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Theorem coprmdvds2 13787
Description: If an integer is divisible by two coprime integers, then it is divisible by their product. (Contributed by Mario Carneiro, 24-Feb-2014.)
Assertion
Ref Expression
coprmdvds2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( M 
||  K  /\  N  ||  K )  ->  ( M  x.  N )  ||  K ) )

Proof of Theorem coprmdvds2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 divides 13535 . . . . . 6  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  ||  K  <->  E. x  e.  ZZ  (
x  x.  N )  =  K ) )
213adant1 1006 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  ||  K  <->  E. x  e.  ZZ  ( x  x.  N )  =  K ) )
32adantr 465 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( N  ||  K 
<->  E. x  e.  ZZ  ( x  x.  N
)  =  K ) )
4 simprr 756 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( ( M  gcd  N )  =  1  /\  x  e.  ZZ ) )  ->  x  e.  ZZ )
5 simpl2 992 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( ( M  gcd  N )  =  1  /\  x  e.  ZZ ) )  ->  N  e.  ZZ )
6 zcn 10649 . . . . . . . . . . . 12  |-  ( x  e.  ZZ  ->  x  e.  CC )
7 zcn 10649 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  N  e.  CC )
8 mulcom 9366 . . . . . . . . . . . 12  |-  ( ( x  e.  CC  /\  N  e.  CC )  ->  ( x  x.  N
)  =  ( N  x.  x ) )
96, 7, 8syl2an 477 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  N  e.  ZZ )  ->  ( x  x.  N
)  =  ( N  x.  x ) )
104, 5, 9syl2anc 661 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( ( M  gcd  N )  =  1  /\  x  e.  ZZ ) )  ->  ( x  x.  N )  =  ( N  x.  x ) )
1110breq2d 4302 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( ( M  gcd  N )  =  1  /\  x  e.  ZZ ) )  ->  ( M  ||  ( x  x.  N
)  <->  M  ||  ( N  x.  x ) ) )
12 simprl 755 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( ( M  gcd  N )  =  1  /\  x  e.  ZZ ) )  ->  ( M  gcd  N )  =  1 )
13 simpl1 991 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( ( M  gcd  N )  =  1  /\  x  e.  ZZ ) )  ->  M  e.  ZZ )
14 coprmdvds 13786 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  x  e.  ZZ )  ->  (
( M  ||  ( N  x.  x )  /\  ( M  gcd  N
)  =  1 )  ->  M  ||  x
) )
1513, 5, 4, 14syl3anc 1218 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( ( M  gcd  N )  =  1  /\  x  e.  ZZ ) )  ->  ( ( M  ||  ( N  x.  x )  /\  ( M  gcd  N )  =  1 )  ->  M  ||  x ) )
1612, 15mpan2d 674 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( ( M  gcd  N )  =  1  /\  x  e.  ZZ ) )  ->  ( M  ||  ( N  x.  x
)  ->  M  ||  x
) )
1711, 16sylbid 215 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( ( M  gcd  N )  =  1  /\  x  e.  ZZ ) )  ->  ( M  ||  ( x  x.  N
)  ->  M  ||  x
) )
18 dvdsmulc 13558 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  x  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  x  ->  ( M  x.  N )  ||  ( x  x.  N
) ) )
1913, 4, 5, 18syl3anc 1218 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( ( M  gcd  N )  =  1  /\  x  e.  ZZ ) )  ->  ( M  ||  x  ->  ( M  x.  N )  ||  (
x  x.  N ) ) )
2017, 19syld 44 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( ( M  gcd  N )  =  1  /\  x  e.  ZZ ) )  ->  ( M  ||  ( x  x.  N
)  ->  ( M  x.  N )  ||  (
x  x.  N ) ) )
21 breq2 4294 . . . . . . . 8  |-  ( ( x  x.  N )  =  K  ->  ( M  ||  ( x  x.  N )  <->  M  ||  K
) )
22 breq2 4294 . . . . . . . 8  |-  ( ( x  x.  N )  =  K  ->  (
( M  x.  N
)  ||  ( x  x.  N )  <->  ( M  x.  N )  ||  K
) )
2321, 22imbi12d 320 . . . . . . 7  |-  ( ( x  x.  N )  =  K  ->  (
( M  ||  (
x  x.  N )  ->  ( M  x.  N )  ||  (
x  x.  N ) )  <->  ( M  ||  K  ->  ( M  x.  N )  ||  K
) ) )
2420, 23syl5ibcom 220 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( ( M  gcd  N )  =  1  /\  x  e.  ZZ ) )  ->  ( (
x  x.  N )  =  K  ->  ( M  ||  K  ->  ( M  x.  N )  ||  K ) ) )
2524anassrs 648 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M  gcd  N )  =  1 )  /\  x  e.  ZZ )  ->  (
( x  x.  N
)  =  K  -> 
( M  ||  K  ->  ( M  x.  N
)  ||  K )
) )
2625rexlimdva 2839 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( E. x  e.  ZZ  ( x  x.  N )  =  K  ->  ( M  ||  K  ->  ( M  x.  N )  ||  K
) ) )
273, 26sylbid 215 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( N  ||  K  ->  ( M  ||  K  ->  ( M  x.  N )  ||  K
) ) )
2827com23 78 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( M  ||  K  ->  ( N  ||  K  ->  ( M  x.  N )  ||  K
) ) )
2928impd 431 1  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( M 
||  K  /\  N  ||  K )  ->  ( M  x.  N )  ||  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2714   class class class wbr 4290  (class class class)co 6089   CCcc 9278   1c1 9281    x. cmul 9285   ZZcz 10644    || cdivides 13533    gcd cgcd 13688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-2nd 6576  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-sup 7689  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-n0 10578  df-z 10645  df-uz 10860  df-rp 10990  df-fl 11640  df-mod 11707  df-seq 11805  df-exp 11864  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-dvds 13534  df-gcd 13689
This theorem is referenced by:  rpmulgcd2  13789  crt  13851  odadd2  16329  ablfac1b  16569  ablfac1eu  16572
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