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Theorem gcdmultiple 13856
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 6211 . . . . . 6  |-  ( k  =  1  ->  ( M  x.  k )  =  ( M  x.  1 ) )
21oveq2d 6219 . . . . 5  |-  ( k  =  1  ->  ( M  gcd  ( M  x.  k ) )  =  ( M  gcd  ( M  x.  1 ) ) )
32eqeq1d 2456 . . . 4  |-  ( k  =  1  ->  (
( M  gcd  ( M  x.  k )
)  =  M  <->  ( M  gcd  ( M  x.  1 ) )  =  M ) )
43imbi2d 316 . . 3  |-  ( k  =  1  ->  (
( M  e.  NN  ->  ( M  gcd  ( M  x.  k )
)  =  M )  <-> 
( M  e.  NN  ->  ( M  gcd  ( M  x.  1 ) )  =  M ) ) )
5 oveq2 6211 . . . . . 6  |-  ( k  =  n  ->  ( M  x.  k )  =  ( M  x.  n ) )
65oveq2d 6219 . . . . 5  |-  ( k  =  n  ->  ( M  gcd  ( M  x.  k ) )  =  ( M  gcd  ( M  x.  n )
) )
76eqeq1d 2456 . . . 4  |-  ( k  =  n  ->  (
( M  gcd  ( M  x.  k )
)  =  M  <->  ( M  gcd  ( M  x.  n
) )  =  M ) )
87imbi2d 316 . . 3  |-  ( k  =  n  ->  (
( M  e.  NN  ->  ( M  gcd  ( M  x.  k )
)  =  M )  <-> 
( M  e.  NN  ->  ( M  gcd  ( M  x.  n )
)  =  M ) ) )
9 oveq2 6211 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( M  x.  k )  =  ( M  x.  ( n  +  1
) ) )
109oveq2d 6219 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  ( M  gcd  ( M  x.  k ) )  =  ( M  gcd  ( M  x.  ( n  +  1 ) ) ) )
1110eqeq1d 2456 . . . 4  |-  ( k  =  ( n  + 
1 )  ->  (
( M  gcd  ( M  x.  k )
)  =  M  <->  ( M  gcd  ( M  x.  (
n  +  1 ) ) )  =  M ) )
1211imbi2d 316 . . 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 6211 . . . . . 6  |-  ( k  =  N  ->  ( M  x.  k )  =  ( M  x.  N ) )
1413oveq2d 6219 . . . . 5  |-  ( k  =  N  ->  ( M  gcd  ( M  x.  k ) )  =  ( M  gcd  ( M  x.  N )
) )
1514eqeq1d 2456 . . . 4  |-  ( k  =  N  ->  (
( M  gcd  ( M  x.  k )
)  =  M  <->  ( M  gcd  ( M  x.  N
) )  =  M ) )
1615imbi2d 316 . . 3  |-  ( k  =  N  ->  (
( M  e.  NN  ->  ( M  gcd  ( M  x.  k )
)  =  M )  <-> 
( M  e.  NN  ->  ( M  gcd  ( M  x.  N )
)  =  M ) ) )
17 nncn 10445 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  CC )
1817mulid1d 9518 . . . . 5  |-  ( M  e.  NN  ->  ( M  x.  1 )  =  M )
1918oveq2d 6219 . . . 4  |-  ( M  e.  NN  ->  ( M  gcd  ( M  x.  1 ) )  =  ( M  gcd  M
) )
20 nnz 10783 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  ZZ )
21 gcdid 13837 . . . . . 6  |-  ( M  e.  ZZ  ->  ( M  gcd  M )  =  ( abs `  M
) )
2220, 21syl 16 . . . . 5  |-  ( M  e.  NN  ->  ( M  gcd  M )  =  ( abs `  M
) )
23 nnre 10444 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  RR )
24 nnnn0 10701 . . . . . . 7  |-  ( M  e.  NN  ->  M  e.  NN0 )
2524nn0ge0d 10754 . . . . . 6  |-  ( M  e.  NN  ->  0  <_  M )
2623, 25absidd 13031 . . . . 5  |-  ( M  e.  NN  ->  ( abs `  M )  =  M )
2722, 26eqtrd 2495 . . . 4  |-  ( M  e.  NN  ->  ( M  gcd  M )  =  M )
2819, 27eqtrd 2495 . . 3  |-  ( M  e.  NN  ->  ( M  gcd  ( M  x.  1 ) )  =  M )
2920adantr 465 . . . . . . . . 9  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  M  e.  ZZ )
30 nnz 10783 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  e.  ZZ )
31 zmulcl 10808 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  n  e.  ZZ )  ->  ( M  x.  n
)  e.  ZZ )
3220, 30, 31syl2an 477 . . . . . . . . 9  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( M  x.  n
)  e.  ZZ )
33 1z 10791 . . . . . . . . . 10  |-  1  e.  ZZ
34 gcdaddm 13835 . . . . . . . . . 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 1302 . . . . . . . . 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 661 . . . . . . . 8  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( M  gcd  ( M  x.  n )
)  =  ( M  gcd  ( ( M  x.  n )  +  ( 1  x.  M
) ) ) )
37 nncn 10445 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  e.  CC )
38 ax-1cn 9455 . . . . . . . . . . . 12  |-  1  e.  CC
39 adddi 9486 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  n  e.  CC  /\  1  e.  CC )  ->  ( M  x.  ( n  +  1 ) )  =  ( ( M  x.  n )  +  ( M  x.  1 ) ) )
4038, 39mp3an3 1304 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  n  e.  CC )  ->  ( M  x.  (
n  +  1 ) )  =  ( ( M  x.  n )  +  ( M  x.  1 ) ) )
41 mulcom 9483 . . . . . . . . . . . . . 14  |-  ( ( M  e.  CC  /\  1  e.  CC )  ->  ( M  x.  1 )  =  ( 1  x.  M ) )
4238, 41mpan2 671 . . . . . . . . . . . . 13  |-  ( M  e.  CC  ->  ( M  x.  1 )  =  ( 1  x.  M ) )
4342adantr 465 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  n  e.  CC )  ->  ( M  x.  1 )  =  ( 1  x.  M ) )
4443oveq2d 6219 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  n  e.  CC )  ->  ( ( M  x.  n )  +  ( M  x.  1 ) )  =  ( ( M  x.  n )  +  ( 1  x.  M ) ) )
4540, 44eqtrd 2495 . . . . . . . . . 10  |-  ( ( M  e.  CC  /\  n  e.  CC )  ->  ( M  x.  (
n  +  1 ) )  =  ( ( M  x.  n )  +  ( 1  x.  M ) ) )
4617, 37, 45syl2an 477 . . . . . . . . 9  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( M  x.  (
n  +  1 ) )  =  ( ( M  x.  n )  +  ( 1  x.  M ) ) )
4746oveq2d 6219 . . . . . . . 8  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( M  gcd  ( M  x.  ( n  +  1 ) ) )  =  ( M  gcd  ( ( M  x.  n )  +  ( 1  x.  M
) ) ) )
4836, 47eqtr4d 2498 . . . . . . 7  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( M  gcd  ( M  x.  n )
)  =  ( M  gcd  ( M  x.  ( n  +  1
) ) ) )
4948eqeq1d 2456 . . . . . 6  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( ( M  gcd  ( M  x.  n
) )  =  M  <-> 
( M  gcd  ( M  x.  ( n  +  1 ) ) )  =  M ) )
5049biimpd 207 . . . . 5  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( ( M  gcd  ( M  x.  n
) )  =  M  ->  ( M  gcd  ( M  x.  (
n  +  1 ) ) )  =  M ) )
5150expcom 435 . . . 4  |-  ( n  e.  NN  ->  ( M  e.  NN  ->  ( ( M  gcd  ( M  x.  n )
)  =  M  -> 
( M  gcd  ( M  x.  ( n  +  1 ) ) )  =  M ) ) )
5251a2d 26 . . 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 10455 . 2  |-  ( N  e.  NN  ->  ( M  e.  NN  ->  ( M  gcd  ( M  x.  N ) )  =  M ) )
5453impcom 430 1  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  gcd  ( M  x.  N )
)  =  M )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   ` cfv 5529  (class class class)co 6203   CCcc 9395   1c1 9398    + caddc 9400    x. cmul 9402   NNcn 10437   ZZcz 10761   abscabs 12845    gcd cgcd 13812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7806  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-seq 11928  df-exp 11987  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-dvds 13658  df-gcd 13813
This theorem is referenced by:  gcdmultiplez  13857  rpmulgcd  13861
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