Theorem eucalg 11729

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.

Step	Hyp	Ref	Expression
1		nn0uz 9353	. . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
2		eucalg.2	. . . . . . . 8 |- R = seq 0 ( ( E o. 1st ) , ( NN0 X. { A } ) )
3		0zd 9059	. . . . . . . 8 |- ( ( M e. NN0 /\ N e. NN0 ) -> 0 e. ZZ )
4		eucalg.3	. . . . . . . . 9 |- A = <. M , N >.
5		opelxpi 4566	. . . . . . . . 9 |- ( ( M e. NN0 /\ N e. NN0 ) -> <. M , N >. e. ( NN0 X. NN0 ) )
6	4, 5	eqeltrid 2224	. . . . . . . 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 ) >. ) )
8	7	eucalgf 11725	. . . . . . . . 9 |- E : ( NN0 X. NN0 ) --> ( NN0 X. NN0 )
9	8	a1i 9	. . . . . . . 8 |- ( ( M e. NN0 /\ N e. NN0 ) -> E : ( NN0 X. NN0 ) --> ( NN0 X. NN0 ) )
10	1, 2, 3, 6, 9	algrf 11715	. . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> R : NN0 --> ( NN0 X. NN0 ) )
11		ffvelrn 5546	. . . . . . 7 |- ( ( R : NN0 --> ( NN0 X. NN0 ) /\ N e. NN0 ) -> ( R ` N ) e. ( NN0 X. NN0 ) )
12	10, 11	sylancom 416	. . . . . 6 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( R ` N ) e. ( NN0 X. NN0 ) )
13		1st2nd2 6066	. . . . . 6 |- ( ( R ` N ) e. ( NN0 X. NN0 ) -> ( R ` N ) = <. ( 1st ` ( R ` N ) ) , ( 2nd ` ( R ` N ) ) >. )
14	12, 13	syl 14	. . . . 5 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( R ` N ) = <. ( 1st ` ( R ` N ) ) , ( 2nd ` ( R ` N ) ) >. )
15	14	fveq2d 5418	. . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( gcd ` ( R ` N ) ) = ( gcd ` <. ( 1st ` ( R ` N ) ) , ( 2nd ` ( R ` N ) ) >. ) )
16		df-ov 5770	. . . 4 |- ( ( 1st ` ( R ` N ) ) gcd ( 2nd ` ( R ` N ) ) ) = ( gcd ` <. ( 1st ` ( R ` N ) ) , ( 2nd ` ( R ` N ) ) >. )
17	15, 16	syl6eqr 2188	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( gcd ` ( R ` N ) ) = ( ( 1st ` ( R ` N ) ) gcd ( 2nd ` ( R ` N ) ) ) )
18	4	fveq2i 5417	. . . . . . . 8 |- ( 2nd ` A ) = ( 2nd ` <. M , N >. )
19		op2ndg 6042	. . . . . . . 8 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 2nd ` <. M , N >. ) = N )
20	18, 19	syl5eq 2182	. . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 2nd ` A ) = N )
21	20	fveq2d 5418	. . . . . 6 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( R ` ( 2nd ` A ) ) = ( R ` N ) )
22	21	fveq2d 5418	. . . . 5 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 2nd ` ( R ` ( 2nd ` A ) ) ) = ( 2nd ` ( R ` N ) ) )
23		xp2nd 6057	. . . . . . . . 9 |- ( A e. ( NN0 X. NN0 ) -> ( 2nd ` A ) e. NN0 )
24	23	nn0zd 9164	. . . . . . . 8 |- ( A e. ( NN0 X. NN0 ) -> ( 2nd ` A ) e. ZZ )
25		uzid 9333	. . . . . . . 8 |- ( ( 2nd ` A ) e. ZZ -> ( 2nd ` A ) e. ( ZZ>= ` ( 2nd ` A ) ) )
26	24, 25	syl 14	. . . . . . 7 |- ( A e. ( NN0 X. NN0 ) -> ( 2nd ` A ) e. ( ZZ>= ` ( 2nd ` A ) ) )
27		eqid 2137	. . . . . . . 8 |- ( 2nd ` A ) = ( 2nd ` A )
28	7, 2, 27	eucalgcvga 11728	. . . . . . 7 |- ( A e. ( NN0 X. NN0 ) -> ( ( 2nd ` A ) e. ( ZZ>= ` ( 2nd ` A ) ) -> ( 2nd ` ( R ` ( 2nd ` A ) ) ) = 0 ) )
29	26, 28	mpd 13	. . . . . 6 |- ( A e. ( NN0 X. NN0 ) -> ( 2nd ` ( R ` ( 2nd ` A ) ) ) = 0 )
30	6, 29	syl 14	. . . . 5 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 2nd ` ( R ` ( 2nd ` A ) ) ) = 0 )
31	22, 30	eqtr3d 2172	. . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 2nd ` ( R ` N ) ) = 0 )
32	31	oveq2d 5783	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( 1st ` ( R ` N ) ) gcd ( 2nd ` ( R ` N ) ) ) = ( ( 1st ` ( R ` N ) ) gcd 0 ) )
33		xp1st 6056	. . . 4 |- ( ( R ` N ) e. ( NN0 X. NN0 ) -> ( 1st ` ( R ` N ) ) e. NN0 )
34		nn0gcdid0 11658	. . . 4 |- ( ( 1st ` ( R ` N ) ) e. NN0 -> ( ( 1st ` ( R ` N ) ) gcd 0 ) = ( 1st ` ( R ` N ) ) )
35	12, 33, 34	3syl 17	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( 1st ` ( R ` N ) ) gcd 0 ) = ( 1st ` ( R ` N ) ) )
36	17, 32, 35	3eqtrrd 2175	. 2 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 1st ` ( R ` N ) ) = ( gcd ` ( R ` N ) ) )
37	7	eucalginv 11726	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( gcd ` ( E ` z ) ) = ( gcd ` z ) )
38	8	ffvelrni 5547	. . . . . . 7 |- ( z e. ( NN0 X. NN0 ) -> ( E ` z ) e. ( NN0 X. NN0 ) )
39		fvres 5438	. . . . . . 7 |- ( ( E ` z ) e. ( NN0 X. NN0 ) -> ( ( gcd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) = ( gcd ` ( E ` z ) ) )
40	38, 39	syl 14	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( ( gcd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) = ( gcd ` ( E ` z ) ) )
41		fvres 5438	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( ( gcd |` ( NN0 X. NN0 ) ) ` z ) = ( gcd ` z ) )
42	37, 40, 41	3eqtr4d 2180	. . . . 5 |- ( z e. ( NN0 X. NN0 ) -> ( ( gcd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) = ( ( gcd |` ( NN0 X. NN0 ) ) ` z ) )
43	2, 8, 42	alginv 11717	. . . 4 |- ( ( A e. ( NN0 X. NN0 ) /\ N e. NN0 ) -> ( ( gcd |` ( NN0 X. NN0 ) ) ` ( R ` N ) ) = ( ( gcd |` ( NN0 X. NN0 ) ) ` ( R ` 0 ) ) )
44	6, 43	sylancom 416	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( gcd |` ( NN0 X. NN0 ) ) ` ( R ` N ) ) = ( ( gcd |` ( NN0 X. NN0 ) ) ` ( R ` 0 ) ) )
45		fvres 5438	. . . 4 |- ( ( R ` N ) e. ( NN0 X. NN0 ) -> ( ( gcd |` ( NN0 X. NN0 ) ) ` ( R ` N ) ) = ( gcd ` ( R ` N ) ) )
46	12, 45	syl 14	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( gcd |` ( NN0 X. NN0 ) ) ` ( R ` N ) ) = ( gcd ` ( R ` N ) ) )
47		0nn0 8985	. . . . 5 |- 0 e. NN0
48		ffvelrn 5546	. . . . 5 |- ( ( R : NN0 --> ( NN0 X. NN0 ) /\ 0 e. NN0 ) -> ( R ` 0 ) e. ( NN0 X. NN0 ) )
49	10, 47, 48	sylancl 409	. . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( R ` 0 ) e. ( NN0 X. NN0 ) )
50		fvres 5438	. . . 4 |- ( ( R ` 0 ) e. ( NN0 X. NN0 ) -> ( ( gcd |` ( NN0 X. NN0 ) ) ` ( R ` 0 ) ) = ( gcd ` ( R ` 0 ) ) )
51	49, 50	syl 14	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( gcd |` ( NN0 X. NN0 ) ) ` ( R ` 0 ) ) = ( gcd ` ( R ` 0 ) ) )
52	44, 46, 51	3eqtr3d 2178	. 2 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( gcd ` ( R ` N ) ) = ( gcd ` ( R ` 0 ) ) )
53	1, 2, 3, 6, 9	ialgr0 11714	. . . . 5 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( R ` 0 ) = A )
54	53, 4	syl6eq 2186	. . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( R ` 0 ) = <. M , N >. )
55	54	fveq2d 5418	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( gcd ` ( R ` 0 ) ) = ( gcd ` <. M , N >. ) )
56		df-ov 5770	. . 3 |- ( M gcd N ) = ( gcd ` <. M , N >. )
57	55, 56	syl6eqr 2188	. 2 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( gcd ` ( R ` 0 ) ) = ( M gcd N ) )
58	36, 52, 57	3eqtrd 2174	1 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 1st ` ( R ` N ) ) = ( M gcd N ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   ifcif 3469   {csn 3522   <.cop 3525   X. cxp 4532   |` cres 4536   o. ccom 4538   -->wf 5114   ` cfv 5118  (class class class)co 5767   e. cmpo 5769   1stc1st 6029   2ndc2nd 6030   0cc0 7613   NN0cn0 8970   ZZcz 9047   ZZ>=cuz 9319   mod cmo 10088   seqcseq 10211   gcd cgcd 11624

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 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733

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 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-sup 6864  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-fz 9784  df-fzo 9913  df-fl 10036  df-mod 10089  df-seqfrec 10212  df-exp 10286  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-dvds 11483  df-gcd 11625

This theorem is referenced by: (None)