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Theorem modgcd 11064
Description: The gcd remains unchanged if one operand is replaced with its remainder modulo the other. (Contributed by Paul Chapman, 31-Mar-2011.)
Assertion
Ref Expression
modgcd  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( M  mod  N )  gcd  N )  =  ( M  gcd  N ) )

Proof of Theorem modgcd
StepHypRef Expression
1 zq 9080 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  QQ )
21adantr 270 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  QQ )
3 nnq 9087 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  QQ )
43adantl 271 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  QQ )
5 nngt0 8419 . . . . . . 7  |-  ( N  e.  NN  ->  0  <  N )
65adantl 271 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  0  <  N )
7 modqval 9696 . . . . . 6  |-  ( ( M  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  ( M  mod  N )  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N
) ) ) ) )
82, 4, 6, 7syl3anc 1174 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  mod  N
)  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N ) ) ) ) )
9 zcn 8725 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
109adantr 270 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  CC )
11 nncn 8402 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  CC )
1211adantl 271 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  CC )
13 znq 9078 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  /  N
)  e.  QQ )
1413flqcld 9649 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( |_ `  ( M  /  N ) )  e.  ZZ )
1514zcnd 8839 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( |_ `  ( M  /  N ) )  e.  CC )
16 mulneg1 7852 . . . . . . . . . . 11  |-  ( ( ( |_ `  ( M  /  N ) )  e.  CC  /\  N  e.  CC )  ->  ( -u ( |_ `  ( M  /  N ) )  x.  N )  = 
-u ( ( |_
`  ( M  /  N ) )  x.  N ) )
17 mulcom 7450 . . . . . . . . . . . 12  |-  ( ( ( |_ `  ( M  /  N ) )  e.  CC  /\  N  e.  CC )  ->  (
( |_ `  ( M  /  N ) )  x.  N )  =  ( N  x.  ( |_ `  ( M  /  N ) ) ) )
1817negeqd 7656 . . . . . . . . . . 11  |-  ( ( ( |_ `  ( M  /  N ) )  e.  CC  /\  N  e.  CC )  ->  -u (
( |_ `  ( M  /  N ) )  x.  N )  = 
-u ( N  x.  ( |_ `  ( M  /  N ) ) ) )
1916, 18eqtrd 2120 . . . . . . . . . 10  |-  ( ( ( |_ `  ( M  /  N ) )  e.  CC  /\  N  e.  CC )  ->  ( -u ( |_ `  ( M  /  N ) )  x.  N )  = 
-u ( N  x.  ( |_ `  ( M  /  N ) ) ) )
2019ancoms 264 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC )  -> 
( -u ( |_ `  ( M  /  N
) )  x.  N
)  =  -u ( N  x.  ( |_ `  ( M  /  N
) ) ) )
21203adant1 961 . . . . . . . 8  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC )  ->  ( -u ( |_ `  ( M  /  N ) )  x.  N )  = 
-u ( N  x.  ( |_ `  ( M  /  N ) ) ) )
2221oveq2d 5650 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC )  ->  ( M  +  ( -u ( |_ `  ( M  /  N ) )  x.  N ) )  =  ( M  +  -u ( N  x.  ( |_ `  ( M  /  N ) ) ) ) )
23 mulcl 7448 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC )  -> 
( N  x.  ( |_ `  ( M  /  N ) ) )  e.  CC )
24 negsub 7709 . . . . . . . . 9  |-  ( ( M  e.  CC  /\  ( N  x.  ( |_ `  ( M  /  N ) ) )  e.  CC )  -> 
( M  +  -u ( N  x.  ( |_ `  ( M  /  N ) ) ) )  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N ) ) ) ) )
2523, 24sylan2 280 . . . . . . . 8  |-  ( ( M  e.  CC  /\  ( N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC ) )  ->  ( M  +  -u ( N  x.  ( |_ `  ( M  /  N ) ) ) )  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N ) ) ) ) )
26253impb 1139 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC )  ->  ( M  +  -u ( N  x.  ( |_ `  ( M  /  N
) ) ) )  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N ) ) ) ) )
2722, 26eqtrd 2120 . . . . . 6  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC )  ->  ( M  +  ( -u ( |_ `  ( M  /  N ) )  x.  N ) )  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N
) ) ) ) )
2810, 12, 15, 27syl3anc 1174 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  +  (
-u ( |_ `  ( M  /  N
) )  x.  N
) )  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N
) ) ) ) )
298, 28eqtr4d 2123 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  mod  N
)  =  ( M  +  ( -u ( |_ `  ( M  /  N ) )  x.  N ) ) )
3029oveq2d 5650 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  ( M  mod  N ) )  =  ( N  gcd  ( M  +  ( -u ( |_ `  ( M  /  N ) )  x.  N ) ) ) )
3114znegcld 8840 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  -> 
-u ( |_ `  ( M  /  N
) )  e.  ZZ )
32 nnz 8739 . . . . 5  |-  ( N  e.  NN  ->  N  e.  ZZ )
3332adantl 271 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  ZZ )
34 simpl 107 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  ZZ )
35 gcdaddm 11057 . . . 4  |-  ( (
-u ( |_ `  ( M  /  N
) )  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  gcd  M
)  =  ( N  gcd  ( M  +  ( -u ( |_ `  ( M  /  N
) )  x.  N
) ) ) )
3631, 33, 34, 35syl3anc 1174 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  M
)  =  ( N  gcd  ( M  +  ( -u ( |_ `  ( M  /  N
) )  x.  N
) ) ) )
3730, 36eqtr4d 2123 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  ( M  mod  N ) )  =  ( N  gcd  M ) )
38 zmodcl 9716 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  mod  N
)  e.  NN0 )
3938nn0zd 8836 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  mod  N
)  e.  ZZ )
40 gcdcom 11047 . . 3  |-  ( ( N  e.  ZZ  /\  ( M  mod  N )  e.  ZZ )  -> 
( N  gcd  ( M  mod  N ) )  =  ( ( M  mod  N )  gcd 
N ) )
4133, 39, 40syl2anc 403 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  ( M  mod  N ) )  =  ( ( M  mod  N )  gcd 
N ) )
42 gcdcom 11047 . . 3  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  gcd  M
)  =  ( M  gcd  N ) )
4333, 34, 42syl2anc 403 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  M
)  =  ( M  gcd  N ) )
4437, 41, 433eqtr3d 2128 1  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( M  mod  N )  gcd  N )  =  ( M  gcd  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 924    = wceq 1289    e. wcel 1438   class class class wbr 3837   ` cfv 5002  (class class class)co 5634   CCcc 7327   0cc0 7329    + caddc 7332    x. cmul 7334    < clt 7501    - cmin 7632   -ucneg 7633    / cdiv 8113   NNcn 8394   ZZcz 8720   QQcq 9073   |_cfl 9640    mod cmo 9694    gcd cgcd 11020
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443  ax-caucvg 7444
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-sup 6658  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fz 9394  df-fzo 9519  df-fl 9642  df-mod 9695  df-iseq 9818  df-seq3 9819  df-exp 9920  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-dvds 10879  df-gcd 11021
This theorem is referenced by:  eucalginv  11120  phimullem  11283
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