Theorem eucalg 12062

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 9565	. . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
2		eucalg.2	. . . . . . . 8 |- R = seq 0 ( ( E o. 1st ) , ( NN0 X. { A } ) )
3		0zd 9268	. . . . . . . 8 |- ( ( M e. NN0 /\ N e. NN0 ) -> 0 e. ZZ )
4		eucalg.3	. . . . . . . . 9 |- A = <. M , N >.
5		opelxpi 4660	. . . . . . . . 9 |- ( ( M e. NN0 /\ N e. NN0 ) -> <. M , N >. e. ( NN0 X. NN0 ) )
6	4, 5	eqeltrid 2264	. . . . . . . 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 12058	. . . . . . . . 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 12048	. . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> R : NN0 --> ( NN0 X. NN0 ) )
11		ffvelcdm 5652	. . . . . . 7 |- ( ( R : NN0 --> ( NN0 X. NN0 ) /\ N e. NN0 ) -> ( R ` N ) e. ( NN0 X. NN0 ) )
12	10, 11	sylancom 420	. . . . . 6 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( R ` N ) e. ( NN0 X. NN0 ) )
13		1st2nd2 6179	. . . . . 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 5521	. . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( gcd ` ( R ` N ) ) = ( gcd ` <. ( 1st ` ( R ` N ) ) , ( 2nd ` ( R ` N ) ) >. ) )
16		df-ov 5881	. . . 4 |- ( ( 1st ` ( R ` N ) ) gcd ( 2nd ` ( R ` N ) ) ) = ( gcd ` <. ( 1st ` ( R ` N ) ) , ( 2nd ` ( R ` N ) ) >. )
17	15, 16	eqtr4di 2228	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( gcd ` ( R ` N ) ) = ( ( 1st ` ( R ` N ) ) gcd ( 2nd ` ( R ` N ) ) ) )
18	4	fveq2i 5520	. . . . . . . 8 |- ( 2nd ` A ) = ( 2nd ` <. M , N >. )
19		op2ndg 6155	. . . . . . . 8 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 2nd ` <. M , N >. ) = N )
20	18, 19	eqtrid 2222	. . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 2nd ` A ) = N )
21	20	fveq2d 5521	. . . . . 6 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( R ` ( 2nd ` A ) ) = ( R ` N ) )
22	21	fveq2d 5521	. . . . 5 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 2nd ` ( R ` ( 2nd ` A ) ) ) = ( 2nd ` ( R ` N ) ) )
23		xp2nd 6170	. . . . . . . . 9 |- ( A e. ( NN0 X. NN0 ) -> ( 2nd ` A ) e. NN0 )
24	23	nn0zd 9376	. . . . . . . 8 |- ( A e. ( NN0 X. NN0 ) -> ( 2nd ` A ) e. ZZ )
25		uzid 9545	. . . . . . . 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 2177	. . . . . . . 8 |- ( 2nd ` A ) = ( 2nd ` A )
28	7, 2, 27	eucalgcvga 12061	. . . . . . 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 2212	. . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 2nd ` ( R ` N ) ) = 0 )
32	31	oveq2d 5894	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( 1st ` ( R ` N ) ) gcd ( 2nd ` ( R ` N ) ) ) = ( ( 1st ` ( R ` N ) ) gcd 0 ) )
33		xp1st 6169	. . . 4 |- ( ( R ` N ) e. ( NN0 X. NN0 ) -> ( 1st ` ( R ` N ) ) e. NN0 )
34		nn0gcdid0 11985	. . . 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 2215	. 2 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 1st ` ( R ` N ) ) = ( gcd ` ( R ` N ) ) )
37	7	eucalginv 12059	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( gcd ` ( E ` z ) ) = ( gcd ` z ) )
38	8	ffvelcdmi 5653	. . . . . . 7 |- ( z e. ( NN0 X. NN0 ) -> ( E ` z ) e. ( NN0 X. NN0 ) )
39		fvres 5541	. . . . . . 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 5541	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( ( gcd |` ( NN0 X. NN0 ) ) ` z ) = ( gcd ` z ) )
42	37, 40, 41	3eqtr4d 2220	. . . . 5 |- ( z e. ( NN0 X. NN0 ) -> ( ( gcd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) = ( ( gcd |` ( NN0 X. NN0 ) ) ` z ) )
43	2, 8, 42	alginv 12050	. . . 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 420	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( gcd |` ( NN0 X. NN0 ) ) ` ( R ` N ) ) = ( ( gcd |` ( NN0 X. NN0 ) ) ` ( R ` 0 ) ) )
45		fvres 5541	. . . 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 9194	. . . . 5 |- 0 e. NN0
48		ffvelcdm 5652	. . . . 5 |- ( ( R : NN0 --> ( NN0 X. NN0 ) /\ 0 e. NN0 ) -> ( R ` 0 ) e. ( NN0 X. NN0 ) )
49	10, 47, 48	sylancl 413	. . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( R ` 0 ) e. ( NN0 X. NN0 ) )
50		fvres 5541	. . . 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 2218	. 2 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( gcd ` ( R ` N ) ) = ( gcd ` ( R ` 0 ) ) )
53	1, 2, 3, 6, 9	ialgr0 12047	. . . . 5 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( R ` 0 ) = A )
54	53, 4	eqtrdi 2226	. . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( R ` 0 ) = <. M , N >. )
55	54	fveq2d 5521	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( gcd ` ( R ` 0 ) ) = ( gcd ` <. M , N >. ) )
56		df-ov 5881	. . 3 |- ( M gcd N ) = ( gcd ` <. M , N >. )
57	55, 56	eqtr4di 2228	. 2 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( gcd ` ( R ` 0 ) ) = ( M gcd N ) )
58	36, 52, 57	3eqtrd 2214	1 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 1st ` ( R ` N ) ) = ( M gcd N ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   ifcif 3536   {csn 3594   <.cop 3597   X. cxp 4626   |` cres 4630   o. ccom 4632   -->wf 5214   ` cfv 5218  (class class class)co 5878   e. cmpo 5880   1stc1st 6142   2ndc2nd 6143   0cc0 7814   NN0cn0 9179   ZZcz 9256   ZZ>=cuz 9531   mod cmo 10325   seqcseq 10448   gcd cgcd 11946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-pre-mulext 7932  ax-arch 7933  ax-caucvg 7934

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-frec 6395  df-sup 6986  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-ap 8542  df-div 8633  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-n0 9180  df-z 9257  df-uz 9532  df-q 9623  df-rp 9657  df-fz 10012  df-fzo 10146  df-fl 10273  df-mod 10326  df-seqfrec 10449  df-exp 10523  df-cj 10854  df-re 10855  df-im 10856  df-rsqrt 11010  df-abs 11011  df-dvds 11798  df-gcd 11947

This theorem is referenced by: (None)