Theorem eucalg 16635

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 12891	. . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘0)
2		eucalg.2	. . . . . . . 8 ⊢ 𝑅 = seq0((𝐸 ∘ 1st ), (ℕ0 × {𝐴}))
3		0zd 12594	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → 0 ∈ ℤ)
4		eucalg.3	. . . . . . . . 9 ⊢ 𝐴 = ⟨𝑀, 𝑁⟩
5		opelxpi 5689	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ⟨𝑀, 𝑁⟩ ∈ (ℕ0 × ℕ0))
6	4, 5	eqeltrid 2869	. . . . . . . 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 16631	. . . . . . . . 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 16621	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → 𝑅:ℕ0⟶(ℕ0 × ℕ0))
11		ffvelcdm 7066	. . . . . . 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 8013	. . . . . 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 6875	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ( gcd ‘(𝑅‘𝑁)) = ( gcd ‘⟨(1st ‘(𝑅‘𝑁)), (2nd ‘(𝑅‘𝑁))⟩))
16		df-ov 7403	. . . 4 ⊢ ((1st ‘(𝑅‘𝑁)) gcd (2nd ‘(𝑅‘𝑁))) = ( gcd ‘⟨(1st ‘(𝑅‘𝑁)), (2nd ‘(𝑅‘𝑁))⟩)
17	15, 16	eqtr4di 2818	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ( gcd ‘(𝑅‘𝑁)) = ((1st ‘(𝑅‘𝑁)) gcd (2nd ‘(𝑅‘𝑁))))
18	4	fveq2i 6874	. . . . . . . 8 ⊢ (2nd ‘𝐴) = (2nd ‘⟨𝑀, 𝑁⟩)
19		op2ndg 7987	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
20	18, 19	eqtrid 2812	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (2nd ‘𝐴) = 𝑁)
21	20	fveq2d 6875	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑅‘(2nd ‘𝐴)) = (𝑅‘𝑁))
22	21	fveq2d 6875	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (2nd ‘(𝑅‘(2nd ‘𝐴))) = (2nd ‘(𝑅‘𝑁)))
23		xp2nd 8007	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (2nd ‘𝐴) ∈ ℕ0)
24	23	nn0zd 12607	. . . . . . . 8 ⊢ (𝐴 ∈ (ℕ0 × ℕ0) → (2nd ‘𝐴) ∈ ℤ)
25		uzid 12868	. . . . . . . 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 2765	. . . . . . . 8 ⊢ (2nd ‘𝐴) = (2nd ‘𝐴)
28	7, 2, 27	eucalgcvga 16634	. . . . . . 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 2802	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (2nd ‘(𝑅‘𝑁)) = 0)
32	31	oveq2d 7416	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((1st ‘(𝑅‘𝑁)) gcd (2nd ‘(𝑅‘𝑁))) = ((1st ‘(𝑅‘𝑁)) gcd 0))
33		xp1st 8006	. . . 4 ⊢ ((𝑅‘𝑁) ∈ (ℕ0 × ℕ0) → (1st ‘(𝑅‘𝑁)) ∈ ℕ0)
34		nn0gcdid0 16569	. . . 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 2805	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (1st ‘(𝑅‘𝑁)) = ( gcd ‘(𝑅‘𝑁)))
37	7	eucalginv 16632	. . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸‘𝑧)) = ( gcd ‘𝑧))
38	8	ffvelcdmi 7068	. . . . . . 7 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (𝐸‘𝑧) ∈ (ℕ0 × ℕ0))
39	38	fvresd 6891	. . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) = ( gcd ‘(𝐸‘𝑧)))
40		fvres 6890	. . . . . 6 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘𝑧) = ( gcd ‘𝑧))
41	37, 39, 40	3eqtr4d 2810	. . . . 5 ⊢ (𝑧 ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝐸‘𝑧)) = (( gcd ↾ (ℕ0 × ℕ0))‘𝑧))
42	2, 8, 41	alginv 16623	. . . 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 6891	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘𝑁)) = ( gcd ‘(𝑅‘𝑁)))
45		0nn0 12510	. . . . 5 ⊢ 0 ∈ ℕ0
46		ffvelcdm 7066	. . . . 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 6891	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘0)) = ( gcd ‘(𝑅‘0)))
49	43, 44, 48	3eqtr3d 2808	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ( gcd ‘(𝑅‘𝑁)) = ( gcd ‘(𝑅‘0)))
50	1, 2, 3, 6	algr0 16620	. . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑅‘0) = 𝐴)
51	50, 4	eqtrdi 2816	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑅‘0) = ⟨𝑀, 𝑁⟩)
52	51	fveq2d 6875	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ( gcd ‘(𝑅‘0)) = ( gcd ‘⟨𝑀, 𝑁⟩))
53		df-ov 7403	. . 3 ⊢ (𝑀 gcd 𝑁) = ( gcd ‘⟨𝑀, 𝑁⟩)
54	52, 53	eqtr4di 2818	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ( gcd ‘(𝑅‘0)) = (𝑀 gcd 𝑁))
55	36, 49, 54	3eqtrd 2804	1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (1st ‘(𝑅‘𝑁)) = (𝑀 gcd 𝑁))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ifcif 4483  {csn 4585  ⟨cop 4591   × cxp 5650   ↾ cres 5654   ∘ ccom 5656  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  1^st c1st 7972  2^nd c2nd 7973  0cc0 11088  ℕ₀cn0 12495  ℤcz 12582  ℤ_≥cuz 12853   mod cmo 13893  seqcseq 14028   gcd cgcd 16542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-sup 9390  df-inf 9391  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12225  df-2 12294  df-3 12295  df-n0 12496  df-z 12583  df-uz 12854  df-rp 13008  df-fz 13527  df-fl 13816  df-mod 13894  df-seq 14029  df-exp 14089  df-cj 15140  df-re 15141  df-im 15142  df-sqrt 15276  df-abs 15277  df-dvds 16301  df-gcd 16543

This theorem is referenced by: (None)