ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eucalg Unicode version

Theorem eucalg 11743
Description: Euclid's Algorithm computes the greatest common divisor of two nonnegative integers by repeatedly replacing the larger of them with its remainder modulo the smaller until the remainder is 0. Theorem 1.15 in [ApostolNT] p. 20.

Upon halting, the 1st member of the final state  ( R `  N ) is equal to the gcd of the values comprising the input state  <. M ,  N >.. This is Metamath 100 proof #69 (greatest common divisor algorithm). (Contributed by Paul Chapman, 31-Mar-2011.) (Proof shortened by Mario Carneiro, 29-May-2014.)

Hypotheses
Ref Expression
eucalgval.1  |-  E  =  ( x  e.  NN0 ,  y  e.  NN0  |->  if ( y  =  0 , 
<. x ,  y >. ,  <. y ,  ( x  mod  y )
>. ) )
eucalg.2  |-  R  =  seq 0 ( ( E  o.  1st ) ,  ( NN0  X.  { A } ) )
eucalg.3  |-  A  = 
<. M ,  N >.
Assertion
Ref Expression
eucalg  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 1st `  ( R `  N )
)  =  ( M  gcd  N ) )
Distinct variable groups:    x, y, M   
x, N, y    x, A, y    x, R
Allowed substitution hints:    R( y)    E( x, y)

Proof of Theorem eucalg
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nn0uz 9363 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
2 eucalg.2 . . . . . . . 8  |-  R  =  seq 0 ( ( E  o.  1st ) ,  ( NN0  X.  { A } ) )
3 0zd 9069 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
0  e.  ZZ )
4 eucalg.3 . . . . . . . . 9  |-  A  = 
<. M ,  N >.
5 opelxpi 4571 . . . . . . . . 9  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  <. M ,  N >.  e.  ( NN0  X.  NN0 ) )
64, 5eqeltrid 2226 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  A  e.  ( NN0  X. 
NN0 ) )
7 eucalgval.1 . . . . . . . . . 10  |-  E  =  ( x  e.  NN0 ,  y  e.  NN0  |->  if ( y  =  0 , 
<. x ,  y >. ,  <. y ,  ( x  mod  y )
>. ) )
87eucalgf 11739 . . . . . . . . 9  |-  E :
( NN0  X.  NN0 ) --> ( NN0  X.  NN0 )
98a1i 9 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  E : ( NN0  X.  NN0 ) --> ( NN0  X.  NN0 ) )
101, 2, 3, 6, 9algrf 11729 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  R : NN0 --> ( NN0 
X.  NN0 ) )
11 ffvelrn 5553 . . . . . . 7  |-  ( ( R : NN0 --> ( NN0 
X.  NN0 )  /\  N  e.  NN0 )  ->  ( R `  N )  e.  ( NN0  X.  NN0 ) )
1210, 11sylancom 416 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( R `  N
)  e.  ( NN0 
X.  NN0 ) )
13 1st2nd2 6073 . . . . . 6  |-  ( ( R `  N )  e.  ( NN0  X.  NN0 )  ->  ( R `
 N )  = 
<. ( 1st `  ( R `  N )
) ,  ( 2nd `  ( R `  N
) ) >. )
1412, 13syl 14 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( R `  N
)  =  <. ( 1st `  ( R `  N ) ) ,  ( 2nd `  ( R `  N )
) >. )
1514fveq2d 5425 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
(  gcd  `  ( R `
 N ) )  =  (  gcd  `  <. ( 1st `  ( R `
 N ) ) ,  ( 2nd `  ( R `  N )
) >. ) )
16 df-ov 5777 . . . 4  |-  ( ( 1st `  ( R `
 N ) )  gcd  ( 2nd `  ( R `  N )
) )  =  (  gcd  `  <. ( 1st `  ( R `  N
) ) ,  ( 2nd `  ( R `
 N ) )
>. )
1715, 16syl6eqr 2190 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
(  gcd  `  ( R `
 N ) )  =  ( ( 1st `  ( R `  N
) )  gcd  ( 2nd `  ( R `  N ) ) ) )
184fveq2i 5424 . . . . . . . 8  |-  ( 2nd `  A )  =  ( 2nd `  <. M ,  N >. )
19 op2ndg 6049 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 2nd `  <. M ,  N >. )  =  N )
2018, 19syl5eq 2184 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 2nd `  A
)  =  N )
2120fveq2d 5425 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( R `  ( 2nd `  A ) )  =  ( R `  N ) )
2221fveq2d 5425 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 2nd `  ( R `  ( 2nd `  A ) ) )  =  ( 2nd `  ( R `  N )
) )
23 xp2nd 6064 . . . . . . . . 9  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  A )  e.  NN0 )
2423nn0zd 9174 . . . . . . . 8  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  A )  e.  ZZ )
25 uzid 9343 . . . . . . . 8  |-  ( ( 2nd `  A )  e.  ZZ  ->  ( 2nd `  A )  e.  ( ZZ>= `  ( 2nd `  A ) ) )
2624, 25syl 14 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  A )  e.  (
ZZ>= `  ( 2nd `  A
) ) )
27 eqid 2139 . . . . . . . 8  |-  ( 2nd `  A )  =  ( 2nd `  A )
287, 2, 27eucalgcvga 11742 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd `  A )  e.  ( ZZ>= `  ( 2nd `  A ) )  ->  ( 2nd `  ( R `  ( 2nd `  A ) ) )  =  0 ) )
2926, 28mpd 13 . . . . . 6  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  ( R `  ( 2nd `  A ) ) )  =  0 )
306, 29syl 14 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 2nd `  ( R `  ( 2nd `  A ) ) )  =  0 )
3122, 30eqtr3d 2174 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 2nd `  ( R `  N )
)  =  0 )
3231oveq2d 5790 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( 1st `  ( R `  N )
)  gcd  ( 2nd `  ( R `  N
) ) )  =  ( ( 1st `  ( R `  N )
)  gcd  0 ) )
33 xp1st 6063 . . . 4  |-  ( ( R `  N )  e.  ( NN0  X.  NN0 )  ->  ( 1st `  ( R `  N
) )  e.  NN0 )
34 nn0gcdid0 11672 . . . 4  |-  ( ( 1st `  ( R `
 N ) )  e.  NN0  ->  ( ( 1st `  ( R `
 N ) )  gcd  0 )  =  ( 1st `  ( R `  N )
) )
3512, 33, 343syl 17 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( 1st `  ( R `  N )
)  gcd  0 )  =  ( 1st `  ( R `  N )
) )
3617, 32, 353eqtrrd 2177 . 2  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 1st `  ( R `  N )
)  =  (  gcd  `  ( R `  N
) ) )
377eucalginv 11740 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  (  gcd  `  ( E `  z
) )  =  (  gcd  `  z )
)
388ffvelrni 5554 . . . . . . 7  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( E `
 z )  e.  ( NN0  X.  NN0 ) )
39 fvres 5445 . . . . . . 7  |-  ( ( E `  z )  e.  ( NN0  X.  NN0 )  ->  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  ( E `  z ) )  =  (  gcd  `  ( E `  z
) ) )
4038, 39syl 14 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  ( E `  z ) )  =  (  gcd  `  ( E `  z
) ) )
41 fvres 5445 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  z )  =  (  gcd  `  z )
)
4237, 40, 413eqtr4d 2182 . . . . 5  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  ( E `  z ) )  =  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  z ) )
432, 8, 42alginv 11731 . . . 4  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  N  e.  NN0 )  ->  (
(  gcd  |`  ( NN0 
X.  NN0 ) ) `  ( R `  N ) )  =  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  ( R `  0 ) ) )
446, 43sylancom 416 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( (  gcd  |`  ( NN0  X.  NN0 ) ) `
 ( R `  N ) )  =  ( (  gcd  |`  ( NN0  X.  NN0 ) ) `
 ( R ` 
0 ) ) )
45 fvres 5445 . . . 4  |-  ( ( R `  N )  e.  ( NN0  X.  NN0 )  ->  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  ( R `  N ) )  =  (  gcd  `  ( R `  N
) ) )
4612, 45syl 14 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( (  gcd  |`  ( NN0  X.  NN0 ) ) `
 ( R `  N ) )  =  (  gcd  `  ( R `  N )
) )
47 0nn0 8995 . . . . 5  |-  0  e.  NN0
48 ffvelrn 5553 . . . . 5  |-  ( ( R : NN0 --> ( NN0 
X.  NN0 )  /\  0  e.  NN0 )  ->  ( R `  0 )  e.  ( NN0  X.  NN0 ) )
4910, 47, 48sylancl 409 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( R `  0
)  e.  ( NN0 
X.  NN0 ) )
50 fvres 5445 . . . 4  |-  ( ( R `  0 )  e.  ( NN0  X.  NN0 )  ->  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  ( R `  0 ) )  =  (  gcd  `  ( R `  0
) ) )
5149, 50syl 14 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( (  gcd  |`  ( NN0  X.  NN0 ) ) `
 ( R ` 
0 ) )  =  (  gcd  `  ( R `  0 )
) )
5244, 46, 513eqtr3d 2180 . 2  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
(  gcd  `  ( R `
 N ) )  =  (  gcd  `  ( R `  0 )
) )
531, 2, 3, 6, 9ialgr0 11728 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( R `  0
)  =  A )
5453, 4syl6eq 2188 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( R `  0
)  =  <. M ,  N >. )
5554fveq2d 5425 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
(  gcd  `  ( R `
 0 ) )  =  (  gcd  `  <. M ,  N >. )
)
56 df-ov 5777 . . 3  |-  ( M  gcd  N )  =  (  gcd  `  <. M ,  N >. )
5755, 56syl6eqr 2190 . 2  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
(  gcd  `  ( R `
 0 ) )  =  ( M  gcd  N ) )
5836, 52, 573eqtrd 2176 1  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 1st `  ( R `  N )
)  =  ( M  gcd  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   ifcif 3474   {csn 3527   <.cop 3530    X. cxp 4537    |` cres 4541    o. ccom 4543   -->wf 5119   ` cfv 5123  (class class class)co 5774    e. cmpo 5776   1stc1st 6036   2ndc2nd 6037   0cc0 7623   NN0cn0 8980   ZZcz 9057   ZZ>=cuz 9329    mod cmo 10098    seqcseq 10221    gcd cgcd 11638
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7714  ax-resscn 7715  ax-1cn 7716  ax-1re 7717  ax-icn 7718  ax-addcl 7719  ax-addrcl 7720  ax-mulcl 7721  ax-mulrcl 7722  ax-addcom 7723  ax-mulcom 7724  ax-addass 7725  ax-mulass 7726  ax-distr 7727  ax-i2m1 7728  ax-0lt1 7729  ax-1rid 7730  ax-0id 7731  ax-rnegex 7732  ax-precex 7733  ax-cnre 7734  ax-pre-ltirr 7735  ax-pre-ltwlin 7736  ax-pre-lttrn 7737  ax-pre-apti 7738  ax-pre-ltadd 7739  ax-pre-mulgt0 7740  ax-pre-mulext 7741  ax-arch 7742  ax-caucvg 7743
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-pnf 7805  df-mnf 7806  df-xr 7807  df-ltxr 7808  df-le 7809  df-sub 7938  df-neg 7939  df-reap 8340  df-ap 8347  df-div 8436  df-inn 8724  df-2 8782  df-3 8783  df-4 8784  df-n0 8981  df-z 9058  df-uz 9330  df-q 9415  df-rp 9445  df-fz 9794  df-fzo 9923  df-fl 10046  df-mod 10099  df-seqfrec 10222  df-exp 10296  df-cj 10617  df-re 10618  df-im 10619  df-rsqrt 10773  df-abs 10774  df-dvds 11497  df-gcd 11639
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator