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Theorem rpdvds 10688
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 942 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  K  e.  ZZ )
2 simpl2 943 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  M  e.  ZZ )
3 gcddvds 10562 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( K  gcd  M )  ||  K  /\  ( K  gcd  M ) 
||  M ) )
41, 2, 3syl2anc 403 . . . . 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 110 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  ||  K
)
64simprd 112 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  ||  M
)
7 simprr 499 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  M  ||  N
)
8 1ne0 8226 . . . . . . . . . . 11  |-  1  =/=  0
9 simprl 498 . . . . . . . . . . . 12  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  N )  =  1 )
109neeq1d 2267 . . . . . . . . . . 11  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( ( K  gcd  N )  =/=  0  <->  1  =/=  0
) )
118, 10mpbiri 166 . . . . . . . . . 10  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  N )  =/=  0
)
1211neneqd 2270 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  -.  ( K  gcd  N )  =  0 )
13 simprl 498 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  K  =  0 )
14 simprr 499 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  M  =  0 )
15 simplrr 503 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  M  ||  N
)
1614, 15eqbrtrrd 3827 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  0  ||  N
)
17 simpll3 980 . . . . . . . . . . . . . 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 10423 . . . . . . . . . . . . . 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 145 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  N  =  0 )
2113, 20jca 300 . . . . . . . . . . 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 113 . . . . . . . . . 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 944 . . . . . . . . . . 11  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  N  e.  ZZ )
24 gcdeq0 10575 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  gcd  N )  =  0  <->  ( K  =  0  /\  N  =  0 ) ) )
251, 23, 24syl2anc 403 . . . . . . . . . 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 167 . . . . . . . . 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 622 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  -.  ( K  =  0  /\  M  =  0 ) )
28 gcdn0cl 10561 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ )  /\  -.  ( K  =  0  /\  M  =  0 ) )  ->  ( K  gcd  M )  e.  NN )
291, 2, 27, 28syl21anc 1169 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  e.  NN )
3029nnzd 8601 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  e.  ZZ )
31 dvdstr 10440 . . . . . 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 1170 . . . . 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 424 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  ||  N
)
3412, 25mtbid 630 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  -.  ( K  =  0  /\  N  =  0 ) )
35 dvdslegcd 10563 . . . . 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 1173 . . . 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 424 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  <_  ( K  gcd  N ) )
3837, 9breqtrd 3829 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  <_  1
)
39 nnle1eq1 8182 . . 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 145 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 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434    =/= wne 2249   class class class wbr 3805  (class class class)co 5563   0cc0 7095   1c1 7096    <_ cle 7268   NNcn 8158   ZZcz 8484    || cdvds 10403    gcd cgcd 10545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209  ax-caucvg 7210
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-sup 6491  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-3 8218  df-4 8219  df-n0 8408  df-z 8485  df-uz 8753  df-q 8838  df-rp 8868  df-fz 9158  df-fzo 9282  df-fl 9404  df-mod 9457  df-iseq 9574  df-iexp 9625  df-cj 9930  df-re 9931  df-im 9932  df-rsqrt 10085  df-abs 10086  df-dvds 10404  df-gcd 10546
This theorem is referenced by: (None)
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