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Theorem modgcd 12991
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 zre 10242 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  RR )
2 nnrp 10577 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  RR+ )
3 modval 11207 . . . . . 6  |-  ( ( M  e.  RR  /\  N  e.  RR+ )  -> 
( M  mod  N
)  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N ) ) ) ) )
41, 2, 3syl2an 464 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  mod  N
)  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N ) ) ) ) )
5 zcn 10243 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
65adantr 452 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  CC )
7 nncn 9964 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  CC )
87adantl 453 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  CC )
9 nnre 9963 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  RR )
10 nnne0 9988 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  =/=  0 )
11 redivcl 9689 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  N  =/=  0 )  ->  ( M  /  N )  e.  RR )
121, 9, 10, 11syl3an 1226 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  N  e.  NN )  ->  ( M  /  N )  e.  RR )
13123anidm23 1243 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  /  N
)  e.  RR )
1413flcld 11162 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( |_ `  ( M  /  N ) )  e.  ZZ )
1514zcnd 10332 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( |_ `  ( M  /  N ) )  e.  CC )
16 mulneg1 9426 . . . . . . . . . . 11  |-  ( ( ( |_ `  ( M  /  N ) )  e.  CC  /\  N  e.  CC )  ->  ( -u ( |_ `  ( M  /  N ) )  x.  N )  = 
-u ( ( |_
`  ( M  /  N ) )  x.  N ) )
17 mulcom 9032 . . . . . . . . . . . 12  |-  ( ( ( |_ `  ( M  /  N ) )  e.  CC  /\  N  e.  CC )  ->  (
( |_ `  ( M  /  N ) )  x.  N )  =  ( N  x.  ( |_ `  ( M  /  N ) ) ) )
1817negeqd 9256 . . . . . . . . . . 11  |-  ( ( ( |_ `  ( M  /  N ) )  e.  CC  /\  N  e.  CC )  ->  -u (
( |_ `  ( M  /  N ) )  x.  N )  = 
-u ( N  x.  ( |_ `  ( M  /  N ) ) ) )
1916, 18eqtrd 2436 . . . . . . . . . 10  |-  ( ( ( |_ `  ( M  /  N ) )  e.  CC  /\  N  e.  CC )  ->  ( -u ( |_ `  ( M  /  N ) )  x.  N )  = 
-u ( N  x.  ( |_ `  ( M  /  N ) ) ) )
2019ancoms 440 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC )  -> 
( -u ( |_ `  ( M  /  N
) )  x.  N
)  =  -u ( N  x.  ( |_ `  ( M  /  N
) ) ) )
21203adant1 975 . . . . . . . 8  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC )  ->  ( -u ( |_ `  ( M  /  N ) )  x.  N )  = 
-u ( N  x.  ( |_ `  ( M  /  N ) ) ) )
2221oveq2d 6056 . . . . . . 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 9030 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC )  -> 
( N  x.  ( |_ `  ( M  /  N ) ) )  e.  CC )
24 negsub 9305 . . . . . . . . 9  |-  ( ( M  e.  CC  /\  ( N  x.  ( |_ `  ( M  /  N ) ) )  e.  CC )  -> 
( M  +  -u ( N  x.  ( |_ `  ( M  /  N ) ) ) )  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N ) ) ) ) )
2523, 24sylan2 461 . . . . . . . 8  |-  ( ( M  e.  CC  /\  ( N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC ) )  ->  ( M  +  -u ( N  x.  ( |_ `  ( M  /  N ) ) ) )  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N ) ) ) ) )
26253impb 1149 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC )  ->  ( M  +  -u ( N  x.  ( |_ `  ( M  /  N
) ) ) )  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N ) ) ) ) )
2722, 26eqtrd 2436 . . . . . 6  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC )  ->  ( M  +  ( -u ( |_ `  ( M  /  N ) )  x.  N ) )  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N
) ) ) ) )
286, 8, 15, 27syl3anc 1184 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  +  (
-u ( |_ `  ( M  /  N
) )  x.  N
) )  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N
) ) ) ) )
294, 28eqtr4d 2439 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  mod  N
)  =  ( M  +  ( -u ( |_ `  ( M  /  N ) )  x.  N ) ) )
3029oveq2d 6056 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  ( M  mod  N ) )  =  ( N  gcd  ( M  +  ( -u ( |_ `  ( M  /  N ) )  x.  N ) ) ) )
3114znegcld 10333 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  -> 
-u ( |_ `  ( M  /  N
) )  e.  ZZ )
32 nnz 10259 . . . . 5  |-  ( N  e.  NN  ->  N  e.  ZZ )
3332adantl 453 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  ZZ )
34 simpl 444 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  ZZ )
35 gcdaddm 12984 . . . 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 1184 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  M
)  =  ( N  gcd  ( M  +  ( -u ( |_ `  ( M  /  N
) )  x.  N
) ) ) )
3730, 36eqtr4d 2439 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  ( M  mod  N ) )  =  ( N  gcd  M ) )
38 zmodcl 11221 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  mod  N
)  e.  NN0 )
3938nn0zd 10329 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  mod  N
)  e.  ZZ )
40 gcdcom 12975 . . 3  |-  ( ( N  e.  ZZ  /\  ( M  mod  N )  e.  ZZ )  -> 
( N  gcd  ( M  mod  N ) )  =  ( ( M  mod  N )  gcd 
N ) )
4133, 39, 40syl2anc 643 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  ( M  mod  N ) )  =  ( ( M  mod  N )  gcd 
N ) )
42 gcdcom 12975 . . 3  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  gcd  M
)  =  ( M  gcd  N ) )
4333, 34, 42syl2anc 643 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  M
)  =  ( M  gcd  N ) )
4437, 41, 433eqtr3d 2444 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946    + caddc 8949    x. cmul 8951    - cmin 9247   -ucneg 9248    / cdiv 9633   NNcn 9956   ZZcz 10238   RR+crp 10568   |_cfl 11156    mod cmo 11205    gcd cgcd 12961
This theorem is referenced by:  eucalginv  13030  phimullem  13123  eulerthlem1  13125  pockthlem  13228  gcdmodi  13365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962
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