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Theorem eucalg 15919
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 (𝑅𝑁) 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 ∈ ℕ0𝑁 ∈ ℕ0) → (1st ‘(𝑅𝑁)) = (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.
StepHypRef Expression
1 nn0uz 12268 . . . . . . . 8 0 = (ℤ‘0)
2 eucalg.2 . . . . . . . 8 𝑅 = seq0((𝐸 ∘ 1st ), (ℕ0 × {𝐴}))
3 0zd 11981 . . . . . . . 8 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 0 ∈ ℤ)
4 eucalg.3 . . . . . . . . 9 𝐴 = ⟨𝑀, 𝑁
5 opelxpi 5585 . . . . . . . . 9 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ⟨𝑀, 𝑁⟩ ∈ (ℕ0 × ℕ0))
64, 5eqeltrid 2914 . . . . . . . 8 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 𝐴 ∈ (ℕ0 × ℕ0))
7 eucalgval.1 . . . . . . . . . 10 𝐸 = (𝑥 ∈ ℕ0, 𝑦 ∈ ℕ0 ↦ if(𝑦 = 0, ⟨𝑥, 𝑦⟩, ⟨𝑦, (𝑥 mod 𝑦)⟩))
87eucalgf 15915 . . . . . . . . 9 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0)
98a1i 11 . . . . . . . 8 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0))
101, 2, 3, 6, 9algrf 15905 . . . . . . 7 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 𝑅:ℕ0⟶(ℕ0 × ℕ0))
11 ffvelrn 6841 . . . . . . 7 ((𝑅:ℕ0⟶(ℕ0 × ℕ0) ∧ 𝑁 ∈ ℕ0) → (𝑅𝑁) ∈ (ℕ0 × ℕ0))
1210, 11sylancom 588 . . . . . 6 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑅𝑁) ∈ (ℕ0 × ℕ0))
13 1st2nd2 7717 . . . . . 6 ((𝑅𝑁) ∈ (ℕ0 × ℕ0) → (𝑅𝑁) = ⟨(1st ‘(𝑅𝑁)), (2nd ‘(𝑅𝑁))⟩)
1412, 13syl 17 . . . . 5 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑅𝑁) = ⟨(1st ‘(𝑅𝑁)), (2nd ‘(𝑅𝑁))⟩)
1514fveq2d 6667 . . . 4 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ( gcd ‘(𝑅𝑁)) = ( gcd ‘⟨(1st ‘(𝑅𝑁)), (2nd ‘(𝑅𝑁))⟩))
16 df-ov 7148 . . . 4 ((1st ‘(𝑅𝑁)) gcd (2nd ‘(𝑅𝑁))) = ( gcd ‘⟨(1st ‘(𝑅𝑁)), (2nd ‘(𝑅𝑁))⟩)
1715, 16syl6eqr 2871 . . 3 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ( gcd ‘(𝑅𝑁)) = ((1st ‘(𝑅𝑁)) gcd (2nd ‘(𝑅𝑁))))
184fveq2i 6666 . . . . . . . 8 (2nd𝐴) = (2nd ‘⟨𝑀, 𝑁⟩)
19 op2ndg 7691 . . . . . . . 8 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
2018, 19syl5eq 2865 . . . . . . 7 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (2nd𝐴) = 𝑁)
2120fveq2d 6667 . . . . . 6 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑅‘(2nd𝐴)) = (𝑅𝑁))
2221fveq2d 6667 . . . . 5 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (2nd ‘(𝑅‘(2nd𝐴))) = (2nd ‘(𝑅𝑁)))
23 xp2nd 7711 . . . . . . . . 9 (𝐴 ∈ (ℕ0 × ℕ0) → (2nd𝐴) ∈ ℕ0)
2423nn0zd 12073 . . . . . . . 8 (𝐴 ∈ (ℕ0 × ℕ0) → (2nd𝐴) ∈ ℤ)
25 uzid 12246 . . . . . . . 8 ((2nd𝐴) ∈ ℤ → (2nd𝐴) ∈ (ℤ‘(2nd𝐴)))
2624, 25syl 17 . . . . . . 7 (𝐴 ∈ (ℕ0 × ℕ0) → (2nd𝐴) ∈ (ℤ‘(2nd𝐴)))
27 eqid 2818 . . . . . . . 8 (2nd𝐴) = (2nd𝐴)
287, 2, 27eucalgcvga 15918 . . . . . . 7 (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd𝐴) ∈ (ℤ‘(2nd𝐴)) → (2nd ‘(𝑅‘(2nd𝐴))) = 0))
2926, 28mpd 15 . . . . . 6 (𝐴 ∈ (ℕ0 × ℕ0) → (2nd ‘(𝑅‘(2nd𝐴))) = 0)
306, 29syl 17 . . . . 5 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (2nd ‘(𝑅‘(2nd𝐴))) = 0)
3122, 30eqtr3d 2855 . . . 4 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (2nd ‘(𝑅𝑁)) = 0)
3231oveq2d 7161 . . 3 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ((1st ‘(𝑅𝑁)) gcd (2nd ‘(𝑅𝑁))) = ((1st ‘(𝑅𝑁)) gcd 0))
33 xp1st 7710 . . . 4 ((𝑅𝑁) ∈ (ℕ0 × ℕ0) → (1st ‘(𝑅𝑁)) ∈ ℕ0)
34 nn0gcdid0 15857 . . . 4 ((1st ‘(𝑅𝑁)) ∈ ℕ0 → ((1st ‘(𝑅𝑁)) gcd 0) = (1st ‘(𝑅𝑁)))
3512, 33, 343syl 18 . . 3 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ((1st ‘(𝑅𝑁)) gcd 0) = (1st ‘(𝑅𝑁)))
3617, 32, 353eqtrrd 2858 . 2 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (1st ‘(𝑅𝑁)) = ( gcd ‘(𝑅𝑁)))
377eucalginv 15916 . . . . . 6 (𝑧 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸𝑧)) = ( gcd ‘𝑧))
388ffvelrni 6842 . . . . . . 7 (𝑧 ∈ (ℕ0 × ℕ0) → (𝐸𝑧) ∈ (ℕ0 × ℕ0))
3938fvresd 6683 . . . . . 6 (𝑧 ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) = ( gcd ‘(𝐸𝑧)))
40 fvres 6682 . . . . . 6 (𝑧 ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘𝑧) = ( gcd ‘𝑧))
4137, 39, 403eqtr4d 2863 . . . . 5 (𝑧 ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) = (( gcd ↾ (ℕ0 × ℕ0))‘𝑧))
422, 8, 41alginv 15907 . . . 4 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝑁 ∈ ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅𝑁)) = (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘0)))
436, 42sylancom 588 . . 3 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅𝑁)) = (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘0)))
4412fvresd 6683 . . 3 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅𝑁)) = ( gcd ‘(𝑅𝑁)))
45 0nn0 11900 . . . . 5 0 ∈ ℕ0
46 ffvelrn 6841 . . . . 5 ((𝑅:ℕ0⟶(ℕ0 × ℕ0) ∧ 0 ∈ ℕ0) → (𝑅‘0) ∈ (ℕ0 × ℕ0))
4710, 45, 46sylancl 586 . . . 4 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑅‘0) ∈ (ℕ0 × ℕ0))
4847fvresd 6683 . . 3 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘0)) = ( gcd ‘(𝑅‘0)))
4943, 44, 483eqtr3d 2861 . 2 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ( gcd ‘(𝑅𝑁)) = ( gcd ‘(𝑅‘0)))
501, 2, 3, 6algr0 15904 . . . . 5 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑅‘0) = 𝐴)
5150, 4syl6eq 2869 . . . 4 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑅‘0) = ⟨𝑀, 𝑁⟩)
5251fveq2d 6667 . . 3 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ( gcd ‘(𝑅‘0)) = ( gcd ‘⟨𝑀, 𝑁⟩))
53 df-ov 7148 . . 3 (𝑀 gcd 𝑁) = ( gcd ‘⟨𝑀, 𝑁⟩)
5452, 53syl6eqr 2871 . 2 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ( gcd ‘(𝑅‘0)) = (𝑀 gcd 𝑁))
5536, 49, 543eqtrd 2857 1 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (1st ‘(𝑅𝑁)) = (𝑀 gcd 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  ifcif 4463  {csn 4557  cop 4563   × cxp 5546  cres 5550  ccom 5552  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  1st c1st 7676  2nd c2nd 7677  0cc0 10525  0cn0 11885  cz 11969  cuz 12231   mod cmo 13225  seqcseq 13357   gcd cgcd 15831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-inf 8895  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-fl 13150  df-mod 13226  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-dvds 15596  df-gcd 15832
This theorem is referenced by: (None)
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