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Theorem coprmdvds2 13095
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 12846 . . . . . 6  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  ||  K  <->  E. x  e.  ZZ  (
x  x.  N )  =  K ) )
213adant1 975 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  ||  K  <->  E. x  e.  ZZ  ( x  x.  N )  =  K ) )
32adantr 452 . . . 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 734 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( ( M  gcd  N )  =  1  /\  x  e.  ZZ ) )  ->  x  e.  ZZ )
5 simpl2 961 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( ( M  gcd  N )  =  1  /\  x  e.  ZZ ) )  ->  N  e.  ZZ )
6 zcn 10279 . . . . . . . . . . . 12  |-  ( x  e.  ZZ  ->  x  e.  CC )
7 zcn 10279 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  N  e.  CC )
8 mulcom 9068 . . . . . . . . . . . 12  |-  ( ( x  e.  CC  /\  N  e.  CC )  ->  ( x  x.  N
)  =  ( N  x.  x ) )
96, 7, 8syl2an 464 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  N  e.  ZZ )  ->  ( x  x.  N
)  =  ( N  x.  x ) )
104, 5, 9syl2anc 643 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( ( M  gcd  N )  =  1  /\  x  e.  ZZ ) )  ->  ( x  x.  N )  =  ( N  x.  x ) )
1110breq2d 4216 . . . . . . . . 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 733 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( ( M  gcd  N )  =  1  /\  x  e.  ZZ ) )  ->  ( M  gcd  N )  =  1 )
13 simpl1 960 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( ( M  gcd  N )  =  1  /\  x  e.  ZZ ) )  ->  M  e.  ZZ )
14 coprmdvds 13094 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  x  e.  ZZ )  ->  (
( M  ||  ( N  x.  x )  /\  ( M  gcd  N
)  =  1 )  ->  M  ||  x
) )
1513, 5, 4, 14syl3anc 1184 . . . . . . . . . 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 656 . . . . . . . . 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 207 . . . . . . . 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 12869 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  x  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  x  ->  ( M  x.  N )  ||  ( x  x.  N
) ) )
1913, 4, 5, 18syl3anc 1184 . . . . . . . 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 42 . . . . . . 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 4208 . . . . . . . 8  |-  ( ( x  x.  N )  =  K  ->  ( M  ||  ( x  x.  N )  <->  M  ||  K
) )
22 breq2 4208 . . . . . . . 8  |-  ( ( x  x.  N )  =  K  ->  (
( M  x.  N
)  ||  ( x  x.  N )  <->  ( M  x.  N )  ||  K
) )
2321, 22imbi12d 312 . . . . . . 7  |-  ( ( x  x.  N )  =  K  ->  (
( M  ||  (
x  x.  N )  ->  ( M  x.  N )  ||  (
x  x.  N ) )  <->  ( M  ||  K  ->  ( M  x.  N )  ||  K
) ) )
2420, 23syl5ibcom 212 . . . . . 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 630 . . . . 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 2822 . . . 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 207 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( N  ||  K  ->  ( M  ||  K  ->  ( M  x.  N )  ||  K
) ) )
2827com23 74 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( M  ||  K  ->  ( N  ||  K  ->  ( M  x.  N )  ||  K
) ) )
2928imp3a 421 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2698   class class class wbr 4204  (class class class)co 6073   CCcc 8980   1c1 8983    x. cmul 8987   ZZcz 10274    || cdivides 12844    gcd cgcd 12998
This theorem is referenced by:  rpmulgcd2  13097  crt  13159  odadd2  15456  ablfac1b  15620  ablfac1eu  15623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999
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