Theorem eucalg 12764

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 ∈ ℕ0, 𝑦 ∈ ℕ0 ↦ if(𝑦 = 0, ⟨𝑥, 𝑦⟩, ⟨𝑦, (𝑥 mod 𝑦)⟩))
eucalg.2	⊢ 𝑅 = seq0((𝐸 ∘ 1st ), (ℕ0 × {𝐴}))
eucalg.3	⊢ 𝐴 = ⟨𝑀, 𝑁⟩

Assertion

Ref	Expression
eucalg	⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (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 9895	. . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘0)
2		eucalg.2	. . . . . . . 8 ⊢ 𝑅 = seq0((𝐸 ∘ 1st ), (ℕ0 × {𝐴}))
3		0zd 9594	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → 0 ∈ ℤ)
4		eucalg.3	. . . . . . . . 9 ⊢ 𝐴 = ⟨𝑀, 𝑁⟩
5		opelxpi 4783	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ⟨𝑀, 𝑁⟩ ∈ (ℕ0 × ℕ0))
6	4, 5	eqeltrid 2321	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → 𝐴 ∈ (ℕ0 × ℕ0))
7		eucalgval.1	. . . . . . . . . 10 ⊢ 𝐸 = (𝑥 ∈ ℕ0, 𝑦 ∈ ℕ0 ↦ if(𝑦 = 0, ⟨𝑥, 𝑦⟩, ⟨𝑦, (𝑥 mod 𝑦)⟩))
8	7	eucalgf 12760	. . . . . . . . 9 ⊢ 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0)
9	8	a1i 9	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0))
10	1, 2, 3, 6, 9	algrf 12750	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → 𝑅:ℕ0⟶(ℕ0 × ℕ0))
11		ffvelcdm 5812	. . . . . . 7 ⊢ ((𝑅:ℕ0⟶(ℕ0 × ℕ0) ∧ 𝑁 ∈ ℕ0) → (𝑅‘𝑁) ∈ (ℕ0 × ℕ0))
12	10, 11	sylancom 420	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑅‘𝑁) ∈ (ℕ0 × ℕ0))
13		1st2nd2 6371	. . . . . 6 ⊢ ((𝑅‘𝑁) ∈ (ℕ0 × ℕ0) → (𝑅‘𝑁) = ⟨(1st ‘(𝑅‘𝑁)), (2nd ‘(𝑅‘𝑁))⟩)
14	12, 13	syl 14	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑅‘𝑁) = ⟨(1st ‘(𝑅‘𝑁)), (2nd ‘(𝑅‘𝑁))⟩)
15	14	fveq2d 5676	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ( gcd ‘(𝑅‘𝑁)) = ( gcd ‘⟨(1st ‘(𝑅‘𝑁)), (2nd ‘(𝑅‘𝑁))⟩))
16		df-ov 6055	. . . 4 ⊢ ((1st ‘(𝑅‘𝑁)) gcd (2nd ‘(𝑅‘𝑁))) = ( gcd ‘⟨(1st ‘(𝑅‘𝑁)), (2nd ‘(𝑅‘𝑁))⟩)
17	15, 16	eqtr4di 2285	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ( gcd ‘(𝑅‘𝑁)) = ((1st ‘(𝑅‘𝑁)) gcd (2nd ‘(𝑅‘𝑁))))
18	4	fveq2i 5675	. . . . . . . 8 ⊢ (2nd ‘𝐴) = (2nd ‘⟨𝑀, 𝑁⟩)
19		op2ndg 6347	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
20	18, 19	eqtrid 2279	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (2nd ‘𝐴) = 𝑁)
21	20	fveq2d 5676	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑅‘(2nd ‘𝐴)) = (𝑅‘𝑁))
22	21	fveq2d 5676	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (2nd ‘(𝑅‘(2nd ‘𝐴))) = (2nd ‘(𝑅‘𝑁)))
23		xp2nd 6362	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (2nd ‘𝐴) ∈ ℕ0)
24	23	nn0zd 9704	. . . . . . . 8 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (2nd ‘𝐴) ∈ ℤ)
25		uzid 9874	. . . . . . . 8 ⊢ ((2nd ‘𝐴) ∈ ℤ → (2nd ‘𝐴) ∈ (ℤ≥‘(2nd ‘𝐴)))
26	24, 25	syl 14	. . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (2nd ‘𝐴) ∈ (ℤ≥‘(2nd ‘𝐴)))
27		eqid 2234	. . . . . . . 8 ⊢ (2nd ‘𝐴) = (2nd ‘𝐴)
28	7, 2, 27	eucalgcvga 12763	. . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd ‘𝐴) ∈ (ℤ≥‘(2nd ‘𝐴)) → (2nd ‘(𝑅‘(2nd ‘𝐴))) = 0))
29	26, 28	mpd 13	. . . . . 6 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (2nd ‘(𝑅‘(2nd ‘𝐴))) = 0)
30	6, 29	syl 14	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (2nd ‘(𝑅‘(2nd ‘𝐴))) = 0)
31	22, 30	eqtr3d 2269	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (2nd ‘(𝑅‘𝑁)) = 0)
32	31	oveq2d 6068	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((1st ‘(𝑅‘𝑁)) gcd (2nd ‘(𝑅‘𝑁))) = ((1st ‘(𝑅‘𝑁)) gcd 0))
33		xp1st 6361	. . . 4 ⊢ ((𝑅‘𝑁) ∈ (ℕ0 × ℕ0) → (1st ‘(𝑅‘𝑁)) ∈ ℕ0)
34		nn0gcdid0 12685	. . . 4 ⊢ ((1st ‘(𝑅‘𝑁)) ∈ ℕ0 → ((1st ‘(𝑅‘𝑁)) gcd 0) = (1st ‘(𝑅‘𝑁)))
35	12, 33, 34	3syl 17	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((1st ‘(𝑅‘𝑁)) gcd 0) = (1st ‘(𝑅‘𝑁)))
36	17, 32, 35	3eqtrrd 2272	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (1st ‘(𝑅‘𝑁)) = ( gcd ‘(𝑅‘𝑁)))
37	7	eucalginv 12761	. . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸‘𝑧)) = ( gcd ‘𝑧))
38	8	ffvelcdmi 5813	. . . . . . 7 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (𝐸‘𝑧) ∈ (ℕ0 × ℕ0))
39		fvres 5696	. . . . . . 7 ⊢ ((𝐸‘𝑧) ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) = ( gcd ‘(𝐸‘𝑧)))
40	38, 39	syl 14	. . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) = ( gcd ‘(𝐸‘𝑧)))
41		fvres 5696	. . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘𝑧) = ( gcd ‘𝑧))
42	37, 40, 41	3eqtr4d 2277	. . . . 5 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) = (( gcd ↾ (ℕ0 × ℕ0))‘𝑧))
43	2, 8, 42	alginv 12752	. . . 4 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝑁 ∈ ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝑁)) = (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘0)))
44	6, 43	sylancom 420	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝑁)) = (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘0)))
45		fvres 5696	. . . 4 ⊢ ((𝑅‘𝑁) ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝑁)) = ( gcd ‘(𝑅‘𝑁)))
46	12, 45	syl 14	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝑁)) = ( gcd ‘(𝑅‘𝑁)))
47		0nn0 9516	. . . . 5 ⊢ 0 ∈ ℕ0
48		ffvelcdm 5812	. . . . 5 ⊢ ((𝑅:ℕ0⟶(ℕ0 × ℕ0) ∧ 0 ∈ ℕ0) → (𝑅‘0) ∈ (ℕ0 × ℕ0))
49	10, 47, 48	sylancl 413	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑅‘0) ∈ (ℕ0 × ℕ0))
50		fvres 5696	. . . 4 ⊢ ((𝑅‘0) ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘0)) = ( gcd ‘(𝑅‘0)))
51	49, 50	syl 14	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘0)) = ( gcd ‘(𝑅‘0)))
52	44, 46, 51	3eqtr3d 2275	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ( gcd ‘(𝑅‘𝑁)) = ( gcd ‘(𝑅‘0)))
53	1, 2, 3, 6, 9	ialgr0 12749	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑅‘0) = 𝐴)
54	53, 4	eqtrdi 2283	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑅‘0) = ⟨𝑀, 𝑁⟩)
55	54	fveq2d 5676	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ( gcd ‘(𝑅‘0)) = ( gcd ‘⟨𝑀, 𝑁⟩))
56		df-ov 6055	. . 3 ⊢ (𝑀 gcd 𝑁) = ( gcd ‘⟨𝑀, 𝑁⟩)
57	55, 56	eqtr4di 2285	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ( gcd ‘(𝑅‘0)) = (𝑀 gcd 𝑁))
58	36, 52, 57	3eqtrd 2271	1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (1st ‘(𝑅‘𝑁)) = (𝑀 gcd 𝑁))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  ifcif 3622  {csn 3691  ⟨cop 3694   × cxp 4749   ↾ cres 4753   ∘ ccom 4755  ⟶wf 5350  ‘cfv 5354  (class class class)co 6052   ∈ cmpo 6054  1^st c1st 6334  2^nd c2nd 6335  0cc0 8132  ℕ₀cn0 9501  ℤcz 9582  ℤ_≥cuz 9859   mod cmo 10691  seqcseq 10816   gcd cgcd 12657

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250  ax-arch 8251  ax-caucvg 8252

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-sup 7277  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-n0 9502  df-z 9583  df-uz 9860  df-q 9958  df-rp 9993  df-fz 10349  df-fzo 10484  df-fl 10637  df-mod 10692  df-seqfrec 10817  df-exp 10908  df-cj 11535  df-re 11536  df-im 11537  df-rsqrt 11691  df-abs 11692  df-dvds 12482  df-gcd 12658

This theorem is referenced by: (None)