MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rpdvds Unicode version

Theorem rpdvds 12803
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 958 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  K  e.  ZZ )
2 simpl2 959 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  M  e.  ZZ )
3 gcddvds 12694 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( K  gcd  M )  ||  K  /\  ( K  gcd  M ) 
||  M ) )
41, 2, 3syl2anc 642 . . . . 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 445 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  ||  K
)
64simprd 449 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  ||  M
)
7 simprr 733 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  M  ||  N
)
8 ax-1ne0 8806 . . . . . . . . . . 11  |-  1  =/=  0
9 simprl 732 . . . . . . . . . . . 12  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  N )  =  1 )
109neeq1d 2459 . . . . . . . . . . 11  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( ( K  gcd  N )  =/=  0  <->  1  =/=  0
) )
118, 10mpbiri 224 . . . . . . . . . 10  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  N )  =/=  0
)
1211neneqd 2462 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  -.  ( K  gcd  N )  =  0 )
13 simprl 732 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  K  =  0 )
14 simprr 733 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  M  =  0 )
15 simplrr 737 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  M  ||  N
)
1614, 15eqbrtrrd 4045 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  0  ||  N
)
17 simpll3 996 . . . . . . . . . . . . . 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 12549 . . . . . . . . . . . . . 14  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
1917, 18syl 15 . . . . . . . . . . . . 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 201 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  N  =  0 )
2113, 20jca 518 . . . . . . . . . . 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 423 . . . . . . . . . 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 960 . . . . . . . . . . 11  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  N  e.  ZZ )
24 gcdeq0 12700 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  gcd  N )  =  0  <->  ( K  =  0  /\  N  =  0 ) ) )
251, 23, 24syl2anc 642 . . . . . . . . . 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 225 . . . . . . . . 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 168 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  -.  ( K  =  0  /\  M  =  0 ) )
28 gcdn0cl 12693 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ )  /\  -.  ( K  =  0  /\  M  =  0 ) )  ->  ( K  gcd  M )  e.  NN )
291, 2, 27, 28syl21anc 1181 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  e.  NN )
3029nnzd 10116 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  e.  ZZ )
31 dvdstr 12562 . . . . . 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 1182 . . . . 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 660 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  ||  N
)
3412, 25mtbid 291 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  -.  ( K  =  0  /\  N  =  0 ) )
35 dvdslegcd 12695 . . . . 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 1185 . . . 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 660 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  <_  ( K  gcd  N ) )
3837, 9breqtrd 4047 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  <_  1
)
39 nnle1eq1 9774 . . 3  |-  ( ( K  gcd  M )  e.  NN  ->  (
( K  gcd  M
)  <_  1  <->  ( K  gcd  M )  =  1 ) )
4029, 39syl 15 . 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 201 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023  (class class class)co 5858   0cc0 8737   1c1 8738    <_ cle 8868   NNcn 9746   ZZcz 10024    || cdivides 12531    gcd cgcd 12685
This theorem is referenced by:  pgpfac1lem2  15310  dvdsmulf1o  20434  lgsquad2lem2  20598
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686
  Copyright terms: Public domain W3C validator