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Theorem gcdmultiplez 13756
Description: Extend gcdmultiple 13755 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 oveq2 6120 . . . 4  |-  ( N  =  0  ->  ( M  x.  N )  =  ( M  x.  0 ) )
21oveq2d 6128 . . 3  |-  ( N  =  0  ->  ( M  gcd  ( M  x.  N ) )  =  ( M  gcd  ( M  x.  0 ) ) )
32eqeq1d 2451 . 2  |-  ( N  =  0  ->  (
( M  gcd  ( M  x.  N )
)  =  M  <->  ( M  gcd  ( M  x.  0 ) )  =  M ) )
4 nncn 10351 . . . . . . 7  |-  ( M  e.  NN  ->  M  e.  CC )
5 zcn 10672 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  CC )
6 absmul 12804 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( abs `  ( M  x.  N )
)  =  ( ( abs `  M )  x.  ( abs `  N
) ) )
74, 5, 6syl2an 477 . . . . . 6  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( abs `  ( M  x.  N )
)  =  ( ( abs `  M )  x.  ( abs `  N
) ) )
8 nnre 10350 . . . . . . . . 9  |-  ( M  e.  NN  ->  M  e.  RR )
9 nnnn0 10607 . . . . . . . . . 10  |-  ( M  e.  NN  ->  M  e.  NN0 )
109nn0ge0d 10660 . . . . . . . . 9  |-  ( M  e.  NN  ->  0  <_  M )
118, 10absidd 12930 . . . . . . . 8  |-  ( M  e.  NN  ->  ( abs `  M )  =  M )
1211oveq1d 6127 . . . . . . 7  |-  ( M  e.  NN  ->  (
( abs `  M
)  x.  ( abs `  N ) )  =  ( M  x.  ( abs `  N ) ) )
1312adantr 465 . . . . . 6  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  x.  ( abs `  N ) )  =  ( M  x.  ( abs `  N ) ) )
147, 13eqtrd 2475 . . . . 5  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( abs `  ( M  x.  N )
)  =  ( M  x.  ( abs `  N
) ) )
1514oveq2d 6128 . . . 4  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( abs `  ( M  x.  N ) ) )  =  ( M  gcd  ( M  x.  ( abs `  N ) ) ) )
1615adantr 465 . . 3  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( M  gcd  ( abs `  ( M  x.  N )
) )  =  ( M  gcd  ( M  x.  ( abs `  N
) ) ) )
17 simpll 753 . . . . 5  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  M  e.  NN )
1817nnzd 10767 . . . 4  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  M  e.  ZZ )
19 nnz 10689 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  ZZ )
20 zmulcl 10714 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )
2119, 20sylan 471 . . . . 5  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )
2221adantr 465 . . . 4  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( M  x.  N )  e.  ZZ )
23 gcdabs2 13740 . . . 4  |-  ( ( M  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ )  ->  ( M  gcd  ( abs `  ( M  x.  N ) ) )  =  ( M  gcd  ( M  x.  N ) ) )
2418, 22, 23syl2anc 661 . . 3  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( M  gcd  ( abs `  ( M  x.  N )
) )  =  ( M  gcd  ( M  x.  N ) ) )
25 nnabscl 12834 . . . . 5  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
26 gcdmultiple 13755 . . . . 5  |-  ( ( M  e.  NN  /\  ( abs `  N )  e.  NN )  -> 
( M  gcd  ( M  x.  ( abs `  N ) ) )  =  M )
2725, 26sylan2 474 . . . 4  |-  ( ( M  e.  NN  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( M  gcd  ( M  x.  ( abs `  N ) ) )  =  M )
2827anassrs 648 . . 3  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( M  gcd  ( M  x.  ( abs `  N ) ) )  =  M )
2916, 24, 283eqtr3d 2483 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( M  gcd  ( M  x.  N
) )  =  M )
30 mul01 9569 . . . . . 6  |-  ( M  e.  CC  ->  ( M  x.  0 )  =  0 )
3130oveq2d 6128 . . . . 5  |-  ( M  e.  CC  ->  ( M  gcd  ( M  x.  0 ) )  =  ( M  gcd  0
) )
324, 31syl 16 . . . 4  |-  ( M  e.  NN  ->  ( M  gcd  ( M  x.  0 ) )  =  ( M  gcd  0
) )
3332adantr 465 . . 3  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  0 ) )  =  ( M  gcd  0 ) )
34 nn0gcdid0 13730 . . . . 5  |-  ( M  e.  NN0  ->  ( M  gcd  0 )  =  M )
359, 34syl 16 . . . 4  |-  ( M  e.  NN  ->  ( M  gcd  0 )  =  M )
3635adantr 465 . . 3  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  0
)  =  M )
3733, 36eqtrd 2475 . 2  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  0 ) )  =  M )
383, 29, 37pm2.61ne 2710 1  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  N )
)  =  M )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2620   ` cfv 5439  (class class class)co 6112   CCcc 9301   0cc0 9303    x. cmul 9308   NNcn 10343   NN0cn0 10600   ZZcz 10667   abscabs 12744    gcd cgcd 13711
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 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-rp 11013  df-seq 11828  df-exp 11887  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-dvds 13557  df-gcd 13712
This theorem is referenced by: (None)
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