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Theorem coprm 13778
Description: A prime number either divides an integer or is coprime to it, but not both. (Contributed by Paul Chapman, 22-Jun-2011.)
Assertion
Ref Expression
coprm  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  P  ||  N  <->  ( P  gcd  N )  =  1 ) )

Proof of Theorem coprm
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 prmz 13759 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  ZZ )
2 gcddvds 13691 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( P  gcd  N )  ||  P  /\  ( P  gcd  N ) 
||  N ) )
31, 2sylan 471 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  (
( P  gcd  N
)  ||  P  /\  ( P  gcd  N ) 
||  N ) )
43simprd 463 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  gcd  N )  ||  N )
5 breq1 4288 . . . . 5  |-  ( ( P  gcd  N )  =  P  ->  (
( P  gcd  N
)  ||  N  <->  P  ||  N
) )
64, 5syl5ibcom 220 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  (
( P  gcd  N
)  =  P  ->  P  ||  N ) )
76con3d 133 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  P  ||  N  ->  -.  ( P  gcd  N
)  =  P ) )
8 0nnn 10345 . . . . . . . . 9  |-  -.  0  e.  NN
9 prmnn 13758 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
10 eleq1 2497 . . . . . . . . . 10  |-  ( P  =  0  ->  ( P  e.  NN  <->  0  e.  NN ) )
119, 10syl5ibcom 220 . . . . . . . . 9  |-  ( P  e.  Prime  ->  ( P  =  0  ->  0  e.  NN ) )
128, 11mtoi 178 . . . . . . . 8  |-  ( P  e.  Prime  ->  -.  P  =  0 )
1312intnanrd 908 . . . . . . 7  |-  ( P  e.  Prime  ->  -.  ( P  =  0  /\  N  =  0 ) )
1413adantr 465 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  -.  ( P  =  0  /\  N  =  0
) )
15 gcdn0cl 13690 . . . . . . . 8  |-  ( ( ( P  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( P  =  0  /\  N  =  0 ) )  ->  ( P  gcd  N )  e.  NN )
1615ex 434 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( P  =  0  /\  N  =  0 )  -> 
( P  gcd  N
)  e.  NN ) )
171, 16sylan 471 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  ( P  =  0  /\  N  =  0 )  ->  ( P  gcd  N )  e.  NN ) )
1814, 17mpd 15 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  gcd  N )  e.  NN )
193simpld 459 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  gcd  N )  ||  P )
20 isprm2 13763 . . . . . . . 8  |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
2120simprbi 464 . . . . . . 7  |-  ( P  e.  Prime  ->  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) )
22 breq1 4288 . . . . . . . . 9  |-  ( z  =  ( P  gcd  N )  ->  ( z  ||  P  <->  ( P  gcd  N )  ||  P ) )
23 eqeq1 2443 . . . . . . . . . 10  |-  ( z  =  ( P  gcd  N )  ->  ( z  =  1  <->  ( P  gcd  N )  =  1 ) )
24 eqeq1 2443 . . . . . . . . . 10  |-  ( z  =  ( P  gcd  N )  ->  ( z  =  P  <->  ( P  gcd  N )  =  P ) )
2523, 24orbi12d 709 . . . . . . . . 9  |-  ( z  =  ( P  gcd  N )  ->  ( (
z  =  1  \/  z  =  P )  <-> 
( ( P  gcd  N )  =  1  \/  ( P  gcd  N
)  =  P ) ) )
2622, 25imbi12d 320 . . . . . . . 8  |-  ( z  =  ( P  gcd  N )  ->  ( (
z  ||  P  ->  ( z  =  1  \/  z  =  P ) )  <->  ( ( P  gcd  N )  ||  P  ->  ( ( P  gcd  N )  =  1  \/  ( P  gcd  N )  =  P ) ) ) )
2726rspcv 3062 . . . . . . 7  |-  ( ( P  gcd  N )  e.  NN  ->  ( A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) )  ->  (
( P  gcd  N
)  ||  P  ->  ( ( P  gcd  N
)  =  1  \/  ( P  gcd  N
)  =  P ) ) ) )
2821, 27syl5com 30 . . . . . 6  |-  ( P  e.  Prime  ->  ( ( P  gcd  N )  e.  NN  ->  (
( P  gcd  N
)  ||  P  ->  ( ( P  gcd  N
)  =  1  \/  ( P  gcd  N
)  =  P ) ) ) )
2928adantr 465 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  (
( P  gcd  N
)  e.  NN  ->  ( ( P  gcd  N
)  ||  P  ->  ( ( P  gcd  N
)  =  1  \/  ( P  gcd  N
)  =  P ) ) ) )
3018, 19, 29mp2d 45 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  (
( P  gcd  N
)  =  1  \/  ( P  gcd  N
)  =  P ) )
31 biorf 405 . . . . 5  |-  ( -.  ( P  gcd  N
)  =  P  -> 
( ( P  gcd  N )  =  1  <->  (
( P  gcd  N
)  =  P  \/  ( P  gcd  N )  =  1 ) ) )
32 orcom 387 . . . . 5  |-  ( ( ( P  gcd  N
)  =  P  \/  ( P  gcd  N )  =  1 )  <->  ( ( P  gcd  N )  =  1  \/  ( P  gcd  N )  =  P ) )
3331, 32syl6bb 261 . . . 4  |-  ( -.  ( P  gcd  N
)  =  P  -> 
( ( P  gcd  N )  =  1  <->  (
( P  gcd  N
)  =  1  \/  ( P  gcd  N
)  =  P ) ) )
3430, 33syl5ibrcom 222 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  ( P  gcd  N
)  =  P  -> 
( P  gcd  N
)  =  1 ) )
357, 34syld 44 . 2  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  P  ||  N  -> 
( P  gcd  N
)  =  1 ) )
36 iddvds 13538 . . . . . . 7  |-  ( P  e.  ZZ  ->  P  ||  P )
371, 36syl 16 . . . . . 6  |-  ( P  e.  Prime  ->  P  ||  P )
3837adantr 465 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  P  ||  P )
39 dvdslegcd 13692 . . . . . . . . 9  |-  ( ( ( P  e.  ZZ  /\  P  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( P  =  0  /\  N  =  0 ) )  -> 
( ( P  ||  P  /\  P  ||  N
)  ->  P  <_  ( P  gcd  N ) ) )
4039ex 434 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  P  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( P  =  0  /\  N  =  0 )  ->  ( ( P  ||  P  /\  P  ||  N )  ->  P  <_  ( P  gcd  N
) ) ) )
41403anidm12 1275 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( P  =  0  /\  N  =  0 )  -> 
( ( P  ||  P  /\  P  ||  N
)  ->  P  <_  ( P  gcd  N ) ) ) )
421, 41sylan 471 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  ( P  =  0  /\  N  =  0 )  ->  ( ( P  ||  P  /\  P  ||  N )  ->  P  <_  ( P  gcd  N
) ) ) )
4314, 42mpd 15 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  (
( P  ||  P  /\  P  ||  N )  ->  P  <_  ( P  gcd  N ) ) )
4438, 43mpand 675 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  ||  N  ->  P  <_  ( P  gcd  N
) ) )
45 prmuz2 13773 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
46 eluz2b1 10918 . . . . . . . 8  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  ZZ  /\  1  < 
P ) )
4745, 46sylib 196 . . . . . . 7  |-  ( P  e.  Prime  ->  ( P  e.  ZZ  /\  1  <  P ) )
4847simprd 463 . . . . . 6  |-  ( P  e.  Prime  ->  1  < 
P )
4948adantr 465 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  1  <  P )
501zred 10739 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  RR )
5150adantr 465 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  P  e.  RR )
5218nnred 10329 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  gcd  N )  e.  RR )
53 1re 9377 . . . . . . 7  |-  1  e.  RR
54 ltletr 9458 . . . . . . 7  |-  ( ( 1  e.  RR  /\  P  e.  RR  /\  ( P  gcd  N )  e.  RR )  ->  (
( 1  <  P  /\  P  <_  ( P  gcd  N ) )  ->  1  <  ( P  gcd  N ) ) )
5553, 54mp3an1 1301 . . . . . 6  |-  ( ( P  e.  RR  /\  ( P  gcd  N )  e.  RR )  -> 
( ( 1  < 
P  /\  P  <_  ( P  gcd  N ) )  ->  1  <  ( P  gcd  N ) ) )
5651, 52, 55syl2anc 661 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  (
( 1  <  P  /\  P  <_  ( P  gcd  N ) )  ->  1  <  ( P  gcd  N ) ) )
5749, 56mpand 675 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  <_  ( P  gcd  N )  ->  1  <  ( P  gcd  N ) ) )
58 ltneOLD 9464 . . . . . 6  |-  ( ( 1  e.  RR  /\  ( P  gcd  N )  e.  RR  /\  1  <  ( P  gcd  N
) )  ->  ( P  gcd  N )  =/=  1 )
59583expia 1189 . . . . 5  |-  ( ( 1  e.  RR  /\  ( P  gcd  N )  e.  RR )  -> 
( 1  <  ( P  gcd  N )  -> 
( P  gcd  N
)  =/=  1 ) )
6053, 52, 59sylancr 663 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  (
1  <  ( P  gcd  N )  ->  ( P  gcd  N )  =/=  1 ) )
6144, 57, 603syld 55 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  ||  N  ->  ( P  gcd  N )  =/=  1 ) )
6261necon2bd 2654 . 2  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  (
( P  gcd  N
)  =  1  ->  -.  P  ||  N ) )
6335, 62impbid 191 1  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  P  ||  N  <->  ( P  gcd  N )  =  1 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2600   A.wral 2709   class class class wbr 4285   ` cfv 5411  (class class class)co 6086   RRcr 9273   0cc0 9274   1c1 9275    < clt 9410    <_ cle 9411   NNcn 10314   2c2 10363   ZZcz 10638   ZZ>=cuz 10853    || cdivides 13527    gcd cgcd 13682   Primecprime 13755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-int 4122  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-dvds 13528  df-gcd 13683  df-prm 13756
This theorem is referenced by:  prmrp  13779  euclemma  13786  phiprmpw  13843  fermltl  13851  prmdiv  13852  prmdiveq  13853  prmpwdvds  13957  1259lem5  14151  2503lem3  14155  4001lem4  14160  gexexlem  16323  ablfac1lem  16555  ablfac1eu  16560  pgpfac1lem3  16564  perfect1  22536  perfectlem1  22537  perfectlem2  22538  lgslem1  22604  lgsqrlem2  22650  lgsqr  22654  lgsquad2lem2  22667  2sqblem  22685  rpvmasumlem  22705  dchrisum0flblem2  22727  nn0prpwlem  28460
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