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Theorem modgcd 11713
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 9444 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  QQ )
21adantr 274 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  QQ )
3 nnq 9451 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  QQ )
43adantl 275 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  QQ )
5 nngt0 8768 . . . . . . 7  |-  ( N  e.  NN  ->  0  <  N )
65adantl 275 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  0  <  N )
7 modqval 10127 . . . . . 6  |-  ( ( M  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  ( M  mod  N )  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N
) ) ) ) )
82, 4, 6, 7syl3anc 1217 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  mod  N
)  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N ) ) ) ) )
9 zcn 9082 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
109adantr 274 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  CC )
11 nncn 8751 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  CC )
1211adantl 275 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  CC )
13 znq 9442 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  /  N
)  e.  QQ )
1413flqcld 10080 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( |_ `  ( M  /  N ) )  e.  ZZ )
1514zcnd 9197 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( |_ `  ( M  /  N ) )  e.  CC )
16 mulneg1 8180 . . . . . . . . . . 11  |-  ( ( ( |_ `  ( M  /  N ) )  e.  CC  /\  N  e.  CC )  ->  ( -u ( |_ `  ( M  /  N ) )  x.  N )  = 
-u ( ( |_
`  ( M  /  N ) )  x.  N ) )
17 mulcom 7772 . . . . . . . . . . . 12  |-  ( ( ( |_ `  ( M  /  N ) )  e.  CC  /\  N  e.  CC )  ->  (
( |_ `  ( M  /  N ) )  x.  N )  =  ( N  x.  ( |_ `  ( M  /  N ) ) ) )
1817negeqd 7980 . . . . . . . . . . 11  |-  ( ( ( |_ `  ( M  /  N ) )  e.  CC  /\  N  e.  CC )  ->  -u (
( |_ `  ( M  /  N ) )  x.  N )  = 
-u ( N  x.  ( |_ `  ( M  /  N ) ) ) )
1916, 18eqtrd 2173 . . . . . . . . . 10  |-  ( ( ( |_ `  ( M  /  N ) )  e.  CC  /\  N  e.  CC )  ->  ( -u ( |_ `  ( M  /  N ) )  x.  N )  = 
-u ( N  x.  ( |_ `  ( M  /  N ) ) ) )
2019ancoms 266 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC )  -> 
( -u ( |_ `  ( M  /  N
) )  x.  N
)  =  -u ( N  x.  ( |_ `  ( M  /  N
) ) ) )
21203adant1 1000 . . . . . . . 8  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC )  ->  ( -u ( |_ `  ( M  /  N ) )  x.  N )  = 
-u ( N  x.  ( |_ `  ( M  /  N ) ) ) )
2221oveq2d 5797 . . . . . . 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 7770 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC )  -> 
( N  x.  ( |_ `  ( M  /  N ) ) )  e.  CC )
24 negsub 8033 . . . . . . . . 9  |-  ( ( M  e.  CC  /\  ( N  x.  ( |_ `  ( M  /  N ) ) )  e.  CC )  -> 
( M  +  -u ( N  x.  ( |_ `  ( M  /  N ) ) ) )  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N ) ) ) ) )
2523, 24sylan2 284 . . . . . . . 8  |-  ( ( M  e.  CC  /\  ( N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC ) )  ->  ( M  +  -u ( N  x.  ( |_ `  ( M  /  N ) ) ) )  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N ) ) ) ) )
26253impb 1178 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC )  ->  ( M  +  -u ( N  x.  ( |_ `  ( M  /  N
) ) ) )  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N ) ) ) ) )
2722, 26eqtrd 2173 . . . . . 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 1217 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  +  (
-u ( |_ `  ( M  /  N
) )  x.  N
) )  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N
) ) ) ) )
298, 28eqtr4d 2176 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  mod  N
)  =  ( M  +  ( -u ( |_ `  ( M  /  N ) )  x.  N ) ) )
3029oveq2d 5797 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  ( M  mod  N ) )  =  ( N  gcd  ( M  +  ( -u ( |_ `  ( M  /  N ) )  x.  N ) ) ) )
3114znegcld 9198 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  -> 
-u ( |_ `  ( M  /  N
) )  e.  ZZ )
32 nnz 9096 . . . . 5  |-  ( N  e.  NN  ->  N  e.  ZZ )
3332adantl 275 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  ZZ )
34 simpl 108 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  ZZ )
35 gcdaddm 11706 . . . 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 1217 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  M
)  =  ( N  gcd  ( M  +  ( -u ( |_ `  ( M  /  N
) )  x.  N
) ) ) )
3730, 36eqtr4d 2176 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  ( M  mod  N ) )  =  ( N  gcd  M ) )
38 zmodcl 10147 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  mod  N
)  e.  NN0 )
3938nn0zd 9194 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  mod  N
)  e.  ZZ )
40 gcdcom 11696 . . 3  |-  ( ( N  e.  ZZ  /\  ( M  mod  N )  e.  ZZ )  -> 
( N  gcd  ( M  mod  N ) )  =  ( ( M  mod  N )  gcd 
N ) )
4133, 39, 40syl2anc 409 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  ( M  mod  N ) )  =  ( ( M  mod  N )  gcd 
N ) )
42 gcdcom 11696 . . 3  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  gcd  M
)  =  ( M  gcd  N ) )
4333, 34, 42syl2anc 409 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  M
)  =  ( M  gcd  N ) )
4437, 41, 433eqtr3d 2181 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 103    /\ w3a 963    = wceq 1332    e. wcel 1481   class class class wbr 3936   ` cfv 5130  (class class class)co 5781   CCcc 7641   0cc0 7643    + caddc 7646    x. cmul 7648    < clt 7823    - cmin 7956   -ucneg 7957    / cdiv 8455   NNcn 8743   ZZcz 9077   QQcq 9437   |_cfl 10071    mod cmo 10125    gcd cgcd 11669
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-mulrcl 7742  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-precex 7753  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759  ax-pre-mulgt0 7760  ax-pre-mulext 7761  ax-arch 7762  ax-caucvg 7763
This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-ilim 4298  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-frec 6295  df-sup 6878  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-reap 8360  df-ap 8367  df-div 8456  df-inn 8744  df-2 8802  df-3 8803  df-4 8804  df-n0 9001  df-z 9078  df-uz 9350  df-q 9438  df-rp 9470  df-fz 9821  df-fzo 9950  df-fl 10073  df-mod 10126  df-seqfrec 10249  df-exp 10323  df-cj 10645  df-re 10646  df-im 10647  df-rsqrt 10801  df-abs 10802  df-dvds 11528  df-gcd 11670
This theorem is referenced by:  eucalginv  11771  phimullem  11935
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