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Theorem coprm 13789
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 13770 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  ZZ )
2 gcddvds 13702 . . . . . . 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 4298 . . . . 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 10356 . . . . . . . . 9  |-  -.  0  e.  NN
9 prmnn 13769 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
10 eleq1 2503 . . . . . . . . . 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 13701 . . . . . . . 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 13774 . . . . . . . 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 4298 . . . . . . . . 9  |-  ( z  =  ( P  gcd  N )  ->  ( z  ||  P  <->  ( P  gcd  N )  ||  P ) )
23 eqeq1 2449 . . . . . . . . . 10  |-  ( z  =  ( P  gcd  N )  ->  ( z  =  1  <->  ( P  gcd  N )  =  1 ) )
24 eqeq1 2449 . . . . . . . . . 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 3072 . . . . . . 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 13549 . . . . . . 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 13703 . . . . . . . . 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 13784 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
46 eluz2b1 10929 . . . . . . . 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 10750 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  RR )
5150adantr 465 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  P  e.  RR )
5218nnred 10340 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  gcd  N )  e.  RR )
53 1re 9388 . . . . . . 7  |-  1  e.  RR
54 ltletr 9469 . . . . . . 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 9475 . . . . . 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 2663 . 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 2609   A.wral 2718   class class class wbr 4295   ` cfv 5421  (class class class)co 6094   RRcr 9284   0cc0 9285   1c1 9286    < clt 9421    <_ cle 9422   NNcn 10325   2c2 10374   ZZcz 10649   ZZ>=cuz 10864    || cdivides 13538    gcd cgcd 13693   Primecprime 13766
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 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-2o 6924  df-oadd 6927  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-sup 7694  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-n0 10583  df-z 10650  df-uz 10865  df-rp 10995  df-seq 11810  df-exp 11869  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-dvds 13539  df-gcd 13694  df-prm 13767
This theorem is referenced by:  prmrp  13790  euclemma  13797  phiprmpw  13854  fermltl  13862  prmdiv  13863  prmdiveq  13864  prmpwdvds  13968  1259lem5  14162  2503lem3  14166  4001lem4  14171  gexexlem  16337  ablfac1lem  16572  ablfac1eu  16577  pgpfac1lem3  16581  perfect1  22570  perfectlem1  22571  perfectlem2  22572  lgslem1  22638  lgsqrlem2  22684  lgsqr  22688  lgsquad2lem2  22701  2sqblem  22719  rpvmasumlem  22739  dchrisum0flblem2  22761  nn0prpwlem  28520
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