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Theorem modgcd 12715
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 10028 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  RR )
2 nnrp 10363 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  RR+ )
3 modval 10975 . . . . . 6  |-  ( ( M  e.  RR  /\  N  e.  RR+ )  -> 
( M  mod  N
)  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N ) ) ) ) )
41, 2, 3syl2an 463 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  mod  N
)  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N ) ) ) ) )
5 zcn 10029 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
65adantr 451 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  CC )
7 nncn 9754 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  CC )
87adantl 452 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  CC )
9 nnre 9753 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  RR )
10 nnne0 9778 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  =/=  0 )
11 redivcl 9479 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  N  =/=  0 )  ->  ( M  /  N )  e.  RR )
121, 9, 10, 11syl3an 1224 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  N  e.  NN )  ->  ( M  /  N )  e.  RR )
13123anidm23 1241 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  /  N
)  e.  RR )
1413flcld 10930 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( |_ `  ( M  /  N ) )  e.  ZZ )
1514zcnd 10118 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( |_ `  ( M  /  N ) )  e.  CC )
16 mulneg1 9216 . . . . . . . . . . 11  |-  ( ( ( |_ `  ( M  /  N ) )  e.  CC  /\  N  e.  CC )  ->  ( -u ( |_ `  ( M  /  N ) )  x.  N )  = 
-u ( ( |_
`  ( M  /  N ) )  x.  N ) )
17 mulcom 8823 . . . . . . . . . . . 12  |-  ( ( ( |_ `  ( M  /  N ) )  e.  CC  /\  N  e.  CC )  ->  (
( |_ `  ( M  /  N ) )  x.  N )  =  ( N  x.  ( |_ `  ( M  /  N ) ) ) )
1817negeqd 9046 . . . . . . . . . . 11  |-  ( ( ( |_ `  ( M  /  N ) )  e.  CC  /\  N  e.  CC )  ->  -u (
( |_ `  ( M  /  N ) )  x.  N )  = 
-u ( N  x.  ( |_ `  ( M  /  N ) ) ) )
1916, 18eqtrd 2315 . . . . . . . . . 10  |-  ( ( ( |_ `  ( M  /  N ) )  e.  CC  /\  N  e.  CC )  ->  ( -u ( |_ `  ( M  /  N ) )  x.  N )  = 
-u ( N  x.  ( |_ `  ( M  /  N ) ) ) )
2019ancoms 439 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC )  -> 
( -u ( |_ `  ( M  /  N
) )  x.  N
)  =  -u ( N  x.  ( |_ `  ( M  /  N
) ) ) )
21203adant1 973 . . . . . . . 8  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC )  ->  ( -u ( |_ `  ( M  /  N ) )  x.  N )  = 
-u ( N  x.  ( |_ `  ( M  /  N ) ) ) )
2221oveq2d 5874 . . . . . . 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 8821 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC )  -> 
( N  x.  ( |_ `  ( M  /  N ) ) )  e.  CC )
24 negsub 9095 . . . . . . . . 9  |-  ( ( M  e.  CC  /\  ( N  x.  ( |_ `  ( M  /  N ) ) )  e.  CC )  -> 
( M  +  -u ( N  x.  ( |_ `  ( M  /  N ) ) ) )  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N ) ) ) ) )
2523, 24sylan2 460 . . . . . . . 8  |-  ( ( M  e.  CC  /\  ( N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC ) )  ->  ( M  +  -u ( N  x.  ( |_ `  ( M  /  N ) ) ) )  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N ) ) ) ) )
26253impb 1147 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC  /\  ( |_ `  ( M  /  N ) )  e.  CC )  ->  ( M  +  -u ( N  x.  ( |_ `  ( M  /  N
) ) ) )  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N ) ) ) ) )
2722, 26eqtrd 2315 . . . . . 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 1182 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  +  (
-u ( |_ `  ( M  /  N
) )  x.  N
) )  =  ( M  -  ( N  x.  ( |_ `  ( M  /  N
) ) ) ) )
294, 28eqtr4d 2318 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  mod  N
)  =  ( M  +  ( -u ( |_ `  ( M  /  N ) )  x.  N ) ) )
3029oveq2d 5874 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  ( M  mod  N ) )  =  ( N  gcd  ( M  +  ( -u ( |_ `  ( M  /  N ) )  x.  N ) ) ) )
3114znegcld 10119 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  -> 
-u ( |_ `  ( M  /  N
) )  e.  ZZ )
32 nnz 10045 . . . . 5  |-  ( N  e.  NN  ->  N  e.  ZZ )
3332adantl 452 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  ZZ )
34 simpl 443 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  ZZ )
35 gcdaddm 12708 . . . 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 1182 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  M
)  =  ( N  gcd  ( M  +  ( -u ( |_ `  ( M  /  N
) )  x.  N
) ) ) )
3730, 36eqtr4d 2318 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  ( M  mod  N ) )  =  ( N  gcd  M ) )
38 zmodcl 10989 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  mod  N
)  e.  NN0 )
3938nn0zd 10115 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  mod  N
)  e.  ZZ )
40 gcdcom 12699 . . 3  |-  ( ( N  e.  ZZ  /\  ( M  mod  N )  e.  ZZ )  -> 
( N  gcd  ( M  mod  N ) )  =  ( ( M  mod  N )  gcd 
N ) )
4133, 39, 40syl2anc 642 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  ( M  mod  N ) )  =  ( ( M  mod  N )  gcd 
N ) )
42 gcdcom 12699 . . 3  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  gcd  M
)  =  ( M  gcd  N ) )
4333, 34, 42syl2anc 642 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  M
)  =  ( M  gcd  N ) )
4437, 41, 433eqtr3d 2323 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   NNcn 9746   ZZcz 10024   RR+crp 10354   |_cfl 10924    mod cmo 10973    gcd cgcd 12685
This theorem is referenced by:  eucalginv  12754  phimullem  12847  eulerthlem1  12849  pockthlem  12952  gcdmodi  13089
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686
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