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Theorem coprm 13092
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 13075 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  ZZ )
2 gcddvds 13007 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( P  gcd  N )  ||  P  /\  ( P  gcd  N ) 
||  N ) )
31, 2sylan 458 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  (
( P  gcd  N
)  ||  P  /\  ( P  gcd  N ) 
||  N ) )
43simprd 450 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  gcd  N )  ||  N )
5 breq1 4207 . . . . 5  |-  ( ( P  gcd  N )  =  P  ->  (
( P  gcd  N
)  ||  N  <->  P  ||  N
) )
64, 5syl5ibcom 212 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  (
( P  gcd  N
)  =  P  ->  P  ||  N ) )
76con3d 127 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  P  ||  N  ->  -.  ( P  gcd  N
)  =  P ) )
8 0nnn 10023 . . . . . . . . 9  |-  -.  0  e.  NN
9 prmnn 13074 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
10 eleq1 2495 . . . . . . . . . 10  |-  ( P  =  0  ->  ( P  e.  NN  <->  0  e.  NN ) )
119, 10syl5ibcom 212 . . . . . . . . 9  |-  ( P  e.  Prime  ->  ( P  =  0  ->  0  e.  NN ) )
128, 11mtoi 171 . . . . . . . 8  |-  ( P  e.  Prime  ->  -.  P  =  0 )
1312intnanrd 884 . . . . . . 7  |-  ( P  e.  Prime  ->  -.  ( P  =  0  /\  N  =  0 ) )
1413adantr 452 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  -.  ( P  =  0  /\  N  =  0
) )
15 gcdn0cl 13006 . . . . . . . 8  |-  ( ( ( P  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( P  =  0  /\  N  =  0 ) )  ->  ( P  gcd  N )  e.  NN )
1615ex 424 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( P  =  0  /\  N  =  0 )  -> 
( P  gcd  N
)  e.  NN ) )
171, 16sylan 458 . . . . . 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 446 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  gcd  N )  ||  P )
20 isprm2 13079 . . . . . . . 8  |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
2120simprbi 451 . . . . . . 7  |-  ( P  e.  Prime  ->  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) )
22 breq1 4207 . . . . . . . . 9  |-  ( z  =  ( P  gcd  N )  ->  ( z  ||  P  <->  ( P  gcd  N )  ||  P ) )
23 eqeq1 2441 . . . . . . . . . 10  |-  ( z  =  ( P  gcd  N )  ->  ( z  =  1  <->  ( P  gcd  N )  =  1 ) )
24 eqeq1 2441 . . . . . . . . . 10  |-  ( z  =  ( P  gcd  N )  ->  ( z  =  P  <->  ( P  gcd  N )  =  P ) )
2523, 24orbi12d 691 . . . . . . . . 9  |-  ( z  =  ( P  gcd  N )  ->  ( (
z  =  1  \/  z  =  P )  <-> 
( ( P  gcd  N )  =  1  \/  ( P  gcd  N
)  =  P ) ) )
2622, 25imbi12d 312 . . . . . . . 8  |-  ( z  =  ( P  gcd  N )  ->  ( (
z  ||  P  ->  ( z  =  1  \/  z  =  P ) )  <->  ( ( P  gcd  N )  ||  P  ->  ( ( P  gcd  N )  =  1  \/  ( P  gcd  N )  =  P ) ) ) )
2726rspcv 3040 . . . . . . 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 28 . . . . . 6  |-  ( P  e.  Prime  ->  ( ( P  gcd  N )  e.  NN  ->  (
( P  gcd  N
)  ||  P  ->  ( ( P  gcd  N
)  =  1  \/  ( P  gcd  N
)  =  P ) ) ) )
2928adantr 452 . . . . 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 43 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  (
( P  gcd  N
)  =  1  \/  ( P  gcd  N
)  =  P ) )
31 biorf 395 . . . . 5  |-  ( -.  ( P  gcd  N
)  =  P  -> 
( ( P  gcd  N )  =  1  <->  (
( P  gcd  N
)  =  P  \/  ( P  gcd  N )  =  1 ) ) )
32 orcom 377 . . . . 5  |-  ( ( ( P  gcd  N
)  =  P  \/  ( P  gcd  N )  =  1 )  <->  ( ( P  gcd  N )  =  1  \/  ( P  gcd  N )  =  P ) )
3331, 32syl6bb 253 . . . 4  |-  ( -.  ( P  gcd  N
)  =  P  -> 
( ( P  gcd  N )  =  1  <->  (
( P  gcd  N
)  =  1  \/  ( P  gcd  N
)  =  P ) ) )
3430, 33syl5ibrcom 214 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  ( P  gcd  N
)  =  P  -> 
( P  gcd  N
)  =  1 ) )
357, 34syld 42 . 2  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  P  ||  N  -> 
( P  gcd  N
)  =  1 ) )
36 iddvds 12855 . . . . . . 7  |-  ( P  e.  ZZ  ->  P  ||  P )
371, 36syl 16 . . . . . 6  |-  ( P  e.  Prime  ->  P  ||  P )
3837adantr 452 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  P  ||  P )
39 dvdslegcd 13008 . . . . . . . . 9  |-  ( ( ( P  e.  ZZ  /\  P  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( P  =  0  /\  N  =  0 ) )  -> 
( ( P  ||  P  /\  P  ||  N
)  ->  P  <_  ( P  gcd  N ) ) )
4039ex 424 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  P  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( P  =  0  /\  N  =  0 )  ->  ( ( P  ||  P  /\  P  ||  N )  ->  P  <_  ( P  gcd  N
) ) ) )
41403anidm12 1241 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( P  =  0  /\  N  =  0 )  -> 
( ( P  ||  P  /\  P  ||  N
)  ->  P  <_  ( P  gcd  N ) ) ) )
421, 41sylan 458 . . . . . 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 657 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  ||  N  ->  P  <_  ( P  gcd  N
) ) )
45 prmuz2 13089 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
46 eluz2b1 10539 . . . . . . . 8  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  ZZ  /\  1  < 
P ) )
4745, 46sylib 189 . . . . . . 7  |-  ( P  e.  Prime  ->  ( P  e.  ZZ  /\  1  <  P ) )
4847simprd 450 . . . . . 6  |-  ( P  e.  Prime  ->  1  < 
P )
4948adantr 452 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  1  <  P )
501zred 10367 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  RR )
5150adantr 452 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  P  e.  RR )
5218nnred 10007 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  gcd  N )  e.  RR )
53 1re 9082 . . . . . . 7  |-  1  e.  RR
54 ltletr 9158 . . . . . . 7  |-  ( ( 1  e.  RR  /\  P  e.  RR  /\  ( P  gcd  N )  e.  RR )  ->  (
( 1  <  P  /\  P  <_  ( P  gcd  N ) )  ->  1  <  ( P  gcd  N ) ) )
5553, 54mp3an1 1266 . . . . . 6  |-  ( ( P  e.  RR  /\  ( P  gcd  N )  e.  RR )  -> 
( ( 1  < 
P  /\  P  <_  ( P  gcd  N ) )  ->  1  <  ( P  gcd  N ) ) )
5651, 52, 55syl2anc 643 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  (
( 1  <  P  /\  P  <_  ( P  gcd  N ) )  ->  1  <  ( P  gcd  N ) ) )
5749, 56mpand 657 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  <_  ( P  gcd  N )  ->  1  <  ( P  gcd  N ) ) )
58 ltneOLD 9163 . . . . . 6  |-  ( ( 1  e.  RR  /\  ( P  gcd  N )  e.  RR  /\  1  <  ( P  gcd  N
) )  ->  ( P  gcd  N )  =/=  1 )
59583expia 1155 . . . . 5  |-  ( ( 1  e.  RR  /\  ( P  gcd  N )  e.  RR )  -> 
( 1  <  ( P  gcd  N )  -> 
( P  gcd  N
)  =/=  1 ) )
6053, 52, 59sylancr 645 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  (
1  <  ( P  gcd  N )  ->  ( P  gcd  N )  =/=  1 ) )
6144, 57, 603syld 53 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  ||  N  ->  ( P  gcd  N )  =/=  1 ) )
6261necon2bd 2647 . 2  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  (
( P  gcd  N
)  =  1  ->  -.  P  ||  N ) )
6335, 62impbid 184 1  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  P  ||  N  <->  ( P  gcd  N )  =  1 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   RRcr 8981   0cc0 8982   1c1 8983    < clt 9112    <_ cle 9113   NNcn 9992   2c2 10041   ZZcz 10274   ZZ>=cuz 10480    || cdivides 12844    gcd cgcd 12998   Primecprime 13071
This theorem is referenced by:  prmrp  13093  euclemma  13100  phiprmpw  13157  fermltl  13165  prmdiv  13166  prmdiveq  13167  prmpwdvds  13264  1259lem5  13446  2503lem3  13450  4001lem4  13455  gexexlem  15459  ablfac1lem  15618  ablfac1eu  15623  pgpfac1lem3  15627  perfect1  21004  perfectlem1  21005  perfectlem2  21006  lgslem1  21072  lgsqrlem2  21118  lgsqr  21122  lgsquad2lem2  21135  2sqblem  21153  rpvmasumlem  21173  dchrisum0flblem2  21195  nn0prpwlem  26316
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999  df-prm 13072
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