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Theorem gcdmultiplez 11636
Description: Extend gcdmultiple 11635 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 9033 . . . 4  |-  0  e.  ZZ
2 zdceq 9094 . . . 4  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
31, 2mpan2 421 . . 3  |-  ( N  e.  ZZ  -> DECID  N  =  0
)
4 exmiddc 806 . . 3  |-  (DECID  N  =  0  ->  ( N  =  0  \/  -.  N  =  0 ) )
5 nncn 8696 . . . . . . . 8  |-  ( M  e.  NN  ->  M  e.  CC )
6 mul01 8119 . . . . . . . . 9  |-  ( M  e.  CC  ->  ( M  x.  0 )  =  0 )
76oveq2d 5758 . . . . . . . 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 274 . . . . . 6  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  0 ) )  =  ( M  gcd  0 ) )
10 nnnn0 8952 . . . . . . . 8  |-  ( M  e.  NN  ->  M  e.  NN0 )
11 nn0gcdid0 11596 . . . . . . . 8  |-  ( M  e.  NN0  ->  ( M  gcd  0 )  =  M )
1210, 11syl 14 . . . . . . 7  |-  ( M  e.  NN  ->  ( M  gcd  0 )  =  M )
1312adantr 274 . . . . . 6  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  0
)  =  M )
149, 13eqtrd 2150 . . . . 5  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  0 ) )  =  M )
15 oveq2 5750 . . . . . . 7  |-  ( N  =  0  ->  ( M  x.  N )  =  ( M  x.  0 ) )
1615oveq2d 5758 . . . . . 6  |-  ( N  =  0  ->  ( M  gcd  ( M  x.  N ) )  =  ( M  gcd  ( M  x.  0 ) ) )
1716eqeq1d 2126 . . . . 5  |-  ( N  =  0  ->  (
( M  gcd  ( M  x.  N )
)  =  M  <->  ( M  gcd  ( M  x.  0 ) )  =  M ) )
1814, 17syl5ibr 155 . . . 4  |-  ( N  =  0  ->  (
( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  N
) )  =  M ) )
19 df-ne 2286 . . . . 5  |-  ( N  =/=  0  <->  -.  N  =  0 )
20 zcn 9027 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
21 absmul 10809 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( abs `  ( M  x.  N )
)  =  ( ( abs `  M )  x.  ( abs `  N
) ) )
225, 20, 21syl2an 287 . . . . . . . . . 10  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( abs `  ( M  x.  N )
)  =  ( ( abs `  M )  x.  ( abs `  N
) ) )
23 nnre 8695 . . . . . . . . . . . . 13  |-  ( M  e.  NN  ->  M  e.  RR )
2410nn0ge0d 9001 . . . . . . . . . . . . 13  |-  ( M  e.  NN  ->  0  <_  M )
2523, 24absidd 10907 . . . . . . . . . . . 12  |-  ( M  e.  NN  ->  ( abs `  M )  =  M )
2625oveq1d 5757 . . . . . . . . . . 11  |-  ( M  e.  NN  ->  (
( abs `  M
)  x.  ( abs `  N ) )  =  ( M  x.  ( abs `  N ) ) )
2726adantr 274 . . . . . . . . . 10  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  x.  ( abs `  N ) )  =  ( M  x.  ( abs `  N ) ) )
2822, 27eqtrd 2150 . . . . . . . . 9  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( abs `  ( M  x.  N )
)  =  ( M  x.  ( abs `  N
) ) )
2928oveq2d 5758 . . . . . . . 8  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( abs `  ( M  x.  N ) ) )  =  ( M  gcd  ( M  x.  ( abs `  N ) ) ) )
3029adantr 274 . . . . . . 7  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( M  gcd  ( abs `  ( M  x.  N )
) )  =  ( M  gcd  ( M  x.  ( abs `  N
) ) ) )
31 simpll 503 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  M  e.  NN )
3231nnzd 9140 . . . . . . . 8  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  M  e.  ZZ )
33 nnz 9041 . . . . . . . . . 10  |-  ( M  e.  NN  ->  M  e.  ZZ )
34 zmulcl 9075 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )
3533, 34sylan 281 . . . . . . . . 9  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )
3635adantr 274 . . . . . . . 8  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( M  x.  N )  e.  ZZ )
37 gcdabs2 11605 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ )  ->  ( M  gcd  ( abs `  ( M  x.  N ) ) )  =  ( M  gcd  ( M  x.  N ) ) )
3832, 36, 37syl2anc 408 . . . . . . 7  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( M  gcd  ( abs `  ( M  x.  N )
) )  =  ( M  gcd  ( M  x.  N ) ) )
39 nnabscl 10840 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
40 gcdmultiple 11635 . . . . . . . . 9  |-  ( ( M  e.  NN  /\  ( abs `  N )  e.  NN )  -> 
( M  gcd  ( M  x.  ( abs `  N ) ) )  =  M )
4139, 40sylan2 284 . . . . . . . 8  |-  ( ( M  e.  NN  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( M  gcd  ( M  x.  ( abs `  N ) ) )  =  M )
4241anassrs 397 . . . . . . 7  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( M  gcd  ( M  x.  ( abs `  N ) ) )  =  M )
4330, 38, 423eqtr3d 2158 . . . . . 6  |-  ( ( ( M  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( M  gcd  ( M  x.  N
) )  =  M )
4443expcom 115 . . . . 5  |-  ( N  =/=  0  ->  (
( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  N
) )  =  M ) )
4519, 44sylbir 134 . . . 4  |-  ( -.  N  =  0  -> 
( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( M  gcd  ( M  x.  N ) )  =  M ) )
4618, 45jaoi 690 . . 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 555 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 103    \/ wo 682  DECID wdc 804    = wceq 1316    e. wcel 1465    =/= wne 2285   ` cfv 5093  (class class class)co 5742   CCcc 7586   0cc0 7588    x. cmul 7593   NNcn 8688   NN0cn0 8945   ZZcz 9022   abscabs 10737    gcd cgcd 11562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-stab 801  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-sup 6839  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-n0 8946  df-z 9023  df-uz 9295  df-q 9380  df-rp 9410  df-fz 9759  df-fzo 9888  df-fl 10011  df-mod 10064  df-seqfrec 10187  df-exp 10261  df-cj 10582  df-re 10583  df-im 10584  df-rsqrt 10738  df-abs 10739  df-dvds 11421  df-gcd 11563
This theorem is referenced by: (None)
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