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Theorem gcdmultiplez 10617
Description: Extend gcdmultiple 10616 so  N can be an integer. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
gcdmultiplez  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  N )
)  =  M )

Proof of Theorem gcdmultiplez
StepHypRef Expression
1 0z 8495 . . . 4  |-  0  e.  ZZ
2 zdceq 8556 . . . 4  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
31, 2mpan2 416 . . 3  |-  ( N  e.  ZZ  -> DECID  N  =  0
)
4 exmiddc 778 . . 3  |-  (DECID  N  =  0  ->  ( N  =  0  \/  -.  N  =  0 ) )
5 nncn 8166 . . . . . . . 8  |-  ( M  e.  NN  ->  M  e.  CC )
6 mul01 7612 . . . . . . . . 9  |-  ( M  e.  CC  ->  ( M  x.  0 )  =  0 )
76oveq2d 5579 . . . . . . . 8  |-  ( M  e.  CC  ->  ( M  gcd  ( M  x.  0 ) )  =  ( M  gcd  0
) )
85, 7syl 14 . . . . . . 7  |-  ( M  e.  NN  ->  ( M  gcd  ( M  x.  0 ) )  =  ( M  gcd  0
) )
98adantr 270 . . . . . 6  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  0 ) )  =  ( M  gcd  0 ) )
10 nnnn0 8414 . . . . . . . 8  |-  ( M  e.  NN  ->  M  e.  NN0 )
11 nn0gcdid0 10579 . . . . . . . 8  |-  ( M  e.  NN0  ->  ( M  gcd  0 )  =  M )
1210, 11syl 14 . . . . . . 7  |-  ( M  e.  NN  ->  ( M  gcd  0 )  =  M )
1312adantr 270 . . . . . 6  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  0
)  =  M )
149, 13eqtrd 2115 . . . . 5  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  0 ) )  =  M )
15 oveq2 5571 . . . . . . 7  |-  ( N  =  0  ->  ( M  x.  N )  =  ( M  x.  0 ) )
1615oveq2d 5579 . . . . . 6  |-  ( N  =  0  ->  ( M  gcd  ( M  x.  N ) )  =  ( M  gcd  ( M  x.  0 ) ) )
1716eqeq1d 2091 . . . . 5  |-  ( N  =  0  ->  (
( M  gcd  ( M  x.  N )
)  =  M  <->  ( M  gcd  ( M  x.  0 ) )  =  M ) )
1814, 17syl5ibr 154 . . . 4  |-  ( N  =  0  ->  (
( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  N
) )  =  M ) )
19 df-ne 2250 . . . . 5  |-  ( N  =/=  0  <->  -.  N  =  0 )
20 zcn 8489 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
21 absmul 10156 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( abs `  ( M  x.  N )
)  =  ( ( abs `  M )  x.  ( abs `  N
) ) )
225, 20, 21syl2an 283 . . . . . . . . . 10  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( abs `  ( M  x.  N )
)  =  ( ( abs `  M )  x.  ( abs `  N
) ) )
23 nnre 8165 . . . . . . . . . . . . 13  |-  ( M  e.  NN  ->  M  e.  RR )
2410nn0ge0d 8463 . . . . . . . . . . . . 13  |-  ( M  e.  NN  ->  0  <_  M )
2523, 24absidd 10254 . . . . . . . . . . . 12  |-  ( M  e.  NN  ->  ( abs `  M )  =  M )
2625oveq1d 5578 . . . . . . . . . . 11  |-  ( M  e.  NN  ->  (
( abs `  M
)  x.  ( abs `  N ) )  =  ( M  x.  ( abs `  N ) ) )
2726adantr 270 . . . . . . . . . 10  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  x.  ( abs `  N ) )  =  ( M  x.  ( abs `  N ) ) )
2822, 27eqtrd 2115 . . . . . . . . 9  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( abs `  ( M  x.  N )
)  =  ( M  x.  ( abs `  N
) ) )
2928oveq2d 5579 . . . . . . . 8  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( abs `  ( M  x.  N ) ) )  =  ( M  gcd  ( M  x.  ( abs `  N ) ) ) )
3029adantr 270 . . . . . . 7  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( M  gcd  ( abs `  ( M  x.  N )
) )  =  ( M  gcd  ( M  x.  ( abs `  N
) ) ) )
31 simpll 496 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  M  e.  NN )
3231nnzd 8601 . . . . . . . 8  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  M  e.  ZZ )
33 nnz 8503 . . . . . . . . . 10  |-  ( M  e.  NN  ->  M  e.  ZZ )
34 zmulcl 8537 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )
3533, 34sylan 277 . . . . . . . . 9  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )
3635adantr 270 . . . . . . . 8  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( M  x.  N )  e.  ZZ )
37 gcdabs2 10588 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ )  ->  ( M  gcd  ( abs `  ( M  x.  N ) ) )  =  ( M  gcd  ( M  x.  N ) ) )
3832, 36, 37syl2anc 403 . . . . . . 7  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( M  gcd  ( abs `  ( M  x.  N )
) )  =  ( M  gcd  ( M  x.  N ) ) )
39 nnabscl 10187 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
40 gcdmultiple 10616 . . . . . . . . 9  |-  ( ( M  e.  NN  /\  ( abs `  N )  e.  NN )  -> 
( M  gcd  ( M  x.  ( abs `  N ) ) )  =  M )
4139, 40sylan2 280 . . . . . . . 8  |-  ( ( M  e.  NN  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( M  gcd  ( M  x.  ( abs `  N ) ) )  =  M )
4241anassrs 392 . . . . . . 7  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( M  gcd  ( M  x.  ( abs `  N ) ) )  =  M )
4330, 38, 423eqtr3d 2123 . . . . . 6  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( M  gcd  ( M  x.  N
) )  =  M )
4443expcom 114 . . . . 5  |-  ( N  =/=  0  ->  (
( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  N
) )  =  M ) )
4519, 44sylbir 133 . . . 4  |-  ( -.  N  =  0  -> 
( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  N ) )  =  M ) )
4618, 45jaoi 669 . . 3  |-  ( ( N  =  0  \/ 
-.  N  =  0 )  ->  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  N ) )  =  M ) )
473, 4, 463syl 17 . 2  |-  ( N  e.  ZZ  ->  (
( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  N
) )  =  M ) )
4847anabsi7 546 1  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  N )
)  =  M )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 662  DECID wdc 776    = wceq 1285    e. wcel 1434    =/= wne 2249   ` cfv 4952  (class class class)co 5563   CCcc 7093   0cc0 7095    x. cmul 7100   NNcn 8158   NN0cn0 8407   ZZcz 8484   abscabs 10084    gcd cgcd 10545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209  ax-caucvg 7210
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-sup 6491  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-3 8218  df-4 8219  df-n0 8408  df-z 8485  df-uz 8753  df-q 8838  df-rp 8868  df-fz 9158  df-fzo 9282  df-fl 9404  df-mod 9457  df-iseq 9574  df-iexp 9625  df-cj 9930  df-re 9931  df-im 9932  df-rsqrt 10085  df-abs 10086  df-dvds 10404  df-gcd 10546
This theorem is referenced by: (None)
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