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Theorem gcdmultiple 12729
Description: The GCD of a multiple of a number is the number itself. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
gcdmultiple  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  gcd  ( M  x.  N )
)  =  M )

Proof of Theorem gcdmultiple
Dummy variables  k  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5866 . . . . . 6  |-  ( k  =  1  ->  ( M  x.  k )  =  ( M  x.  1 ) )
21oveq2d 5874 . . . . 5  |-  ( k  =  1  ->  ( M  gcd  ( M  x.  k ) )  =  ( M  gcd  ( M  x.  1 ) ) )
32eqeq1d 2291 . . . 4  |-  ( k  =  1  ->  (
( M  gcd  ( M  x.  k )
)  =  M  <->  ( M  gcd  ( M  x.  1 ) )  =  M ) )
43imbi2d 307 . . 3  |-  ( k  =  1  ->  (
( M  e.  NN  ->  ( M  gcd  ( M  x.  k )
)  =  M )  <-> 
( M  e.  NN  ->  ( M  gcd  ( M  x.  1 ) )  =  M ) ) )
5 oveq2 5866 . . . . . 6  |-  ( k  =  n  ->  ( M  x.  k )  =  ( M  x.  n ) )
65oveq2d 5874 . . . . 5  |-  ( k  =  n  ->  ( M  gcd  ( M  x.  k ) )  =  ( M  gcd  ( M  x.  n )
) )
76eqeq1d 2291 . . . 4  |-  ( k  =  n  ->  (
( M  gcd  ( M  x.  k )
)  =  M  <->  ( M  gcd  ( M  x.  n
) )  =  M ) )
87imbi2d 307 . . 3  |-  ( k  =  n  ->  (
( M  e.  NN  ->  ( M  gcd  ( M  x.  k )
)  =  M )  <-> 
( M  e.  NN  ->  ( M  gcd  ( M  x.  n )
)  =  M ) ) )
9 oveq2 5866 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( M  x.  k )  =  ( M  x.  ( n  +  1
) ) )
109oveq2d 5874 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  ( M  gcd  ( M  x.  k ) )  =  ( M  gcd  ( M  x.  ( n  +  1 ) ) ) )
1110eqeq1d 2291 . . . 4  |-  ( k  =  ( n  + 
1 )  ->  (
( M  gcd  ( M  x.  k )
)  =  M  <->  ( M  gcd  ( M  x.  (
n  +  1 ) ) )  =  M ) )
1211imbi2d 307 . . 3  |-  ( k  =  ( n  + 
1 )  ->  (
( M  e.  NN  ->  ( M  gcd  ( M  x.  k )
)  =  M )  <-> 
( M  e.  NN  ->  ( M  gcd  ( M  x.  ( n  +  1 ) ) )  =  M ) ) )
13 oveq2 5866 . . . . . 6  |-  ( k  =  N  ->  ( M  x.  k )  =  ( M  x.  N ) )
1413oveq2d 5874 . . . . 5  |-  ( k  =  N  ->  ( M  gcd  ( M  x.  k ) )  =  ( M  gcd  ( M  x.  N )
) )
1514eqeq1d 2291 . . . 4  |-  ( k  =  N  ->  (
( M  gcd  ( M  x.  k )
)  =  M  <->  ( M  gcd  ( M  x.  N
) )  =  M ) )
1615imbi2d 307 . . 3  |-  ( k  =  N  ->  (
( M  e.  NN  ->  ( M  gcd  ( M  x.  k )
)  =  M )  <-> 
( M  e.  NN  ->  ( M  gcd  ( M  x.  N )
)  =  M ) ) )
17 nncn 9754 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  CC )
1817mulid1d 8852 . . . . 5  |-  ( M  e.  NN  ->  ( M  x.  1 )  =  M )
1918oveq2d 5874 . . . 4  |-  ( M  e.  NN  ->  ( M  gcd  ( M  x.  1 ) )  =  ( M  gcd  M
) )
20 nnz 10045 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  ZZ )
21 gcdid 12710 . . . . . 6  |-  ( M  e.  ZZ  ->  ( M  gcd  M )  =  ( abs `  M
) )
2220, 21syl 15 . . . . 5  |-  ( M  e.  NN  ->  ( M  gcd  M )  =  ( abs `  M
) )
23 nnre 9753 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  RR )
24 nnnn0 9972 . . . . . . 7  |-  ( M  e.  NN  ->  M  e.  NN0 )
2524nn0ge0d 10021 . . . . . 6  |-  ( M  e.  NN  ->  0  <_  M )
2623, 25absidd 11905 . . . . 5  |-  ( M  e.  NN  ->  ( abs `  M )  =  M )
2722, 26eqtrd 2315 . . . 4  |-  ( M  e.  NN  ->  ( M  gcd  M )  =  M )
2819, 27eqtrd 2315 . . 3  |-  ( M  e.  NN  ->  ( M  gcd  ( M  x.  1 ) )  =  M )
2920adantr 451 . . . . . . . . 9  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  M  e.  ZZ )
30 nnz 10045 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  e.  ZZ )
31 zmulcl 10066 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  n  e.  ZZ )  ->  ( M  x.  n
)  e.  ZZ )
3220, 30, 31syl2an 463 . . . . . . . . 9  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( M  x.  n
)  e.  ZZ )
33 1z 10053 . . . . . . . . . 10  |-  1  e.  ZZ
34 gcdaddm 12708 . . . . . . . . . 10  |-  ( ( 1  e.  ZZ  /\  M  e.  ZZ  /\  ( M  x.  n )  e.  ZZ )  ->  ( M  gcd  ( M  x.  n ) )  =  ( M  gcd  (
( M  x.  n
)  +  ( 1  x.  M ) ) ) )
3533, 34mp3an1 1264 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  ( M  x.  n
)  e.  ZZ )  ->  ( M  gcd  ( M  x.  n
) )  =  ( M  gcd  ( ( M  x.  n )  +  ( 1  x.  M ) ) ) )
3629, 32, 35syl2anc 642 . . . . . . . 8  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( M  gcd  ( M  x.  n )
)  =  ( M  gcd  ( ( M  x.  n )  +  ( 1  x.  M
) ) ) )
37 nncn 9754 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  e.  CC )
38 ax-1cn 8795 . . . . . . . . . . . 12  |-  1  e.  CC
39 adddi 8826 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  n  e.  CC  /\  1  e.  CC )  ->  ( M  x.  ( n  +  1 ) )  =  ( ( M  x.  n )  +  ( M  x.  1 ) ) )
4038, 39mp3an3 1266 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  n  e.  CC )  ->  ( M  x.  (
n  +  1 ) )  =  ( ( M  x.  n )  +  ( M  x.  1 ) ) )
41 mulcom 8823 . . . . . . . . . . . . . 14  |-  ( ( M  e.  CC  /\  1  e.  CC )  ->  ( M  x.  1 )  =  ( 1  x.  M ) )
4238, 41mpan2 652 . . . . . . . . . . . . 13  |-  ( M  e.  CC  ->  ( M  x.  1 )  =  ( 1  x.  M ) )
4342adantr 451 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  n  e.  CC )  ->  ( M  x.  1 )  =  ( 1  x.  M ) )
4443oveq2d 5874 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  n  e.  CC )  ->  ( ( M  x.  n )  +  ( M  x.  1 ) )  =  ( ( M  x.  n )  +  ( 1  x.  M ) ) )
4540, 44eqtrd 2315 . . . . . . . . . 10  |-  ( ( M  e.  CC  /\  n  e.  CC )  ->  ( M  x.  (
n  +  1 ) )  =  ( ( M  x.  n )  +  ( 1  x.  M ) ) )
4617, 37, 45syl2an 463 . . . . . . . . 9  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( M  x.  (
n  +  1 ) )  =  ( ( M  x.  n )  +  ( 1  x.  M ) ) )
4746oveq2d 5874 . . . . . . . 8  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( M  gcd  ( M  x.  ( n  +  1 ) ) )  =  ( M  gcd  ( ( M  x.  n )  +  ( 1  x.  M
) ) ) )
4836, 47eqtr4d 2318 . . . . . . 7  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( M  gcd  ( M  x.  n )
)  =  ( M  gcd  ( M  x.  ( n  +  1
) ) ) )
4948eqeq1d 2291 . . . . . 6  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( ( M  gcd  ( M  x.  n
) )  =  M  <-> 
( M  gcd  ( M  x.  ( n  +  1 ) ) )  =  M ) )
5049biimpd 198 . . . . 5  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( ( M  gcd  ( M  x.  n
) )  =  M  ->  ( M  gcd  ( M  x.  (
n  +  1 ) ) )  =  M ) )
5150expcom 424 . . . 4  |-  ( n  e.  NN  ->  ( M  e.  NN  ->  ( ( M  gcd  ( M  x.  n )
)  =  M  -> 
( M  gcd  ( M  x.  ( n  +  1 ) ) )  =  M ) ) )
5251a2d 23 . . 3  |-  ( n  e.  NN  ->  (
( M  e.  NN  ->  ( M  gcd  ( M  x.  n )
)  =  M )  ->  ( M  e.  NN  ->  ( M  gcd  ( M  x.  (
n  +  1 ) ) )  =  M ) ) )
534, 8, 12, 16, 28, 52nnind 9764 . 2  |-  ( N  e.  NN  ->  ( M  e.  NN  ->  ( M  gcd  ( M  x.  N ) )  =  M ) )
5453impcom 419 1  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  gcd  ( M  x.  N )
)  =  M )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    x. cmul 8742   NNcn 9746   ZZcz 10024   abscabs 11719    gcd cgcd 12685
This theorem is referenced by:  gcdmultiplez  12730  rpmulgcd  12734
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686
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