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Theorem coprm 14656
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 14625 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  ZZ )
2 gcddvds 14476 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( P  gcd  N )  ||  P  /\  ( P  gcd  N ) 
||  N ) )
31, 2sylan 473 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  (
( P  gcd  N
)  ||  P  /\  ( P  gcd  N ) 
||  N ) )
43simprd 464 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  gcd  N )  ||  N )
5 breq1 4426 . . . . 5  |-  ( ( P  gcd  N )  =  P  ->  (
( P  gcd  N
)  ||  N  <->  P  ||  N
) )
64, 5syl5ibcom 223 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  (
( P  gcd  N
)  =  P  ->  P  ||  N ) )
76con3d 138 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  P  ||  N  ->  -.  ( P  gcd  N
)  =  P ) )
8 0nnn 10648 . . . . . . . . 9  |-  -.  0  e.  NN
9 prmnn 14624 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
10 eleq1 2495 . . . . . . . . . 10  |-  ( P  =  0  ->  ( P  e.  NN  <->  0  e.  NN ) )
119, 10syl5ibcom 223 . . . . . . . . 9  |-  ( P  e.  Prime  ->  ( P  =  0  ->  0  e.  NN ) )
128, 11mtoi 181 . . . . . . . 8  |-  ( P  e.  Prime  ->  -.  P  =  0 )
1312intnanrd 925 . . . . . . 7  |-  ( P  e.  Prime  ->  -.  ( P  =  0  /\  N  =  0 ) )
1413adantr 466 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  -.  ( P  =  0  /\  N  =  0
) )
15 gcdn0cl 14475 . . . . . . . 8  |-  ( ( ( P  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( P  =  0  /\  N  =  0 ) )  ->  ( P  gcd  N )  e.  NN )
1615ex 435 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( P  =  0  /\  N  =  0 )  -> 
( P  gcd  N
)  e.  NN ) )
171, 16sylan 473 . . . . . 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 460 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  gcd  N )  ||  P )
20 isprm2 14631 . . . . . . . 8  |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
2120simprbi 465 . . . . . . 7  |-  ( P  e.  Prime  ->  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) )
22 breq1 4426 . . . . . . . . 9  |-  ( z  =  ( P  gcd  N )  ->  ( z  ||  P  <->  ( P  gcd  N )  ||  P ) )
23 eqeq1 2426 . . . . . . . . . 10  |-  ( z  =  ( P  gcd  N )  ->  ( z  =  1  <->  ( P  gcd  N )  =  1 ) )
24 eqeq1 2426 . . . . . . . . . 10  |-  ( z  =  ( P  gcd  N )  ->  ( z  =  P  <->  ( P  gcd  N )  =  P ) )
2523, 24orbi12d 714 . . . . . . . . 9  |-  ( z  =  ( P  gcd  N )  ->  ( (
z  =  1  \/  z  =  P )  <-> 
( ( P  gcd  N )  =  1  \/  ( P  gcd  N
)  =  P ) ) )
2622, 25imbi12d 321 . . . . . . . 8  |-  ( z  =  ( P  gcd  N )  ->  ( (
z  ||  P  ->  ( z  =  1  \/  z  =  P ) )  <->  ( ( P  gcd  N )  ||  P  ->  ( ( P  gcd  N )  =  1  \/  ( P  gcd  N )  =  P ) ) ) )
2726rspcv 3178 . . . . . . 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 31 . . . . . 6  |-  ( P  e.  Prime  ->  ( ( P  gcd  N )  e.  NN  ->  (
( P  gcd  N
)  ||  P  ->  ( ( P  gcd  N
)  =  1  \/  ( P  gcd  N
)  =  P ) ) ) )
2928adantr 466 . . . . 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 46 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  (
( P  gcd  N
)  =  1  \/  ( P  gcd  N
)  =  P ) )
31 biorf 406 . . . . 5  |-  ( -.  ( P  gcd  N
)  =  P  -> 
( ( P  gcd  N )  =  1  <->  (
( P  gcd  N
)  =  P  \/  ( P  gcd  N )  =  1 ) ) )
32 orcom 388 . . . . 5  |-  ( ( ( P  gcd  N
)  =  P  \/  ( P  gcd  N )  =  1 )  <->  ( ( P  gcd  N )  =  1  \/  ( P  gcd  N )  =  P ) )
3331, 32syl6bb 264 . . . 4  |-  ( -.  ( P  gcd  N
)  =  P  -> 
( ( P  gcd  N )  =  1  <->  (
( P  gcd  N
)  =  1  \/  ( P  gcd  N
)  =  P ) ) )
3430, 33syl5ibrcom 225 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  ( P  gcd  N
)  =  P  -> 
( P  gcd  N
)  =  1 ) )
357, 34syld 45 . 2  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  P  ||  N  -> 
( P  gcd  N
)  =  1 ) )
36 iddvds 14315 . . . . . . 7  |-  ( P  e.  ZZ  ->  P  ||  P )
371, 36syl 17 . . . . . 6  |-  ( P  e.  Prime  ->  P  ||  P )
3837adantr 466 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  P  ||  P )
39 dvdslegcd 14477 . . . . . . . . 9  |-  ( ( ( P  e.  ZZ  /\  P  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( P  =  0  /\  N  =  0 ) )  -> 
( ( P  ||  P  /\  P  ||  N
)  ->  P  <_  ( P  gcd  N ) ) )
4039ex 435 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  P  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( P  =  0  /\  N  =  0 )  ->  ( ( P  ||  P  /\  P  ||  N )  ->  P  <_  ( P  gcd  N
) ) ) )
41403anidm12 1321 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( P  =  0  /\  N  =  0 )  -> 
( ( P  ||  P  /\  P  ||  N
)  ->  P  <_  ( P  gcd  N ) ) ) )
421, 41sylan 473 . . . . . 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 679 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  ||  N  ->  P  <_  ( P  gcd  N
) ) )
45 prmgt1 14642 . . . . . 6  |-  ( P  e.  Prime  ->  1  < 
P )
4645adantr 466 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  1  <  P )
471zred 11047 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  RR )
4847adantr 466 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  P  e.  RR )
4918nnred 10631 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  gcd  N )  e.  RR )
50 1re 9649 . . . . . . 7  |-  1  e.  RR
51 ltletr 9732 . . . . . . 7  |-  ( ( 1  e.  RR  /\  P  e.  RR  /\  ( P  gcd  N )  e.  RR )  ->  (
( 1  <  P  /\  P  <_  ( P  gcd  N ) )  ->  1  <  ( P  gcd  N ) ) )
5250, 51mp3an1 1347 . . . . . 6  |-  ( ( P  e.  RR  /\  ( P  gcd  N )  e.  RR )  -> 
( ( 1  < 
P  /\  P  <_  ( P  gcd  N ) )  ->  1  <  ( P  gcd  N ) ) )
5348, 49, 52syl2anc 665 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  (
( 1  <  P  /\  P  <_  ( P  gcd  N ) )  ->  1  <  ( P  gcd  N ) ) )
5446, 53mpand 679 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  <_  ( P  gcd  N )  ->  1  <  ( P  gcd  N ) ) )
55 ltneOLD 9738 . . . . . 6  |-  ( ( 1  e.  RR  /\  ( P  gcd  N )  e.  RR  /\  1  <  ( P  gcd  N
) )  ->  ( P  gcd  N )  =/=  1 )
56553expia 1207 . . . . 5  |-  ( ( 1  e.  RR  /\  ( P  gcd  N )  e.  RR )  -> 
( 1  <  ( P  gcd  N )  -> 
( P  gcd  N
)  =/=  1 ) )
5750, 49, 56sylancr 667 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  (
1  <  ( P  gcd  N )  ->  ( P  gcd  N )  =/=  1 ) )
5844, 54, 573syld 57 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( P  ||  N  ->  ( P  gcd  N )  =/=  1 ) )
5958necon2bd 2635 . 2  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  (
( P  gcd  N
)  =  1  ->  -.  P  ||  N ) )
6035, 59impbid 193 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 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   class class class wbr 4423   ` cfv 5601  (class class class)co 6305   RRcr 9545   0cc0 9546   1c1 9547    < clt 9682    <_ cle 9683   NNcn 10616   2c2 10666   ZZcz 10944   ZZ>=cuz 11166    || cdvds 14304    gcd cgcd 14467   Primecprime 14621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-2o 7194  df-oadd 7197  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-sup 7965  df-inf 7966  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-seq 12220  df-exp 12279  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-dvds 14305  df-gcd 14468  df-prm 14622
This theorem is referenced by:  prmrp  14657  euclemma  14664  phiprmpw  14723  fermltl  14731  prmdiv  14732  prmdiveq  14733  vfermltl  14751  prmpwdvds  14847  1259lem5  15105  2503lem3  15109  4001lem4  15114  gexexlem  17489  ablfac1lem  17700  ablfac1eu  17705  pgpfac1lem3  17709  perfect1  24154  perfectlem1  24155  perfectlem2  24156  lgslem1  24222  lgsqrlem2  24268  lgsqr  24272  lgsquad2lem2  24285  2sqblem  24303  rpvmasumlem  24323  dchrisum0flblem2  24345  nn0prpwlem  30983  isodd7  38665
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