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Theorem prime 8815
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 8739 . . . . . . 7  |-  ( x  e.  NN  ->  x  e.  ZZ )
2 1z 8746 . . . . . . . 8  |-  1  e.  ZZ
3 zdceq 8792 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  1  e.  ZZ )  -> DECID  x  =  1 )
42, 3mpan2 416 . . . . . . 7  |-  ( x  e.  ZZ  -> DECID  x  =  1
)
5 dfordc 829 . . . . . . . 8  |-  (DECID  x  =  1  ->  ( (
x  =  1  \/  x  =  A )  <-> 
( -.  x  =  1  ->  x  =  A ) ) )
6 df-ne 2256 . . . . . . . . 9  |-  ( x  =/=  1  <->  -.  x  =  1 )
76imbi1i 236 . . . . . . . 8  |-  ( ( x  =/=  1  ->  x  =  A )  <->  ( -.  x  =  1  ->  x  =  A ) )
85, 7syl6bbr 196 . . . . . . 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 228 . . . . 5  |-  ( x  e.  NN  ->  (
( ( A  /  x )  e.  NN  ->  ( x  =  1  \/  x  =  A ) )  <->  ( ( A  /  x )  e.  NN  ->  ( x  =/=  1  ->  x  =  A ) ) ) )
11 impexp 259 . . . . . 6  |-  ( ( ( x  =/=  1  /\  ( A  /  x
)  e.  NN )  ->  x  =  A )  <->  ( x  =/=  1  ->  ( ( A  /  x )  e.  NN  ->  x  =  A ) ) )
12 bi2.04 246 . . . . . 6  |-  ( ( x  =/=  1  -> 
( ( A  /  x )  e.  NN  ->  x  =  A ) )  <->  ( ( A  /  x )  e.  NN  ->  ( x  =/=  1  ->  x  =  A ) ) )
1311, 12bitri 182 . . . . 5  |-  ( ( ( x  =/=  1  /\  ( A  /  x
)  e.  NN )  ->  x  =  A )  <->  ( ( A  /  x )  e.  NN  ->  ( x  =/=  1  ->  x  =  A ) ) )
1410, 13syl6bbr 196 . . . 4  |-  ( x  e.  NN  ->  (
( ( A  /  x )  e.  NN  ->  ( x  =  1  \/  x  =  A ) )  <->  ( (
x  =/=  1  /\  ( A  /  x
)  e.  NN )  ->  x  =  A ) ) )
1514adantl 271 . . 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 8428 . . . . . . 7  |-  ( x  e.  NN  ->  (
1  <  x  <->  x  =/=  1 ) )
1716adantl 271 . . . . . 6  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( 1  <  x  <->  x  =/=  1 ) )
1817anbi1d 453 . . . . 5  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( 1  < 
x  /\  ( A  /  x )  e.  NN ) 
<->  ( x  =/=  1  /\  ( A  /  x
)  e.  NN ) ) )
19 nnz 8739 . . . . . . . . 9  |-  ( ( A  /  x )  e.  NN  ->  ( A  /  x )  e.  ZZ )
20 nnre 8401 . . . . . . . . . . . . 13  |-  ( x  e.  NN  ->  x  e.  RR )
21 gtndiv 8811 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  A  e.  NN  /\  A  <  x )  ->  -.  ( A  /  x
)  e.  ZZ )
22213expia 1145 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  A  e.  NN )  ->  ( A  <  x  ->  -.  ( A  /  x )  e.  ZZ ) )
2320, 22sylan 277 . . . . . . . . . . . 12  |-  ( ( x  e.  NN  /\  A  e.  NN )  ->  ( A  <  x  ->  -.  ( A  /  x )  e.  ZZ ) )
2423con2d 589 . . . . . . . . . . 11  |-  ( ( x  e.  NN  /\  A  e.  NN )  ->  ( ( A  /  x )  e.  ZZ  ->  -.  A  <  x
) )
25 nnre 8401 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  e.  RR )
26 lenlt 7540 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  A  e.  RR )  ->  ( x  <_  A  <->  -.  A  <  x ) )
2720, 25, 26syl2an 283 . . . . . . . . . . 11  |-  ( ( x  e.  NN  /\  A  e.  NN )  ->  ( x  <_  A  <->  -.  A  <  x ) )
2824, 27sylibrd 167 . . . . . . . . . 10  |-  ( ( x  e.  NN  /\  A  e.  NN )  ->  ( ( A  /  x )  e.  ZZ  ->  x  <_  A )
)
2928ancoms 264 . . . . . . . . 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 386 . . . . . . 7  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( A  /  x )  e.  NN  <->  ( x  <_  A  /\  ( A  /  x
)  e.  NN ) ) )
3231anbi2d 452 . . . . . 6  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( 1  < 
x  /\  ( A  /  x )  e.  NN ) 
<->  ( 1  <  x  /\  ( x  <_  A  /\  ( A  /  x
)  e.  NN ) ) ) )
33 3anass 928 . . . . . 6  |-  ( ( 1  <  x  /\  x  <_  A  /\  ( A  /  x )  e.  NN )  <->  ( 1  <  x  /\  (
x  <_  A  /\  ( A  /  x
)  e.  NN ) ) )
3432, 33syl6bbr 196 . . . . 5  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( 1  < 
x  /\  ( A  /  x )  e.  NN ) 
<->  ( 1  <  x  /\  x  <_  A  /\  ( A  /  x
)  e.  NN ) ) )
3518, 34bitr3d 188 . . . 4  |-  ( ( A  e.  NN  /\  x  e.  NN )  ->  ( ( x  =/=  1  /\  ( A  /  x )  e.  NN )  <->  ( 1  <  x  /\  x  <_  A  /\  ( A  /  x )  e.  NN ) ) )
3635imbi1d 229 . . 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 186 . 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 2376 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 102    <-> wb 103    \/ wo 664  DECID wdc 780    /\ w3a 924    = wceq 1289    e. wcel 1438    =/= wne 2255   A.wral 2359   class class class wbr 3837  (class class class)co 5634   RRcr 7328   1c1 7330    < clt 7501    <_ cle 7502    / cdiv 8113   NNcn 8394   ZZcz 8720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-id 4111  df-po 4114  df-iso 4115  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-n0 8644  df-z 8721
This theorem is referenced by: (None)
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