ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rpdvds Unicode version

Theorem rpdvds 12131
Description: If  K is relatively prime to  N then it is also relatively prime to any divisor  M of  N. (Contributed by Mario Carneiro, 19-Jun-2015.)
Assertion
Ref Expression
rpdvds  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  =  1 )

Proof of Theorem rpdvds
StepHypRef Expression
1 simpl1 1002 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  K  e.  ZZ )
2 simpl2 1003 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  M  e.  ZZ )
3 gcddvds 11996 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( K  gcd  M )  ||  K  /\  ( K  gcd  M ) 
||  M ) )
41, 2, 3syl2anc 411 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( ( K  gcd  M )  ||  K  /\  ( K  gcd  M )  ||  M ) )
54simpld 112 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  ||  K
)
64simprd 114 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  ||  M
)
7 simprr 531 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  M  ||  N
)
8 1ne0 9017 . . . . . . . . . . 11  |-  1  =/=  0
9 simprl 529 . . . . . . . . . . . 12  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  N )  =  1 )
109neeq1d 2378 . . . . . . . . . . 11  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( ( K  gcd  N )  =/=  0  <->  1  =/=  0
) )
118, 10mpbiri 168 . . . . . . . . . 10  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  N )  =/=  0
)
1211neneqd 2381 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  -.  ( K  gcd  N )  =  0 )
13 simprl 529 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  K  =  0 )
14 simprr 531 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  M  =  0 )
15 simplrr 536 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  M  ||  N
)
1614, 15eqbrtrrd 4042 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  0  ||  N
)
17 simpll3 1040 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  N  e.  ZZ )
18 0dvds 11850 . . . . . . . . . . . . . 14  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
1917, 18syl 14 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  ( 0  ||  N 
<->  N  =  0 ) )
2016, 19mpbid 147 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  N  =  0 )
2113, 20jca 306 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  ( K  =  0  /\  N  =  0 ) )
2221ex 115 . . . . . . . . . 10  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( ( K  =  0  /\  M  =  0 )  ->  ( K  =  0  /\  N  =  0 ) ) )
23 simpl3 1004 . . . . . . . . . . 11  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  N  e.  ZZ )
24 gcdeq0 12010 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  gcd  N )  =  0  <->  ( K  =  0  /\  N  =  0 ) ) )
251, 23, 24syl2anc 411 . . . . . . . . . 10  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( ( K  gcd  N )  =  0  <->  ( K  =  0  /\  N  =  0 ) ) )
2622, 25sylibrd 169 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( ( K  =  0  /\  M  =  0 )  ->  ( K  gcd  N )  =  0 ) )
2712, 26mtod 664 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  -.  ( K  =  0  /\  M  =  0 ) )
28 gcdn0cl 11995 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ )  /\  -.  ( K  =  0  /\  M  =  0 ) )  ->  ( K  gcd  M )  e.  NN )
291, 2, 27, 28syl21anc 1248 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  e.  NN )
3029nnzd 9404 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  e.  ZZ )
31 dvdstr 11867 . . . . . 6  |-  ( ( ( K  gcd  M
)  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  gcd  M )  ||  M  /\  M  ||  N )  -> 
( K  gcd  M
)  ||  N )
)
3230, 2, 23, 31syl3anc 1249 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( (
( K  gcd  M
)  ||  M  /\  M  ||  N )  -> 
( K  gcd  M
)  ||  N )
)
336, 7, 32mp2and 433 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  ||  N
)
3412, 25mtbid 673 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  -.  ( K  =  0  /\  N  =  0 ) )
35 dvdslegcd 11997 . . . . 5  |-  ( ( ( ( K  gcd  M )  e.  ZZ  /\  K  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( K  =  0  /\  N  =  0
) )  ->  (
( ( K  gcd  M )  ||  K  /\  ( K  gcd  M ) 
||  N )  -> 
( K  gcd  M
)  <_  ( K  gcd  N ) ) )
3630, 1, 23, 34, 35syl31anc 1252 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( (
( K  gcd  M
)  ||  K  /\  ( K  gcd  M ) 
||  N )  -> 
( K  gcd  M
)  <_  ( K  gcd  N ) ) )
375, 33, 36mp2and 433 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  <_  ( K  gcd  N ) )
3837, 9breqtrd 4044 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  <_  1
)
39 nnle1eq1 8973 . . 3  |-  ( ( K  gcd  M )  e.  NN  ->  (
( K  gcd  M
)  <_  1  <->  ( K  gcd  M )  =  1 ) )
4029, 39syl 14 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( ( K  gcd  M )  <_ 
1  <->  ( K  gcd  M )  =  1 ) )
4138, 40mpbid 147 1  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  =  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160    =/= wne 2360   class class class wbr 4018  (class class class)co 5896   0cc0 7841   1c1 7842    <_ cle 8023   NNcn 8949   ZZcz 9283    || cdvds 11826    gcd cgcd 11975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959  ax-arch 7960  ax-caucvg 7961
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-frec 6416  df-sup 7013  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-2 9008  df-3 9009  df-4 9010  df-n0 9207  df-z 9284  df-uz 9559  df-q 9650  df-rp 9684  df-fz 10039  df-fzo 10173  df-fl 10301  df-mod 10354  df-seqfrec 10477  df-exp 10551  df-cj 10883  df-re 10884  df-im 10885  df-rsqrt 11039  df-abs 11040  df-dvds 11827  df-gcd 11976
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator