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

Theorem rpmul 14635
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 14621 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( ( K  gcd  M )  x.  ( K  gcd  N ) ) )
2 oveq12 6305 . . . . 5  |-  ( ( ( K  gcd  M
)  =  1  /\  ( K  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  x.  ( K  gcd  N
) )  =  ( 1  x.  1 ) )
3 1t1e1 10746 . . . . 5  |-  ( 1  x.  1 )  =  1
42, 3syl6eq 2477 . . . 4  |-  ( ( ( K  gcd  M
)  =  1  /\  ( K  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  x.  ( K  gcd  N
) )  =  1 )
54breq2d 4429 . . 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 223 . 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 1005 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
8 zmulcl 10974 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )
983adant1 1023 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N )  e.  ZZ )
107, 9gcdcld 14445 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  e. 
NN0 )
11 dvds1 14320 . . 3  |-  ( ( K  gcd  ( M  x.  N ) )  e.  NN0  ->  ( ( K  gcd  ( M  x.  N ) ) 
||  1  <->  ( K  gcd  ( M  x.  N
) )  =  1 ) )
1210, 11syl 17 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  ( M  x.  N )
)  ||  1  <->  ( K  gcd  ( M  x.  N
) )  =  1 ) )
136, 12sylibd 217 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   class class class wbr 4417  (class class class)co 6296   1c1 9529    x. cmul 9533   NN0cn0 10858   ZZcz 10926    || cdvds 14272    gcd cgcd 14431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-sup 7953  df-inf 7954  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-n0 10859  df-z 10927  df-uz 11149  df-rp 11292  df-fl 12014  df-mod 12083  df-seq 12200  df-exp 12259  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-dvds 14273  df-gcd 14432
This theorem is referenced by:  phimullem  14687  eulerthlem1  14689  eulerthlem2  14690
  Copyright terms: Public domain W3C validator