Theorem eucalg 16641

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 first 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 12896	. . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘0)
2		eucalg.2	. . . . . . . 8 ⊢ 𝑅 = seq0((𝐸 ∘ 1st ), (ℕ0 × {𝐴}))
3		0zd 12599	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → 0 ∈ ℤ)
4		eucalg.3	. . . . . . . . 9 ⊢ 𝐴 = ⟨𝑀, 𝑁⟩
5		opelxpi 5696	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ⟨𝑀, 𝑁⟩ ∈ (ℕ0 × ℕ0))
6	4, 5	eqeltrid 2873	. . . . . . . 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 16637	. . . . . . . . 9 ⊢ 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0)
9	8	a1i 11	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0))
10	1, 2, 3, 6, 9	algrf 16627	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → 𝑅:ℕ0⟶(ℕ0 × ℕ0))
11		ffvelcdm 7074	. . . . . . 7 ⊢ ((𝑅:ℕ0⟶(ℕ0 × ℕ0) ∧ 𝑁 ∈ ℕ0) → (𝑅‘𝑁) ∈ (ℕ0 × ℕ0))
12	10, 11	sylancom 599	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑅‘𝑁) ∈ (ℕ0 × ℕ0))
13		1st2nd2 8021	. . . . . 6 ⊢ ((𝑅‘𝑁) ∈ (ℕ0 × ℕ0) → (𝑅‘𝑁) = ⟨(1st ‘(𝑅‘𝑁)), (2nd ‘(𝑅‘𝑁))⟩)
14	12, 13	syl 18	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑅‘𝑁) = ⟨(1st ‘(𝑅‘𝑁)), (2nd ‘(𝑅‘𝑁))⟩)
15	14	fveq2d 6883	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ( gcd ‘(𝑅‘𝑁)) = ( gcd ‘⟨(1st ‘(𝑅‘𝑁)), (2nd ‘(𝑅‘𝑁))⟩))
16		df-ov 7411	. . . 4 ⊢ ((1st ‘(𝑅‘𝑁)) gcd (2nd ‘(𝑅‘𝑁))) = ( gcd ‘⟨(1st ‘(𝑅‘𝑁)), (2nd ‘(𝑅‘𝑁))⟩)
17	15, 16	eqtr4di 2822	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ( gcd ‘(𝑅‘𝑁)) = ((1st ‘(𝑅‘𝑁)) gcd (2nd ‘(𝑅‘𝑁))))
18	4	fveq2i 6882	. . . . . . . 8 ⊢ (2nd ‘𝐴) = (2nd ‘⟨𝑀, 𝑁⟩)
19		op2ndg 7995	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
20	18, 19	eqtrid 2816	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (2nd ‘𝐴) = 𝑁)
21	20	fveq2d 6883	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑅‘(2nd ‘𝐴)) = (𝑅‘𝑁))
22	21	fveq2d 6883	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (2nd ‘(𝑅‘(2nd ‘𝐴))) = (2nd ‘(𝑅‘𝑁)))
23		xp2nd 8015	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (2nd ‘𝐴) ∈ ℕ0)
24	23	nn0zd 12612	. . . . . . . 8 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (2nd ‘𝐴) ∈ ℤ)
25		uzid 12873	. . . . . . . 8 ⊢ ((2nd ‘𝐴) ∈ ℤ → (2nd ‘𝐴) ∈ (ℤ≥‘(2nd ‘𝐴)))
26	24, 25	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (2nd ‘𝐴) ∈ (ℤ≥‘(2nd ‘𝐴)))
27		eqid 2769	. . . . . . . 8 ⊢ (2nd ‘𝐴) = (2nd ‘𝐴)
28	7, 2, 27	eucalgcvga 16640	. . . . . . 7 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd ‘𝐴) ∈ (ℤ≥‘(2nd ‘𝐴)) → (2nd ‘(𝑅‘(2nd ‘𝐴))) = 0))
29	26, 28	mpd 16	. . . . . 6 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (2nd ‘(𝑅‘(2nd ‘𝐴))) = 0)
30	6, 29	syl 18	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (2nd ‘(𝑅‘(2nd ‘𝐴))) = 0)
31	22, 30	eqtr3d 2806	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (2nd ‘(𝑅‘𝑁)) = 0)
32	31	oveq2d 7424	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((1st ‘(𝑅‘𝑁)) gcd (2nd ‘(𝑅‘𝑁))) = ((1st ‘(𝑅‘𝑁)) gcd 0))
33		xp1st 8014	. . . 4 ⊢ ((𝑅‘𝑁) ∈ (ℕ0 × ℕ0) → (1st ‘(𝑅‘𝑁)) ∈ ℕ0)
34		nn0gcdid0 16575	. . . 4 ⊢ ((1st ‘(𝑅‘𝑁)) ∈ ℕ0 → ((1st ‘(𝑅‘𝑁)) gcd 0) = (1st ‘(𝑅‘𝑁)))
35	12, 33, 34	3syl 19	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((1st ‘(𝑅‘𝑁)) gcd 0) = (1st ‘(𝑅‘𝑁)))
36	17, 32, 35	3eqtrrd 2809	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (1st ‘(𝑅‘𝑁)) = ( gcd ‘(𝑅‘𝑁)))
37	7	eucalginv 16638	. . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸‘𝑧)) = ( gcd ‘𝑧))
38	8	ffvelcdmi 7076	. . . . . . 7 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (𝐸‘𝑧) ∈ (ℕ0 × ℕ0))
39	38	fvresd 6899	. . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) = ( gcd ‘(𝐸‘𝑧)))
40		fvres 6898	. . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘𝑧) = ( gcd ‘𝑧))
41	37, 39, 40	3eqtr4d 2814	. . . . 5 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) = (( gcd ↾ (ℕ0 × ℕ0))‘𝑧))
42	2, 8, 41	alginv 16629	. . . 4 ⊢ ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝑁 ∈ ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝑁)) = (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘0)))
43	6, 42	sylancom 599	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝑁)) = (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘0)))
44	12	fvresd 6899	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝑁)) = ( gcd ‘(𝑅‘𝑁)))
45		0nn0 12515	. . . . 5 ⊢ 0 ∈ ℕ0
46		ffvelcdm 7074	. . . . 5 ⊢ ((𝑅:ℕ0⟶(ℕ0 × ℕ0) ∧ 0 ∈ ℕ0) → (𝑅‘0) ∈ (ℕ0 × ℕ0))
47	10, 45, 46	sylancl 597	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑅‘0) ∈ (ℕ0 × ℕ0))
48	47	fvresd 6899	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘0)) = ( gcd ‘(𝑅‘0)))
49	43, 44, 48	3eqtr3d 2812	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ( gcd ‘(𝑅‘𝑁)) = ( gcd ‘(𝑅‘0)))
50	1, 2, 3, 6	algr0 16626	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑅‘0) = 𝐴)
51	50, 4	eqtrdi 2820	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑅‘0) = ⟨𝑀, 𝑁⟩)
52	51	fveq2d 6883	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ( gcd ‘(𝑅‘0)) = ( gcd ‘⟨𝑀, 𝑁⟩))
53		df-ov 7411	. . 3 ⊢ (𝑀 gcd 𝑁) = ( gcd ‘⟨𝑀, 𝑁⟩)
54	52, 53	eqtr4di 2822	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ( gcd ‘(𝑅‘0)) = (𝑀 gcd 𝑁))
55	36, 49, 54	3eqtrd 2808	1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (1st ‘(𝑅‘𝑁)) = (𝑀 gcd 𝑁))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ifcif 4489  {csn 4591  ⟨cop 4597   × cxp 5657   ↾ cres 5661   ∘ ccom 5663  ⟶wf 6529  ‘cfv 6533  (class class class)co 7408   ∈ cmpo 7410  1^st c1st 7980  2^nd c2nd 7981  0cc0 11096  ℕ₀cn0 12500  ℤcz 12587  ℤ_≥cuz 12858   mod cmo 13898  seqcseq 14033   gcd cgcd 16548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173  ax-pre-sup 11174

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9398  df-inf 9399  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-div 11868  df-nn 12230  df-2 12299  df-3 12300  df-n0 12501  df-z 12588  df-uz 12859  df-rp 13013  df-fz 13532  df-fl 13821  df-mod 13899  df-seq 14034  df-exp 14094  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-dvds 16307  df-gcd 16549

This theorem is referenced by: (None)