5  |-  ( ( x  =  1  \/  x  =  A )  <-> 
( x  =/=  1  ->  x  =  A ) )
43imbi2i 303 . . . 4  |-  ( ( ( A  /  x
)  e.  NN  ->  ( x  =  1  \/  x  =  A ) )  <->  ( ( A  /  x )  e.  NN  ->  ( x  =/=  1  ->  x  =  A ) ) )
51, 2, 43bitr4ri 269 . . 3  |-  ( ( ( A  /  x
)  e.  NN  ->  ( x  =  1  \/  x  =  A ) )  <->  ( ( x  =/=  1  /\  ( A  /  x )  e.  NN )  ->  x  =  A ) )
6 nngt1ne1 9818 . . . . . . 7  |-  ( x  e.  NN  ->  (
1  <  x  <->  x  =/=  1 ) )
76adantl 452 . . . . . 6  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( 1  <  x  <->  x  =/=  1 ) )
87anbi1d 685 . . . . 5  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( 1  < 
x  /\  ( A  /  x )  e.  NN ) 
<->  ( x  =/=  1  /\  ( A  /  x
)  e.  NN ) ) )
9 nnz 10092 . . . . . . . . 9  |-  ( ( A  /  x )  e.  NN  ->  ( A  /  x )  e.  ZZ )
10 nnre 9798 . . . . . . . . . . . . 13  |-  ( x  e.  NN  ->  x  e.  RR )
11 gtndiv 10136 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  A  e.  NN  /\  A  <  x )  ->  -.  ( A  /  x
)  e.  ZZ )
12113expia 1153 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  A  e.  NN )  ->  ( A  <  x  ->  -.  ( A  /  x )  e.  ZZ ) )
1310, 12sylan 457 . . . . . . . . . . . 12  |-  ( ( x  e.  NN  /\  A  e.  NN )  ->  ( A  <  x  ->  -.  ( A  /  x )  e.  ZZ ) )
1413con2d 107 . . . . . . . . . . 11  |-  ( ( x  e.  NN  /\  A  e.  NN )  ->  ( ( A  /  x )  e.  ZZ  ->  -.  A  <  x
) )
15 nnre 9798 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  e.  RR )
16 lenlt 8946 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  A  e.  RR )  ->  ( x  <_  A  <->  -.  A  <  x ) )
1710, 15, 16syl2an 463 . . . . . . . . . . 11  |-  ( ( x  e.  NN  /\  A  e.  NN )  ->  ( x  <_  A  <->  -.  A  <  x ) )
1814, 17sylibrd 225 . . . . . . . . . 10  |-  ( ( x  e.  NN  /\  A  e.  NN )  ->  ( ( A  /  x )  e.  ZZ  ->  x  <_  A )
)
1918ancoms 439 . . . . . . . . 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 616 . . . . . . 7  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( A  /  x )  e.  NN  <->  ( x  <_  A  /\  ( A  /  x
)  e.  NN ) ) )
2221anbi2d 684 . . . . . 6  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( 1  < 
x  /\  ( A  /  x )  e.  NN ) 
<->  ( 1  <  x  /\  ( x  <_  A  /\  ( A  /  x
)  e.  NN ) ) ) )
23 3anass 938 . . . . . 6  |-  ( ( 1  <  x  /\  x  <_  A  /\  ( A  /  x )  e.  NN )  <->  ( 1  <  x  /\  (
x  <_  A  /\  ( A  /  x
)  e.  NN ) ) )
2422, 23syl6bbr 254 . . . . 5  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( 1  < 
x  /\  ( A  /  x )  e.  NN ) 
<->  ( 1  <  x  /\  x  <_  A  /\  ( A  /  x
)  e.  NN ) ) )
258, 24bitr3d 246 . . . 4  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( x  =/=  1  /\  ( A  /  x )  e.  NN )  <->  ( 1  <  x  /\  x  <_  A  /\  ( A  /  x )  e.  NN ) ) )
2625imbi1d 308 . . 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 248 . 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 2593 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   A.wral 2577   class class class wbr 4060  (class class class)co 5900   RRcr 8781   1c1 8783    < clt 8912    <_ cle 8913    / cdiv 9468   NNcn 9791   ZZcz 10071
This theorem is referenced by:  infpnlem1  13004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-n0 10013  df-z 10072
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