Theorem eucalg 12630

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 9790	. . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
2		eucalg.2	. . . . . . . 8 |- R = seq 0 ( ( E o. 1st ) , ( NN0 X. { A } ) )
3		0zd 9490	. . . . . . . 8 |- ( ( M e. NN0 /\ N e. NN0 ) -> 0 e. ZZ )
4		eucalg.3	. . . . . . . . 9 |- A = <. M , N >.
5		opelxpi 4757	. . . . . . . . 9 |- ( ( M e. NN0 /\ N e. NN0 ) -> <. M , N >. e. ( NN0 X. NN0 ) )
6	4, 5	eqeltrid 2318	. . . . . . . 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 12626	. . . . . . . . 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 12616	. . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> R : NN0 --> ( NN0 X. NN0 ) )
11		ffvelcdm 5780	. . . . . . 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 6337	. . . . . 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 5643	. . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( gcd ` ( R ` N ) ) = ( gcd ` <. ( 1st ` ( R ` N ) ) , ( 2nd ` ( R ` N ) ) >. ) )
16		df-ov 6020	. . . 4 |- ( ( 1st ` ( R ` N ) ) gcd ( 2nd ` ( R ` N ) ) ) = ( gcd ` <. ( 1st ` ( R ` N ) ) , ( 2nd ` ( R ` N ) ) >. )
17	15, 16	eqtr4di 2282	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( gcd ` ( R ` N ) ) = ( ( 1st ` ( R ` N ) ) gcd ( 2nd ` ( R ` N ) ) ) )
18	4	fveq2i 5642	. . . . . . . 8 |- ( 2nd ` A ) = ( 2nd ` <. M , N >. )
19		op2ndg 6313	. . . . . . . 8 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 2nd ` <. M , N >. ) = N )
20	18, 19	eqtrid 2276	. . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 2nd ` A ) = N )
21	20	fveq2d 5643	. . . . . 6 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( R ` ( 2nd ` A ) ) = ( R ` N ) )
22	21	fveq2d 5643	. . . . 5 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 2nd ` ( R ` ( 2nd ` A ) ) ) = ( 2nd ` ( R ` N ) ) )
23		xp2nd 6328	. . . . . . . . 9 |- ( A e. ( NN0 X. NN0 ) -> ( 2nd ` A ) e. NN0 )
24	23	nn0zd 9599	. . . . . . . 8 |- ( A e. ( NN0 X. NN0 ) -> ( 2nd ` A ) e. ZZ )
25		uzid 9769	. . . . . . . 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 2231	. . . . . . . 8 |- ( 2nd ` A ) = ( 2nd ` A )
28	7, 2, 27	eucalgcvga 12629	. . . . . . 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 2266	. . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 2nd ` ( R ` N ) ) = 0 )
32	31	oveq2d 6033	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( 1st ` ( R ` N ) ) gcd ( 2nd ` ( R ` N ) ) ) = ( ( 1st ` ( R ` N ) ) gcd 0 ) )
33		xp1st 6327	. . . 4 |- ( ( R ` N ) e. ( NN0 X. NN0 ) -> ( 1st ` ( R ` N ) ) e. NN0 )
34		nn0gcdid0 12551	. . . 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 2269	. 2 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 1st ` ( R ` N ) ) = ( gcd ` ( R ` N ) ) )
37	7	eucalginv 12627	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( gcd ` ( E ` z ) ) = ( gcd ` z ) )
38	8	ffvelcdmi 5781	. . . . . . 7 |- ( z e. ( NN0 X. NN0 ) -> ( E ` z ) e. ( NN0 X. NN0 ) )
39		fvres 5663	. . . . . . 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 5663	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( ( gcd |` ( NN0 X. NN0 ) ) ` z ) = ( gcd ` z ) )
42	37, 40, 41	3eqtr4d 2274	. . . . 5 |- ( z e. ( NN0 X. NN0 ) -> ( ( gcd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) = ( ( gcd |` ( NN0 X. NN0 ) ) ` z ) )
43	2, 8, 42	alginv 12618	. . . 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 5663	. . . 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 9416	. . . . 5 |- 0 e. NN0
48		ffvelcdm 5780	. . . . 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 5663	. . . 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 2272	. 2 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( gcd ` ( R ` N ) ) = ( gcd ` ( R ` 0 ) ) )
53	1, 2, 3, 6, 9	ialgr0 12615	. . . . 5 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( R ` 0 ) = A )
54	53, 4	eqtrdi 2280	. . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( R ` 0 ) = <. M , N >. )
55	54	fveq2d 5643	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( gcd ` ( R ` 0 ) ) = ( gcd ` <. M , N >. ) )
56		df-ov 6020	. . 3 |- ( M gcd N ) = ( gcd ` <. M , N >. )
57	55, 56	eqtr4di 2282	. 2 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( gcd ` ( R ` 0 ) ) = ( M gcd N ) )
58	36, 52, 57	3eqtrd 2268	1 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 1st ` ( R ` N ) ) = ( M gcd N ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   ifcif 3605   {csn 3669   <.cop 3672   X. cxp 4723   |` cres 4727   o. ccom 4729   -->wf 5322   ` cfv 5326  (class class class)co 6017   e. cmpo 6019   1stc1st 6300   2ndc2nd 6301   0cc0 8031   NN0cn0 9401   ZZcz 9478   ZZ>=cuz 9754   mod cmo 10583   seqcseq 10708   gcd cgcd 12523

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-sup 7182  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-dvds 12348  df-gcd 12524

This theorem is referenced by: (None)