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Theorem coprm 14089
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 14069 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  ZZ )
2 gcddvds 14001 . . . . . . 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 4443 . . . . 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 10556 . . . . . . . . 9  |-  -.  0  e.  NN
9 prmnn 14068 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
10 eleq1 2532 . . . . . . . . . 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 910 . . . . . . 7  |-  ( P  e.  Prime  ->  -.  ( P  =  0  /\  N  =  0 ) )
1413adantr 465 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  -.  ( P  =  0  /\  N  =  0
) )
15 gcdn0cl 14000 . . . . . . . 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 14073 . . . . . . . 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 4443 . . . . . . . . 9  |-  ( z  =  ( P  gcd  N )  ->  ( z  ||  P  <->  ( P  gcd  N )  ||  P ) )
23 eqeq1 2464 . . . . . . . . . 10  |-  ( z  =  ( P  gcd  N )  ->  ( z  =  1  <->  ( P  gcd  N )  =  1 ) )
24 eqeq1 2464 . . . . . . . . . 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 3203 . . . . . . 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 13847 . . . . . . 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 14002 . . . . . . . . 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 1280 . . . . . . 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 14083 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
46 eluz2b1 11142 . . . . . . . 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 10955 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  RR )
5150adantr 465 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  P  e.  RR )
5218nnred 10540 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  gcd  N )  e.  RR )
53 1re 9584 . . . . . . 7  |-  1  e.  RR
54 ltletr 9665 . . . . . . 7  |-  ( ( 1  e.  RR  /\  P  e.  RR  /\  ( P  gcd  N )  e.  RR )  ->  (
( 1  <  P  /\  P  <_  ( P  gcd  N ) )  ->  1  <  ( P  gcd  N ) ) )
5553, 54mp3an1 1306 . . . . . 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 9671 . . . . . 6  |-  ( ( 1  e.  RR  /\  ( P  gcd  N )  e.  RR  /\  1  <  ( P  gcd  N
) )  ->  ( P  gcd  N )  =/=  1 )
59583expia 1193 . . . . 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 2675 . 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 968    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   RRcr 9480   0cc0 9481   1c1 9482    < clt 9617    <_ cle 9618   NNcn 10525   2c2 10574   ZZcz 10853   ZZ>=cuz 11071    || cdivides 13836    gcd cgcd 13992   Primecprime 14065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-dvds 13837  df-gcd 13993  df-prm 14066
This theorem is referenced by:  prmrp  14090  euclemma  14097  phiprmpw  14154  fermltl  14162  prmdiv  14163  prmdiveq  14164  prmpwdvds  14270  1259lem5  14464  2503lem3  14468  4001lem4  14473  gexexlem  16644  ablfac1lem  16902  ablfac1eu  16907  pgpfac1lem3  16911  perfect1  23224  perfectlem1  23225  perfectlem2  23226  lgslem1  23292  lgsqrlem2  23338  lgsqr  23342  lgsquad2lem2  23355  2sqblem  23373  rpvmasumlem  23393  dchrisum0flblem2  23415  nn0prpwlem  29704
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