MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gcdmultiplez Unicode version

Theorem gcdmultiplez 12979
Description: Extend gcdmultiple 12978 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 6029 . . . 4  |-  ( N  =  0  ->  ( M  x.  N )  =  ( M  x.  0 ) )
21oveq2d 6037 . . 3  |-  ( N  =  0  ->  ( M  gcd  ( M  x.  N ) )  =  ( M  gcd  ( M  x.  0 ) ) )
32eqeq1d 2396 . 2  |-  ( N  =  0  ->  (
( M  gcd  ( M  x.  N )
)  =  M  <->  ( M  gcd  ( M  x.  0 ) )  =  M ) )
4 nncn 9941 . . . . . . 7  |-  ( M  e.  NN  ->  M  e.  CC )
5 zcn 10220 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  CC )
6 absmul 12027 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( abs `  ( M  x.  N )
)  =  ( ( abs `  M )  x.  ( abs `  N
) ) )
74, 5, 6syl2an 464 . . . . . 6  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( abs `  ( M  x.  N )
)  =  ( ( abs `  M )  x.  ( abs `  N
) ) )
8 nnre 9940 . . . . . . . . 9  |-  ( M  e.  NN  ->  M  e.  RR )
9 nnnn0 10161 . . . . . . . . . 10  |-  ( M  e.  NN  ->  M  e.  NN0 )
109nn0ge0d 10210 . . . . . . . . 9  |-  ( M  e.  NN  ->  0  <_  M )
118, 10absidd 12153 . . . . . . . 8  |-  ( M  e.  NN  ->  ( abs `  M )  =  M )
1211oveq1d 6036 . . . . . . 7  |-  ( M  e.  NN  ->  (
( abs `  M
)  x.  ( abs `  N ) )  =  ( M  x.  ( abs `  N ) ) )
1312adantr 452 . . . . . 6  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  x.  ( abs `  N ) )  =  ( M  x.  ( abs `  N ) ) )
147, 13eqtrd 2420 . . . . 5  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( abs `  ( M  x.  N )
)  =  ( M  x.  ( abs `  N
) ) )
1514oveq2d 6037 . . . 4  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( abs `  ( M  x.  N ) ) )  =  ( M  gcd  ( M  x.  ( abs `  N ) ) ) )
1615adantr 452 . . 3  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( M  gcd  ( abs `  ( M  x.  N )
) )  =  ( M  gcd  ( M  x.  ( abs `  N
) ) ) )
17 simpll 731 . . . . 5  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  M  e.  NN )
1817nnzd 10307 . . . 4  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  M  e.  ZZ )
19 nnz 10236 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  ZZ )
20 zmulcl 10257 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )
2119, 20sylan 458 . . . . 5  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )
2221adantr 452 . . . 4  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( M  x.  N )  e.  ZZ )
23 gcdabs2 12963 . . . 4  |-  ( ( M  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ )  ->  ( M  gcd  ( abs `  ( M  x.  N ) ) )  =  ( M  gcd  ( M  x.  N ) ) )
2418, 22, 23syl2anc 643 . . 3  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( M  gcd  ( abs `  ( M  x.  N )
) )  =  ( M  gcd  ( M  x.  N ) ) )
25 nnabscl 12057 . . . . 5  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
26 gcdmultiple 12978 . . . . 5  |-  ( ( M  e.  NN  /\  ( abs `  N )  e.  NN )  -> 
( M  gcd  ( M  x.  ( abs `  N ) ) )  =  M )
2725, 26sylan2 461 . . . 4  |-  ( ( M  e.  NN  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( M  gcd  ( M  x.  ( abs `  N ) ) )  =  M )
2827anassrs 630 . . 3  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( M  gcd  ( M  x.  ( abs `  N ) ) )  =  M )
2916, 24, 283eqtr3d 2428 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( M  gcd  ( M  x.  N
) )  =  M )
30 mul01 9178 . . . . . 6  |-  ( M  e.  CC  ->  ( M  x.  0 )  =  0 )
3130oveq2d 6037 . . . . 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 452 . . 3  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  0 ) )  =  ( M  gcd  0 ) )
34 nn0gcdid0 12953 . . . . 5  |-  ( M  e.  NN0  ->  ( M  gcd  0 )  =  M )
359, 34syl 16 . . . 4  |-  ( M  e.  NN  ->  ( M  gcd  0 )  =  M )
3635adantr 452 . . 3  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  0
)  =  M )
3733, 36eqtrd 2420 . 2  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  0 ) )  =  M )
383, 29, 37pm2.61ne 2626 1  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  N )
)  =  M )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551   ` cfv 5395  (class class class)co 6021   CCcc 8922   0cc0 8924    x. cmul 8929   NNcn 9933   NN0cn0 10154   ZZcz 10215   abscabs 11967    gcd cgcd 12934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-seq 11252  df-exp 11311  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-dvds 12781  df-gcd 12935
  Copyright terms: Public domain W3C validator