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

Theorem rpmul 14126
Description: If  K is relatively prime to  M and to  N, it is also relatively prime to their product. (Contributed by Mario Carneiro, 24-Feb-2014.) (Proof shortened by Mario Carneiro, 2-Jul-2015.)
Assertion
Ref Expression
rpmul  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  gcd  M )  =  1  /\  ( K  gcd  N
)  =  1 )  ->  ( K  gcd  ( M  x.  N
) )  =  1 ) )

Proof of Theorem rpmul
StepHypRef Expression
1 mulgcddvds 14107 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( ( K  gcd  M )  x.  ( K  gcd  N ) ) )
2 oveq12 6294 . . . . 5  |-  ( ( ( K  gcd  M
)  =  1  /\  ( K  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  x.  ( K  gcd  N
) )  =  ( 1  x.  1 ) )
3 1t1e1 10684 . . . . 5  |-  ( 1  x.  1 )  =  1
42, 3syl6eq 2524 . . . 4  |-  ( ( ( K  gcd  M
)  =  1  /\  ( K  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  x.  ( K  gcd  N
) )  =  1 )
54breq2d 4459 . . 3  |-  ( ( ( K  gcd  M
)  =  1  /\  ( K  gcd  N
)  =  1 )  ->  ( ( K  gcd  ( M  x.  N ) )  ||  ( ( K  gcd  M )  x.  ( K  gcd  N ) )  <-> 
( K  gcd  ( M  x.  N )
)  ||  1 ) )
61, 5syl5ibcom 220 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  gcd  M )  =  1  /\  ( K  gcd  N
)  =  1 )  ->  ( K  gcd  ( M  x.  N
) )  ||  1
) )
7 simp1 996 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
8 zmulcl 10912 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )
983adant1 1014 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N )  e.  ZZ )
107, 9gcdcld 14018 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  e. 
NN0 )
11 dvds1 13896 . . 3  |-  ( ( K  gcd  ( M  x.  N ) )  e.  NN0  ->  ( ( K  gcd  ( M  x.  N ) ) 
||  1  <->  ( K  gcd  ( M  x.  N
) )  =  1 ) )
1210, 11syl 16 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  ( M  x.  N )
)  ||  1  <->  ( K  gcd  ( M  x.  N
) )  =  1 ) )
136, 12sylibd 214 1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  gcd  M )  =  1  /\  ( K  gcd  N
)  =  1 )  ->  ( K  gcd  ( M  x.  N
) )  =  1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447  (class class class)co 6285   1c1 9494    x. cmul 9498   NN0cn0 10796   ZZcz 10865    || cdivides 13850    gcd cgcd 14006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-fl 11898  df-mod 11966  df-seq 12077  df-exp 12136  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-dvds 13851  df-gcd 14007
This theorem is referenced by:  phimullem  14171  eulerthlem1  14173  eulerthlem2  14174
  Copyright terms: Public domain W3C validator