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Theorem gcdmultiplez 11284
Description: Extend gcdmultiple 11283 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 8759 . . . 4  |-  0  e.  ZZ
2 zdceq 8820 . . . 4  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
31, 2mpan2 416 . . 3  |-  ( N  e.  ZZ  -> DECID  N  =  0
)
4 exmiddc 782 . . 3  |-  (DECID  N  =  0  ->  ( N  =  0  \/  -.  N  =  0 ) )
5 nncn 8428 . . . . . . . 8  |-  ( M  e.  NN  ->  M  e.  CC )
6 mul01 7865 . . . . . . . . 9  |-  ( M  e.  CC  ->  ( M  x.  0 )  =  0 )
76oveq2d 5668 . . . . . . . 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 8678 . . . . . . . 8  |-  ( M  e.  NN  ->  M  e.  NN0 )
11 nn0gcdid0 11246 . . . . . . . 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 2120 . . . . 5  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  0 ) )  =  M )
15 oveq2 5660 . . . . . . 7  |-  ( N  =  0  ->  ( M  x.  N )  =  ( M  x.  0 ) )
1615oveq2d 5668 . . . . . 6  |-  ( N  =  0  ->  ( M  gcd  ( M  x.  N ) )  =  ( M  gcd  ( M  x.  0 ) ) )
1716eqeq1d 2096 . . . . 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 2256 . . . . 5  |-  ( N  =/=  0  <->  -.  N  =  0 )
20 zcn 8753 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
21 absmul 10498 . . . . . . . . . . 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 8427 . . . . . . . . . . . . 13  |-  ( M  e.  NN  ->  M  e.  RR )
2410nn0ge0d 8727 . . . . . . . . . . . . 13  |-  ( M  e.  NN  ->  0  <_  M )
2523, 24absidd 10596 . . . . . . . . . . . 12  |-  ( M  e.  NN  ->  ( abs `  M )  =  M )
2625oveq1d 5667 . . . . . . . . . . 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 2120 . . . . . . . . 9  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( abs `  ( M  x.  N )
)  =  ( M  x.  ( abs `  N
) ) )
2928oveq2d 5668 . . . . . . . 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 8865 . . . . . . . 8  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  M  e.  ZZ )
33 nnz 8767 . . . . . . . . . 10  |-  ( M  e.  NN  ->  M  e.  ZZ )
34 zmulcl 8801 . . . . . . . . . 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 11255 . . . . . . . 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 10529 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
40 gcdmultiple 11283 . . . . . . . . 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 2128 . . . . . 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 671 . . 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 548 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 664  DECID wdc 780    = wceq 1289    e. wcel 1438    =/= wne 2255   ` cfv 5015  (class class class)co 5652   CCcc 7346   0cc0 7348    x. cmul 7353   NNcn 8420   NN0cn0 8671   ZZcz 8748   abscabs 10426    gcd cgcd 11212
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-mulrcl 7442  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-mulass 7446  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-1rid 7450  ax-0id 7451  ax-rnegex 7452  ax-precex 7453  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459  ax-pre-mulgt0 7460  ax-pre-mulext 7461  ax-arch 7462  ax-caucvg 7463
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-sup 6677  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-reap 8050  df-ap 8057  df-div 8138  df-inn 8421  df-2 8479  df-3 8480  df-4 8481  df-n0 8672  df-z 8749  df-uz 9018  df-q 9103  df-rp 9133  df-fz 9423  df-fzo 9550  df-fl 9673  df-mod 9726  df-iseq 9849  df-seq3 9850  df-exp 9951  df-cj 10272  df-re 10273  df-im 10274  df-rsqrt 10427  df-abs 10428  df-dvds 11071  df-gcd 11213
This theorem is referenced by: (None)
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