MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prime Unicode version

Theorem prime 10094
Description: Two ways to express " A is a prime number (or 1)." See also isprm 12762. (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 2532 . . . . 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 9775 . . . . . . 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 10047 . . . . . . . . 9  |-  ( ( A  /  x )  e.  NN  ->  ( A  /  x )  e.  ZZ )
10 nnre 9755 . . . . . . . . . . . . 13  |-  ( x  e.  NN  ->  x  e.  RR )
11 gtndiv 10091 . . . . . . . . . . . . . 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 9755 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  e.  RR )
16 lenlt 8903 . . . . . . . . . . . 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 2561 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 1625    e. wcel 1686    =/= wne 2448   A.wral 2545   class class class wbr 4025  (class class class)co 5860   RRcr 8738   1c1 8740    < clt 8869    <_ cle 8870    / cdiv 9425   NNcn 9748   ZZcz 10026
This theorem is referenced by:  infpnlem1  12959
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-riota 6306  df-recs 6390  df-rdg 6425  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-n0 9968  df-z 10027
  Copyright terms: Public domain W3C validator