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Theorem eucalg 12705
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.

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 >.. (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
StepHypRef Expression
1 nn0uz 10215 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
2 eucalg.2 . . . . . . . 8  |-  R  =  seq  0 ( ( E  o.  1st ) ,  ( NN0  X.  { A } ) )
3 0z 9988 . . . . . . . . 9  |-  0  e.  ZZ
43a1i 12 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
0  e.  ZZ )
5 eucalg.3 . . . . . . . . 9  |-  A  = 
<. M ,  N >.
6 opelxpi 4695 . . . . . . . . 9  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  <. M ,  N >.  e.  ( NN0  X.  NN0 ) )
75, 6syl5eqel 2340 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  A  e.  ( NN0  X. 
NN0 ) )
8 eucalgval.1 . . . . . . . . . 10  |-  E  =  ( x  e.  NN0 ,  y  e.  NN0  |->  if ( y  =  0 , 
<. x ,  y >. ,  <. y ,  ( x  mod  y )
>. ) )
98eucalgf 12701 . . . . . . . . 9  |-  E :
( NN0  X.  NN0 ) --> ( NN0  X.  NN0 )
109a1i 12 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  E : ( NN0  X.  NN0 ) --> ( NN0  X.  NN0 ) )
111, 2, 4, 7, 10algrf 12691 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  R : NN0 --> ( NN0 
X.  NN0 ) )
12 ffvelrn 5583 . . . . . . 7  |-  ( ( R : NN0 --> ( NN0 
X.  NN0 )  /\  N  e.  NN0 )  ->  ( R `  N )  e.  ( NN0  X.  NN0 ) )
1311, 12sylancom 651 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( R `  N
)  e.  ( NN0 
X.  NN0 ) )
14 1st2nd2 6079 . . . . . 6  |-  ( ( R `  N )  e.  ( NN0  X.  NN0 )  ->  ( R `
 N )  = 
<. ( 1st `  ( R `  N )
) ,  ( 2nd `  ( R `  N
) ) >. )
1513, 14syl 17 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( R `  N
)  =  <. ( 1st `  ( R `  N ) ) ,  ( 2nd `  ( R `  N )
) >. )
1615fveq2d 5448 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
(  gcd  `  ( R `
 N ) )  =  (  gcd  `  <. ( 1st `  ( R `
 N ) ) ,  ( 2nd `  ( R `  N )
) >. ) )
17 df-ov 5781 . . . 4  |-  ( ( 1st `  ( R `
 N ) )  gcd  ( 2nd `  ( R `  N )
) )  =  (  gcd  `  <. ( 1st `  ( R `  N
) ) ,  ( 2nd `  ( R `
 N ) )
>. )
1816, 17syl6eqr 2306 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
(  gcd  `  ( R `
 N ) )  =  ( ( 1st `  ( R `  N
) )  gcd  ( 2nd `  ( R `  N ) ) ) )
195fveq2i 5447 . . . . . . . 8  |-  ( 2nd `  A )  =  ( 2nd `  <. M ,  N >. )
20 op2ndg 6053 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 2nd `  <. M ,  N >. )  =  N )
2119, 20syl5eq 2300 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 2nd `  A
)  =  N )
2221fveq2d 5448 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( R `  ( 2nd `  A ) )  =  ( R `  N ) )
2322fveq2d 5448 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 2nd `  ( R `  ( 2nd `  A ) ) )  =  ( 2nd `  ( R `  N )
) )
24 xp2nd 6070 . . . . . . . . 9  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  A )  e.  NN0 )
2524nn0zd 10068 . . . . . . . 8  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  A )  e.  ZZ )
26 uzid 10195 . . . . . . . 8  |-  ( ( 2nd `  A )  e.  ZZ  ->  ( 2nd `  A )  e.  ( ZZ>= `  ( 2nd `  A ) ) )
2725, 26syl 17 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  A )  e.  (
ZZ>= `  ( 2nd `  A
) ) )
28 eqid 2256 . . . . . . . 8  |-  ( 2nd `  A )  =  ( 2nd `  A )
298, 2, 28eucalgcvga 12704 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd `  A )  e.  ( ZZ>= `  ( 2nd `  A ) )  ->  ( 2nd `  ( R `  ( 2nd `  A ) ) )  =  0 ) )
3027, 29mpd 16 . . . . . 6  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  ( R `  ( 2nd `  A ) ) )  =  0 )
317, 30syl 17 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 2nd `  ( R `  ( 2nd `  A ) ) )  =  0 )
3223, 31eqtr3d 2290 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 2nd `  ( R `  N )
)  =  0 )
3332oveq2d 5794 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( 1st `  ( R `  N )
)  gcd  ( 2nd `  ( R `  N
) ) )  =  ( ( 1st `  ( R `  N )
)  gcd  0 ) )
34 xp1st 6069 . . . 4  |-  ( ( R `  N )  e.  ( NN0  X.  NN0 )  ->  ( 1st `  ( R `  N
) )  e.  NN0 )
35 nn0gcdid0 12652 . . . 4  |-  ( ( 1st `  ( R `
 N ) )  e.  NN0  ->  ( ( 1st `  ( R `
 N ) )  gcd  0 )  =  ( 1st `  ( R `  N )
) )
3613, 34, 353syl 20 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( 1st `  ( R `  N )
)  gcd  0 )  =  ( 1st `  ( R `  N )
) )
3718, 33, 363eqtrrd 2293 . 2  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 1st `  ( R `  N )
)  =  (  gcd  `  ( R `  N
) ) )
38 gcdf 12646 . . . . . . 7  |-  gcd  :
( ZZ  X.  ZZ )
--> NN0
39 ffn 5313 . . . . . . 7  |-  (  gcd 
: ( ZZ  X.  ZZ ) --> NN0  ->  gcd  Fn  ( ZZ  X.  ZZ ) )
4038, 39ax-mp 10 . . . . . 6  |-  gcd  Fn  ( ZZ  X.  ZZ )
41 nn0ssz 9997 . . . . . . 7  |-  NN0  C_  ZZ
42 xpss12 4766 . . . . . . 7  |-  ( ( NN0  C_  ZZ  /\  NN0  C_  ZZ )  ->  ( NN0  X.  NN0 )  C_  ( ZZ  X.  ZZ ) )
4341, 41, 42mp2an 656 . . . . . 6  |-  ( NN0 
X.  NN0 )  C_  ( ZZ  X.  ZZ )
44 fnssres 5281 . . . . . 6  |-  ( (  gcd  Fn  ( ZZ 
X.  ZZ )  /\  ( NN0  X.  NN0 )  C_  ( ZZ  X.  ZZ ) )  ->  (  gcd  |`  ( NN0  X.  NN0 ) )  Fn  ( NN0  X.  NN0 ) )
4540, 43, 44mp2an 656 . . . . 5  |-  (  gcd  |`  ( NN0  X.  NN0 ) )  Fn  ( NN0  X.  NN0 )
468eucalginv 12702 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  (  gcd  `  ( E `  z
) )  =  (  gcd  `  z )
)
479ffvelrni 5584 . . . . . . 7  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( E `
 z )  e.  ( NN0  X.  NN0 ) )
48 fvres 5461 . . . . . . 7  |-  ( ( E `  z )  e.  ( NN0  X.  NN0 )  ->  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  ( E `  z ) )  =  (  gcd  `  ( E `  z
) ) )
4947, 48syl 17 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  ( E `  z ) )  =  (  gcd  `  ( E `  z
) ) )
50 fvres 5461 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  z )  =  (  gcd  `  z )
)
5146, 49, 503eqtr4d 2298 . . . . 5  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  ( E `  z ) )  =  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  z ) )
522, 9, 45, 51alginv 12693 . . . 4  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  N  e.  NN0 )  ->  (
(  gcd  |`  ( NN0 
X.  NN0 ) ) `  ( R `  N ) )  =  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  ( R `  0 ) ) )
537, 52sylancom 651 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( (  gcd  |`  ( NN0  X.  NN0 ) ) `
 ( R `  N ) )  =  ( (  gcd  |`  ( NN0  X.  NN0 ) ) `
 ( R ` 
0 ) ) )
54 fvres 5461 . . . 4  |-  ( ( R `  N )  e.  ( NN0  X.  NN0 )  ->  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  ( R `  N ) )  =  (  gcd  `  ( R `  N
) ) )
5513, 54syl 17 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( (  gcd  |`  ( NN0  X.  NN0 ) ) `
 ( R `  N ) )  =  (  gcd  `  ( R `  N )
) )
56 0nn0 9933 . . . . 5  |-  0  e.  NN0
57 ffvelrn 5583 . . . . 5  |-  ( ( R : NN0 --> ( NN0 
X.  NN0 )  /\  0  e.  NN0 )  ->  ( R `  0 )  e.  ( NN0  X.  NN0 ) )
5811, 56, 57sylancl 646 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( R `  0
)  e.  ( NN0 
X.  NN0 ) )
59 fvres 5461 . . . 4  |-  ( ( R `  0 )  e.  ( NN0  X.  NN0 )  ->  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  ( R `  0 ) )  =  (  gcd  `  ( R `  0
) ) )
6058, 59syl 17 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( (  gcd  |`  ( NN0  X.  NN0 ) ) `
 ( R ` 
0 ) )  =  (  gcd  `  ( R `  0 )
) )
6153, 55, 603eqtr3d 2296 . 2  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
(  gcd  `  ( R `
 N ) )  =  (  gcd  `  ( R `  0 )
) )
621, 2, 4, 7algr0 12690 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( R `  0
)  =  A )
6362, 5syl6eq 2304 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( R `  0
)  =  <. M ,  N >. )
6463fveq2d 5448 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
(  gcd  `  ( R `
 0 ) )  =  (  gcd  `  <. M ,  N >. )
)
65 df-ov 5781 . . 3  |-  ( M  gcd  N )  =  (  gcd  `  <. M ,  N >. )
6664, 65syl6eqr 2306 . 2  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
(  gcd  `  ( R `
 0 ) )  =  ( M  gcd  N ) )
6737, 61, 663eqtrd 2292 1  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 1st `  ( R `  N )
)  =  ( M  gcd  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    C_ wss 3113   ifcif 3525   {csn 3600   <.cop 3603    X. cxp 4645    |` cres 4649    o. ccom 4651    Fn wfn 4654   -->wf 4655   ` cfv 4659  (class class class)co 5778    e. cmpt2 5780   1stc1st 6040   2ndc2nd 6041   0cc0 8691   NN0cn0 9918   ZZcz 9977   ZZ>=cuz 10183    mod cmo 10925    seq cseq 10998    gcd cgcd 12633
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-sup 7148  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-n0 9919  df-z 9978  df-uz 10184  df-rp 10308  df-fz 10735  df-fl 10877  df-mod 10926  df-seq 10999  df-exp 11057  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-divides 12480  df-gcd 12634
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