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Theorem gcdmultiple 11964
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 5859 . . . . . 6  |-  ( k  =  1  ->  ( M  x.  k )  =  ( M  x.  1 ) )
21oveq2d 5867 . . . . 5  |-  ( k  =  1  ->  ( M  gcd  ( M  x.  k ) )  =  ( M  gcd  ( M  x.  1 ) ) )
32eqeq1d 2179 . . . 4  |-  ( k  =  1  ->  (
( M  gcd  ( M  x.  k )
)  =  M  <->  ( M  gcd  ( M  x.  1 ) )  =  M ) )
43imbi2d 229 . . 3  |-  ( k  =  1  ->  (
( M  e.  NN  ->  ( M  gcd  ( M  x.  k )
)  =  M )  <-> 
( M  e.  NN  ->  ( M  gcd  ( M  x.  1 ) )  =  M ) ) )
5 oveq2 5859 . . . . . 6  |-  ( k  =  n  ->  ( M  x.  k )  =  ( M  x.  n ) )
65oveq2d 5867 . . . . 5  |-  ( k  =  n  ->  ( M  gcd  ( M  x.  k ) )  =  ( M  gcd  ( M  x.  n )
) )
76eqeq1d 2179 . . . 4  |-  ( k  =  n  ->  (
( M  gcd  ( M  x.  k )
)  =  M  <->  ( M  gcd  ( M  x.  n
) )  =  M ) )
87imbi2d 229 . . 3  |-  ( k  =  n  ->  (
( M  e.  NN  ->  ( M  gcd  ( M  x.  k )
)  =  M )  <-> 
( M  e.  NN  ->  ( M  gcd  ( M  x.  n )
)  =  M ) ) )
9 oveq2 5859 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( M  x.  k )  =  ( M  x.  ( n  +  1
) ) )
109oveq2d 5867 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  ( M  gcd  ( M  x.  k ) )  =  ( M  gcd  ( M  x.  ( n  +  1 ) ) ) )
1110eqeq1d 2179 . . . 4  |-  ( k  =  ( n  + 
1 )  ->  (
( M  gcd  ( M  x.  k )
)  =  M  <->  ( M  gcd  ( M  x.  (
n  +  1 ) ) )  =  M ) )
1211imbi2d 229 . . 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 5859 . . . . . 6  |-  ( k  =  N  ->  ( M  x.  k )  =  ( M  x.  N ) )
1413oveq2d 5867 . . . . 5  |-  ( k  =  N  ->  ( M  gcd  ( M  x.  k ) )  =  ( M  gcd  ( M  x.  N )
) )
1514eqeq1d 2179 . . . 4  |-  ( k  =  N  ->  (
( M  gcd  ( M  x.  k )
)  =  M  <->  ( M  gcd  ( M  x.  N
) )  =  M ) )
1615imbi2d 229 . . 3  |-  ( k  =  N  ->  (
( M  e.  NN  ->  ( M  gcd  ( M  x.  k )
)  =  M )  <-> 
( M  e.  NN  ->  ( M  gcd  ( M  x.  N )
)  =  M ) ) )
17 nncn 8875 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  CC )
1817mulid1d 7926 . . . . 5  |-  ( M  e.  NN  ->  ( M  x.  1 )  =  M )
1918oveq2d 5867 . . . 4  |-  ( M  e.  NN  ->  ( M  gcd  ( M  x.  1 ) )  =  ( M  gcd  M
) )
20 nnz 9220 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  ZZ )
21 gcdid 11930 . . . . . 6  |-  ( M  e.  ZZ  ->  ( M  gcd  M )  =  ( abs `  M
) )
2220, 21syl 14 . . . . 5  |-  ( M  e.  NN  ->  ( M  gcd  M )  =  ( abs `  M
) )
23 nnre 8874 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  RR )
24 nnnn0 9131 . . . . . . 7  |-  ( M  e.  NN  ->  M  e.  NN0 )
2524nn0ge0d 9180 . . . . . 6  |-  ( M  e.  NN  ->  0  <_  M )
2623, 25absidd 11120 . . . . 5  |-  ( M  e.  NN  ->  ( abs `  M )  =  M )
2722, 26eqtrd 2203 . . . 4  |-  ( M  e.  NN  ->  ( M  gcd  M )  =  M )
2819, 27eqtrd 2203 . . 3  |-  ( M  e.  NN  ->  ( M  gcd  ( M  x.  1 ) )  =  M )
2920adantr 274 . . . . . . . . 9  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  M  e.  ZZ )
30 nnz 9220 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  e.  ZZ )
31 zmulcl 9254 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  n  e.  ZZ )  ->  ( M  x.  n
)  e.  ZZ )
3220, 30, 31syl2an 287 . . . . . . . . 9  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( M  x.  n
)  e.  ZZ )
33 1z 9227 . . . . . . . . . 10  |-  1  e.  ZZ
34 gcdaddm 11928 . . . . . . . . . 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 1319 . . . . . . . . 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 409 . . . . . . . 8  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( M  gcd  ( M  x.  n )
)  =  ( M  gcd  ( ( M  x.  n )  +  ( 1  x.  M
) ) ) )
37 nncn 8875 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  e.  CC )
38 ax-1cn 7856 . . . . . . . . . . . 12  |-  1  e.  CC
39 adddi 7895 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  n  e.  CC  /\  1  e.  CC )  ->  ( M  x.  ( n  +  1 ) )  =  ( ( M  x.  n )  +  ( M  x.  1 ) ) )
4038, 39mp3an3 1321 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  n  e.  CC )  ->  ( M  x.  (
n  +  1 ) )  =  ( ( M  x.  n )  +  ( M  x.  1 ) ) )
41 mulcom 7892 . . . . . . . . . . . . . 14  |-  ( ( M  e.  CC  /\  1  e.  CC )  ->  ( M  x.  1 )  =  ( 1  x.  M ) )
4238, 41mpan2 423 . . . . . . . . . . . . 13  |-  ( M  e.  CC  ->  ( M  x.  1 )  =  ( 1  x.  M ) )
4342adantr 274 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  n  e.  CC )  ->  ( M  x.  1 )  =  ( 1  x.  M ) )
4443oveq2d 5867 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  n  e.  CC )  ->  ( ( M  x.  n )  +  ( M  x.  1 ) )  =  ( ( M  x.  n )  +  ( 1  x.  M ) ) )
4540, 44eqtrd 2203 . . . . . . . . . 10  |-  ( ( M  e.  CC  /\  n  e.  CC )  ->  ( M  x.  (
n  +  1 ) )  =  ( ( M  x.  n )  +  ( 1  x.  M ) ) )
4617, 37, 45syl2an 287 . . . . . . . . 9  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( M  x.  (
n  +  1 ) )  =  ( ( M  x.  n )  +  ( 1  x.  M ) ) )
4746oveq2d 5867 . . . . . . . 8  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( M  gcd  ( M  x.  ( n  +  1 ) ) )  =  ( M  gcd  ( ( M  x.  n )  +  ( 1  x.  M
) ) ) )
4836, 47eqtr4d 2206 . . . . . . 7  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( M  gcd  ( M  x.  n )
)  =  ( M  gcd  ( M  x.  ( n  +  1
) ) ) )
4948eqeq1d 2179 . . . . . 6  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( ( M  gcd  ( M  x.  n
) )  =  M  <-> 
( M  gcd  ( M  x.  ( n  +  1 ) ) )  =  M ) )
5049biimpd 143 . . . . 5  |-  ( ( M  e.  NN  /\  n  e.  NN )  ->  ( ( M  gcd  ( M  x.  n
) )  =  M  ->  ( M  gcd  ( M  x.  (
n  +  1 ) ) )  =  M ) )
5150expcom 115 . . . 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 8883 . 2  |-  ( N  e.  NN  ->  ( M  e.  NN  ->  ( M  gcd  ( M  x.  N ) )  =  M ) )
5453impcom 124 1  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  gcd  ( M  x.  N )
)  =  M )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   ` cfv 5196  (class class class)co 5851   CCcc 7761   1c1 7764    + caddc 7766    x. cmul 7768   NNcn 8867   ZZcz 9201   abscabs 10950    gcd cgcd 11886
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7854  ax-resscn 7855  ax-1cn 7856  ax-1re 7857  ax-icn 7858  ax-addcl 7859  ax-addrcl 7860  ax-mulcl 7861  ax-mulrcl 7862  ax-addcom 7863  ax-mulcom 7864  ax-addass 7865  ax-mulass 7866  ax-distr 7867  ax-i2m1 7868  ax-0lt1 7869  ax-1rid 7870  ax-0id 7871  ax-rnegex 7872  ax-precex 7873  ax-cnre 7874  ax-pre-ltirr 7875  ax-pre-ltwlin 7876  ax-pre-lttrn 7877  ax-pre-apti 7878  ax-pre-ltadd 7879  ax-pre-mulgt0 7880  ax-pre-mulext 7881  ax-arch 7882  ax-caucvg 7883
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-frec 6368  df-sup 6958  df-pnf 7945  df-mnf 7946  df-xr 7947  df-ltxr 7948  df-le 7949  df-sub 8081  df-neg 8082  df-reap 8483  df-ap 8490  df-div 8579  df-inn 8868  df-2 8926  df-3 8927  df-4 8928  df-n0 9125  df-z 9202  df-uz 9477  df-q 9568  df-rp 9600  df-fz 9955  df-fzo 10088  df-fl 10215  df-mod 10268  df-seqfrec 10391  df-exp 10465  df-cj 10795  df-re 10796  df-im 10797  df-rsqrt 10951  df-abs 10952  df-dvds 11739  df-gcd 11887
This theorem is referenced by:  gcdmultiplez  11965  rpmulgcd  11970
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