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Theorem eucalg 14625
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 >.. 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 11217 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
2 eucalg.2 . . . . . . . 8  |-  R  =  seq 0 ( ( E  o.  1st ) ,  ( NN0  X.  { A } ) )
3 0zd 10973 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
0  e.  ZZ )
4 eucalg.3 . . . . . . . . 9  |-  A  = 
<. M ,  N >.
5 opelxpi 4871 . . . . . . . . 9  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  <. M ,  N >.  e.  ( NN0  X.  NN0 ) )
64, 5syl5eqel 2553 . . . . . . . 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 14621 . . . . . . . . 9  |-  E :
( NN0  X.  NN0 ) --> ( NN0  X.  NN0 )
98a1i 11 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  E : ( NN0  X.  NN0 ) --> ( NN0  X.  NN0 ) )
101, 2, 3, 6, 9algrf 14611 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  R : NN0 --> ( NN0 
X.  NN0 ) )
11 ffvelrn 6035 . . . . . . 7  |-  ( ( R : NN0 --> ( NN0 
X.  NN0 )  /\  N  e.  NN0 )  ->  ( R `  N )  e.  ( NN0  X.  NN0 ) )
1210, 11sylancom 680 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( R `  N
)  e.  ( NN0 
X.  NN0 ) )
13 1st2nd2 6849 . . . . . 6  |-  ( ( R `  N )  e.  ( NN0  X.  NN0 )  ->  ( R `
 N )  = 
<. ( 1st `  ( R `  N )
) ,  ( 2nd `  ( R `  N
) ) >. )
1412, 13syl 17 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( R `  N
)  =  <. ( 1st `  ( R `  N ) ) ,  ( 2nd `  ( R `  N )
) >. )
1514fveq2d 5883 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
(  gcd  `  ( R `
 N ) )  =  (  gcd  `  <. ( 1st `  ( R `
 N ) ) ,  ( 2nd `  ( R `  N )
) >. ) )
16 df-ov 6311 . . . 4  |-  ( ( 1st `  ( R `
 N ) )  gcd  ( 2nd `  ( R `  N )
) )  =  (  gcd  `  <. ( 1st `  ( R `  N
) ) ,  ( 2nd `  ( R `
 N ) )
>. )
1715, 16syl6eqr 2523 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
(  gcd  `  ( R `
 N ) )  =  ( ( 1st `  ( R `  N
) )  gcd  ( 2nd `  ( R `  N ) ) ) )
184fveq2i 5882 . . . . . . . 8  |-  ( 2nd `  A )  =  ( 2nd `  <. M ,  N >. )
19 op2ndg 6825 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 2nd `  <. M ,  N >. )  =  N )
2018, 19syl5eq 2517 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 2nd `  A
)  =  N )
2120fveq2d 5883 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( R `  ( 2nd `  A ) )  =  ( R `  N ) )
2221fveq2d 5883 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 2nd `  ( R `  ( 2nd `  A ) ) )  =  ( 2nd `  ( R `  N )
) )
23 xp2nd 6843 . . . . . . . . 9  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  A )  e.  NN0 )
2423nn0zd 11061 . . . . . . . 8  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  A )  e.  ZZ )
25 uzid 11197 . . . . . . . 8  |-  ( ( 2nd `  A )  e.  ZZ  ->  ( 2nd `  A )  e.  ( ZZ>= `  ( 2nd `  A ) ) )
2624, 25syl 17 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  A )  e.  (
ZZ>= `  ( 2nd `  A
) ) )
27 eqid 2471 . . . . . . . 8  |-  ( 2nd `  A )  =  ( 2nd `  A )
287, 2, 27eucalgcvga 14624 . . . . . . 7  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd `  A )  e.  ( ZZ>= `  ( 2nd `  A ) )  ->  ( 2nd `  ( R `  ( 2nd `  A ) ) )  =  0 ) )
2926, 28mpd 15 . . . . . 6  |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( 2nd `  ( R `  ( 2nd `  A ) ) )  =  0 )
306, 29syl 17 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 2nd `  ( R `  ( 2nd `  A ) ) )  =  0 )
3122, 30eqtr3d 2507 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 2nd `  ( R `  N )
)  =  0 )
3231oveq2d 6324 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( 1st `  ( R `  N )
)  gcd  ( 2nd `  ( R `  N
) ) )  =  ( ( 1st `  ( R `  N )
)  gcd  0 ) )
33 xp1st 6842 . . . 4  |-  ( ( R `  N )  e.  ( NN0  X.  NN0 )  ->  ( 1st `  ( R `  N
) )  e.  NN0 )
34 nn0gcdid0 14568 . . . 4  |-  ( ( 1st `  ( R `
 N ) )  e.  NN0  ->  ( ( 1st `  ( R `
 N ) )  gcd  0 )  =  ( 1st `  ( R `  N )
) )
3512, 33, 343syl 18 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( 1st `  ( R `  N )
)  gcd  0 )  =  ( 1st `  ( R `  N )
) )
3617, 32, 353eqtrrd 2510 . 2  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 1st `  ( R `  N )
)  =  (  gcd  `  ( R `  N
) ) )
37 gcdf 14562 . . . . . . 7  |-  gcd  :
( ZZ  X.  ZZ )
--> NN0
38 ffn 5739 . . . . . . 7  |-  (  gcd 
: ( ZZ  X.  ZZ ) --> NN0  ->  gcd  Fn  ( ZZ  X.  ZZ ) )
3937, 38ax-mp 5 . . . . . 6  |-  gcd  Fn  ( ZZ  X.  ZZ )
40 nn0ssz 10982 . . . . . . 7  |-  NN0  C_  ZZ
41 xpss12 4945 . . . . . . 7  |-  ( ( NN0  C_  ZZ  /\  NN0  C_  ZZ )  ->  ( NN0  X.  NN0 )  C_  ( ZZ  X.  ZZ ) )
4240, 40, 41mp2an 686 . . . . . 6  |-  ( NN0 
X.  NN0 )  C_  ( ZZ  X.  ZZ )
43 fnssres 5699 . . . . . 6  |-  ( (  gcd  Fn  ( ZZ 
X.  ZZ )  /\  ( NN0  X.  NN0 )  C_  ( ZZ  X.  ZZ ) )  ->  (  gcd  |`  ( NN0  X.  NN0 ) )  Fn  ( NN0  X.  NN0 ) )
4439, 42, 43mp2an 686 . . . . 5  |-  (  gcd  |`  ( NN0  X.  NN0 ) )  Fn  ( NN0  X.  NN0 )
457eucalginv 14622 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  (  gcd  `  ( E `  z
) )  =  (  gcd  `  z )
)
468ffvelrni 6036 . . . . . . 7  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( E `
 z )  e.  ( NN0  X.  NN0 ) )
47 fvres 5893 . . . . . . 7  |-  ( ( E `  z )  e.  ( NN0  X.  NN0 )  ->  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  ( E `  z ) )  =  (  gcd  `  ( E `  z
) ) )
4846, 47syl 17 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  ( E `  z ) )  =  (  gcd  `  ( E `  z
) ) )
49 fvres 5893 . . . . . 6  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  z )  =  (  gcd  `  z )
)
5045, 48, 493eqtr4d 2515 . . . . 5  |-  ( z  e.  ( NN0  X.  NN0 )  ->  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  ( E `  z ) )  =  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  z ) )
512, 8, 44, 50alginv 14613 . . . 4  |-  ( ( A  e.  ( NN0 
X.  NN0 )  /\  N  e.  NN0 )  ->  (
(  gcd  |`  ( NN0 
X.  NN0 ) ) `  ( R `  N ) )  =  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  ( R `  0 ) ) )
526, 51sylancom 680 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( (  gcd  |`  ( NN0  X.  NN0 ) ) `
 ( R `  N ) )  =  ( (  gcd  |`  ( NN0  X.  NN0 ) ) `
 ( R ` 
0 ) ) )
53 fvres 5893 . . . 4  |-  ( ( R `  N )  e.  ( NN0  X.  NN0 )  ->  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  ( R `  N ) )  =  (  gcd  `  ( R `  N
) ) )
5412, 53syl 17 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( (  gcd  |`  ( NN0  X.  NN0 ) ) `
 ( R `  N ) )  =  (  gcd  `  ( R `  N )
) )
55 0nn0 10908 . . . . 5  |-  0  e.  NN0
56 ffvelrn 6035 . . . . 5  |-  ( ( R : NN0 --> ( NN0 
X.  NN0 )  /\  0  e.  NN0 )  ->  ( R `  0 )  e.  ( NN0  X.  NN0 ) )
5710, 55, 56sylancl 675 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( R `  0
)  e.  ( NN0 
X.  NN0 ) )
58 fvres 5893 . . . 4  |-  ( ( R `  0 )  e.  ( NN0  X.  NN0 )  ->  ( (  gcd  |`  ( NN0  X. 
NN0 ) ) `  ( R `  0 ) )  =  (  gcd  `  ( R `  0
) ) )
5957, 58syl 17 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( (  gcd  |`  ( NN0  X.  NN0 ) ) `
 ( R ` 
0 ) )  =  (  gcd  `  ( R `  0 )
) )
6052, 54, 593eqtr3d 2513 . 2  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
(  gcd  `  ( R `
 N ) )  =  (  gcd  `  ( R `  0 )
) )
611, 2, 3, 6algr0 14610 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( R `  0
)  =  A )
6261, 4syl6eq 2521 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( R `  0
)  =  <. M ,  N >. )
6362fveq2d 5883 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
(  gcd  `  ( R `
 0 ) )  =  (  gcd  `  <. M ,  N >. )
)
64 df-ov 6311 . . 3  |-  ( M  gcd  N )  =  (  gcd  `  <. M ,  N >. )
6563, 64syl6eqr 2523 . 2  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
(  gcd  `  ( R `
 0 ) )  =  ( M  gcd  N ) )
6636, 60, 653eqtrd 2509 1  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 1st `  ( R `  N )
)  =  ( M  gcd  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    C_ wss 3390   ifcif 3872   {csn 3959   <.cop 3965    X. cxp 4837    |` cres 4841    o. ccom 4843    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   1stc1st 6810   2ndc2nd 6811   0cc0 9557   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182    mod cmo 12129    seqcseq 12251    gcd cgcd 14547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383  df-gcd 14548
This theorem is referenced by: (None)
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