Theorem eucalg 12856

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 9967	. . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
2		eucalg.2	. . . . . . . 8 |- R = seq 0 ( ( E o. 1st ) , ( NN0 X. { A } ) )
3		0zd 9661	. . . . . . . 8 |- ( ( M e. NN0 /\ N e. NN0 ) -> 0 e. ZZ )
4		eucalg.3	. . . . . . . . 9 |- A = <. M , N >.
5		opelxpi 4806	. . . . . . . . 9 |- ( ( M e. NN0 /\ N e. NN0 ) -> <. M , N >. e. ( NN0 X. NN0 ) )
6	4, 5	eqeltrid 2325	. . . . . . . 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 12852	. . . . . . . . 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 12842	. . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> R : NN0 --> ( NN0 X. NN0 ) )
11		ffvelcdm 5841	. . . . . . 7 |- ( ( R : NN0 --> ( NN0 X. NN0 ) /\ N e. NN0 ) -> ( R ` N ) e. ( NN0 X. NN0 ) )
12	10, 11	sylancom 424	. . . . . 6 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( R ` N ) e. ( NN0 X. NN0 ) )
13		1st2nd2 6409	. . . . . 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 5699	. . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( gcd ` ( R ` N ) ) = ( gcd ` <. ( 1st ` ( R ` N ) ) , ( 2nd ` ( R ` N ) ) >. ) )
16		df-ov 6088	. . . 4 |- ( ( 1st ` ( R ` N ) ) gcd ( 2nd ` ( R ` N ) ) ) = ( gcd ` <. ( 1st ` ( R ` N ) ) , ( 2nd ` ( R ` N ) ) >. )
17	15, 16	eqtr4di 2289	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( gcd ` ( R ` N ) ) = ( ( 1st ` ( R ` N ) ) gcd ( 2nd ` ( R ` N ) ) ) )
18	4	fveq2i 5698	. . . . . . . 8 |- ( 2nd ` A ) = ( 2nd ` <. M , N >. )
19		op2ndg 6385	. . . . . . . 8 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 2nd ` <. M , N >. ) = N )
20	18, 19	eqtrid 2283	. . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 2nd ` A ) = N )
21	20	fveq2d 5699	. . . . . 6 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( R ` ( 2nd ` A ) ) = ( R ` N ) )
22	21	fveq2d 5699	. . . . 5 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 2nd ` ( R ` ( 2nd ` A ) ) ) = ( 2nd ` ( R ` N ) ) )
23		xp2nd 6400	. . . . . . . . 9 |- ( A e. ( NN0 X. NN0 ) -> ( 2nd ` A ) e. NN0 )
24	23	nn0zd 9771	. . . . . . . 8 |- ( A e. ( NN0 X. NN0 ) -> ( 2nd ` A ) e. ZZ )
25		uzid 9946	. . . . . . . 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 2238	. . . . . . . 8 |- ( 2nd ` A ) = ( 2nd ` A )
28	7, 2, 27	eucalgcvga 12855	. . . . . . 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 2273	. . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 2nd ` ( R ` N ) ) = 0 )
32	31	oveq2d 6101	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( 1st ` ( R ` N ) ) gcd ( 2nd ` ( R ` N ) ) ) = ( ( 1st ` ( R ` N ) ) gcd 0 ) )
33		xp1st 6399	. . . 4 |- ( ( R ` N ) e. ( NN0 X. NN0 ) -> ( 1st ` ( R ` N ) ) e. NN0 )
34		nn0gcdid0 12777	. . . 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 2276	. 2 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 1st ` ( R ` N ) ) = ( gcd ` ( R ` N ) ) )
37	7	eucalginv 12853	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( gcd ` ( E ` z ) ) = ( gcd ` z ) )
38	8	ffvelcdmi 5842	. . . . . . 7 |- ( z e. ( NN0 X. NN0 ) -> ( E ` z ) e. ( NN0 X. NN0 ) )
39		fvres 5719	. . . . . . 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 5719	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( ( gcd |` ( NN0 X. NN0 ) ) ` z ) = ( gcd ` z ) )
42	37, 40, 41	3eqtr4d 2281	. . . . 5 |- ( z e. ( NN0 X. NN0 ) -> ( ( gcd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) = ( ( gcd |` ( NN0 X. NN0 ) ) ` z ) )
43	2, 8, 42	alginv 12844	. . . 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 424	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( gcd |` ( NN0 X. NN0 ) ) ` ( R ` N ) ) = ( ( gcd |` ( NN0 X. NN0 ) ) ` ( R ` 0 ) ) )
45		fvres 5719	. . . 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 9583	. . . . 5 |- 0 e. NN0
48		ffvelcdm 5841	. . . . 5 |- ( ( R : NN0 --> ( NN0 X. NN0 ) /\ 0 e. NN0 ) -> ( R ` 0 ) e. ( NN0 X. NN0 ) )
49	10, 47, 48	sylancl 417	. . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( R ` 0 ) e. ( NN0 X. NN0 ) )
50		fvres 5719	. . . 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 2279	. 2 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( gcd ` ( R ` N ) ) = ( gcd ` ( R ` 0 ) ) )
53	1, 2, 3, 6, 9	ialgr0 12841	. . . . 5 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( R ` 0 ) = A )
54	53, 4	eqtrdi 2287	. . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( R ` 0 ) = <. M , N >. )
55	54	fveq2d 5699	. . 3 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( gcd ` ( R ` 0 ) ) = ( gcd ` <. M , N >. ) )
56		df-ov 6088	. . 3 |- ( M gcd N ) = ( gcd ` <. M , N >. )
57	55, 56	eqtr4di 2289	. 2 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( gcd ` ( R ` 0 ) ) = ( M gcd N ) )
58	36, 52, 57	3eqtrd 2275	1 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 1st ` ( R ` N ) ) = ( M gcd N ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   ifcif 3638   {csn 3709   <.cop 3712   X. cxp 4772   |` cres 4776   o. ccom 4778   -->wf 5373   ` cfv 5377  (class class class)co 6085   e. cmpo 6087   1stc1st 6372   2ndc2nd 6373   0cc0 8180   NN0cn0 9568   ZZcz 9649   ZZ>=cuz 9931   mod cmo 10774   seqcseq 10899   gcd cgcd 12749

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297  ax-pre-mulext 8298  ax-arch 8299  ax-caucvg 8300

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-sup 7325  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913  df-div 9006  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-n0 9569  df-z 9650  df-uz 9932  df-q 10030  df-rp 10066  df-fz 10423  df-fzo 10561  df-fl 10716  df-mod 10775  df-seqfrec 10900  df-exp 10991  df-cj 11623  df-re 11624  df-im 11625  df-rsqrt 11780  df-abs 11781  df-dvds 12574  df-gcd 12750

This theorem is used by: (None)