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Theorem rpdvds 11707
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 969 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  K  e.  ZZ )
2 simpl2 970 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  M  e.  ZZ )
3 gcddvds 11579 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( K  gcd  M )  ||  K  /\  ( K  gcd  M ) 
||  M ) )
41, 2, 3syl2anc 408 . . . . 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 111 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  ||  K
)
64simprd 113 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  ||  M
)
7 simprr 506 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  M  ||  N
)
8 1ne0 8756 . . . . . . . . . . 11  |-  1  =/=  0
9 simprl 505 . . . . . . . . . . . 12  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  N )  =  1 )
109neeq1d 2303 . . . . . . . . . . 11  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( ( K  gcd  N )  =/=  0  <->  1  =/=  0
) )
118, 10mpbiri 167 . . . . . . . . . 10  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  N )  =/=  0
)
1211neneqd 2306 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  -.  ( K  gcd  N )  =  0 )
13 simprl 505 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  K  =  0 )
14 simprr 506 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  M  =  0 )
15 simplrr 510 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  M  ||  N
)
1614, 15eqbrtrrd 3922 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  0  ||  N
)
17 simpll3 1007 . . . . . . . . . . . . . 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 11440 . . . . . . . . . . . . . 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 146 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  N  =  0 )
2113, 20jca 304 . . . . . . . . . . 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 114 . . . . . . . . . 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 971 . . . . . . . . . . 11  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  N  e.  ZZ )
24 gcdeq0 11592 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  gcd  N )  =  0  <->  ( K  =  0  /\  N  =  0 ) ) )
251, 23, 24syl2anc 408 . . . . . . . . . 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 168 . . . . . . . . 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 637 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  -.  ( K  =  0  /\  M  =  0 ) )
28 gcdn0cl 11578 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ )  /\  -.  ( K  =  0  /\  M  =  0 ) )  ->  ( K  gcd  M )  e.  NN )
291, 2, 27, 28syl21anc 1200 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  e.  NN )
3029nnzd 9140 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  e.  ZZ )
31 dvdstr 11457 . . . . . 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 1201 . . . . 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 429 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  ||  N
)
3412, 25mtbid 646 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  -.  ( K  =  0  /\  N  =  0 ) )
35 dvdslegcd 11580 . . . . 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 1204 . . . 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 429 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  <_  ( K  gcd  N ) )
3837, 9breqtrd 3924 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  <_  1
)
39 nnle1eq1 8712 . . 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 146 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 103    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465    =/= wne 2285   class class class wbr 3899  (class class class)co 5742   0cc0 7588   1c1 7589    <_ cle 7769   NNcn 8688   ZZcz 9022    || cdvds 11420    gcd cgcd 11562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-stab 801  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-sup 6839  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-n0 8946  df-z 9023  df-uz 9295  df-q 9380  df-rp 9410  df-fz 9759  df-fzo 9888  df-fl 10011  df-mod 10064  df-seqfrec 10187  df-exp 10261  df-cj 10582  df-re 10583  df-im 10584  df-rsqrt 10738  df-abs 10739  df-dvds 11421  df-gcd 11563
This theorem is referenced by: (None)
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