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Theorem rpdvds 14277
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 999 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  K  e.  ZZ )
2 simpl2 1000 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  M  e.  ZZ )
3 gcddvds 14165 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( K  gcd  M )  ||  K  /\  ( K  gcd  M ) 
||  M ) )
41, 2, 3syl2anc 661 . . . . 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 459 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  ||  K
)
64simprd 463 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  ||  M
)
7 simprr 757 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  M  ||  N
)
8 ax-1ne0 9578 . . . . . . . . . . 11  |-  1  =/=  0
9 simprl 756 . . . . . . . . . . . 12  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  N )  =  1 )
109neeq1d 2734 . . . . . . . . . . 11  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( ( K  gcd  N )  =/=  0  <->  1  =/=  0
) )
118, 10mpbiri 233 . . . . . . . . . 10  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  N )  =/=  0
)
1211neneqd 2659 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  -.  ( K  gcd  N )  =  0 )
13 simprl 756 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  K  =  0 )
14 simprr 757 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  M  =  0 )
15 simplrr 762 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  M  ||  N
)
1614, 15eqbrtrrd 4478 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  0  ||  N
)
17 simpll3 1037 . . . . . . . . . . . . . 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 14016 . . . . . . . . . . . . . 14  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
1917, 18syl 16 . . . . . . . . . . . . 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 210 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  (
( K  gcd  N
)  =  1  /\  M  ||  N ) )  /\  ( K  =  0  /\  M  =  0 ) )  ->  N  =  0 )
2113, 20jca 532 . . . . . . . . . . 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 434 . . . . . . . . . 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 1001 . . . . . . . . . . 11  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  N  e.  ZZ )
24 gcdeq0 14171 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  gcd  N )  =  0  <->  ( K  =  0  /\  N  =  0 ) ) )
251, 23, 24syl2anc 661 . . . . . . . . . 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 234 . . . . . . . . 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 177 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  -.  ( K  =  0  /\  M  =  0 ) )
28 gcdn0cl 14164 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ )  /\  -.  ( K  =  0  /\  M  =  0 ) )  ->  ( K  gcd  M )  e.  NN )
291, 2, 27, 28syl21anc 1227 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  e.  NN )
3029nnzd 10989 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  e.  ZZ )
31 dvdstr 14030 . . . . . 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 1228 . . . . 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 679 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  ||  N
)
3412, 25mtbid 300 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  -.  ( K  =  0  /\  N  =  0 ) )
35 dvdslegcd 14166 . . . . 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 1231 . . . 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 679 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  <_  ( K  gcd  N ) )
3837, 9breqtrd 4480 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  <_  1
)
39 nnle1eq1 10584 . . 3  |-  ( ( K  gcd  M )  e.  NN  ->  (
( K  gcd  M
)  <_  1  <->  ( K  gcd  M )  =  1 ) )
4029, 39syl 16 . 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 210 1  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  =  1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456  (class class class)co 6296   0cc0 9509   1c1 9510    <_ cle 9646   NNcn 10556   ZZcz 10885    || cdvds 13998    gcd cgcd 14156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-dvds 13999  df-gcd 14157
This theorem is referenced by:  pgpfac1lem2  17253  dvdsmulf1o  23596  lgsquad2lem2  23760
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