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Theorem eucalg 11751
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 9372 . . . . . . . 8 0 = (ℤ‘0)
2 eucalg.2 . . . . . . . 8 𝑅 = seq0((𝐸 ∘ 1st ), (ℕ0 × {𝐴}))
3 0zd 9078 . . . . . . . 8 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 0 ∈ ℤ)
4 eucalg.3 . . . . . . . . 9 𝐴 = ⟨𝑀, 𝑁
5 opelxpi 4571 . . . . . . . . 9 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ⟨𝑀, 𝑁⟩ ∈ (ℕ0 × ℕ0))
64, 5eqeltrid 2226 . . . . . . . 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 11747 . . . . . . . . 9 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0)
98a1i 9 . . . . . . . 8 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 𝐸:(ℕ0 × ℕ0)⟶(ℕ0 × ℕ0))
101, 2, 3, 6, 9algrf 11737 . . . . . . 7 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 𝑅:ℕ0⟶(ℕ0 × ℕ0))
11 ffvelrn 5553 . . . . . . 7 ((𝑅:ℕ0⟶(ℕ0 × ℕ0) ∧ 𝑁 ∈ ℕ0) → (𝑅𝑁) ∈ (ℕ0 × ℕ0))
1210, 11sylancom 416 . . . . . 6 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑅𝑁) ∈ (ℕ0 × ℕ0))
13 1st2nd2 6073 . . . . . 6 ((𝑅𝑁) ∈ (ℕ0 × ℕ0) → (𝑅𝑁) = ⟨(1st ‘(𝑅𝑁)), (2nd ‘(𝑅𝑁))⟩)
1412, 13syl 14 . . . . 5 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑅𝑁) = ⟨(1st ‘(𝑅𝑁)), (2nd ‘(𝑅𝑁))⟩)
1514fveq2d 5425 . . . 4 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ( gcd ‘(𝑅𝑁)) = ( gcd ‘⟨(1st ‘(𝑅𝑁)), (2nd ‘(𝑅𝑁))⟩))
16 df-ov 5777 . . . 4 ((1st ‘(𝑅𝑁)) gcd (2nd ‘(𝑅𝑁))) = ( gcd ‘⟨(1st ‘(𝑅𝑁)), (2nd ‘(𝑅𝑁))⟩)
1715, 16syl6eqr 2190 . . 3 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ( gcd ‘(𝑅𝑁)) = ((1st ‘(𝑅𝑁)) gcd (2nd ‘(𝑅𝑁))))
184fveq2i 5424 . . . . . . . 8 (2nd𝐴) = (2nd ‘⟨𝑀, 𝑁⟩)
19 op2ndg 6049 . . . . . . . 8 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
2018, 19syl5eq 2184 . . . . . . 7 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (2nd𝐴) = 𝑁)
2120fveq2d 5425 . . . . . 6 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑅‘(2nd𝐴)) = (𝑅𝑁))
2221fveq2d 5425 . . . . 5 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (2nd ‘(𝑅‘(2nd𝐴))) = (2nd ‘(𝑅𝑁)))
23 xp2nd 6064 . . . . . . . . 9 (𝐴 ∈ (ℕ0 × ℕ0) → (2nd𝐴) ∈ ℕ0)
2423nn0zd 9183 . . . . . . . 8 (𝐴 ∈ (ℕ0 × ℕ0) → (2nd𝐴) ∈ ℤ)
25 uzid 9352 . . . . . . . 8 ((2nd𝐴) ∈ ℤ → (2nd𝐴) ∈ (ℤ‘(2nd𝐴)))
2624, 25syl 14 . . . . . . 7 (𝐴 ∈ (ℕ0 × ℕ0) → (2nd𝐴) ∈ (ℤ‘(2nd𝐴)))
27 eqid 2139 . . . . . . . 8 (2nd𝐴) = (2nd𝐴)
287, 2, 27eucalgcvga 11750 . . . . . . 7 (𝐴 ∈ (ℕ0 × ℕ0) → ((2nd𝐴) ∈ (ℤ‘(2nd𝐴)) → (2nd ‘(𝑅‘(2nd𝐴))) = 0))
2926, 28mpd 13 . . . . . 6 (𝐴 ∈ (ℕ0 × ℕ0) → (2nd ‘(𝑅‘(2nd𝐴))) = 0)
306, 29syl 14 . . . . 5 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (2nd ‘(𝑅‘(2nd𝐴))) = 0)
3122, 30eqtr3d 2174 . . . 4 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (2nd ‘(𝑅𝑁)) = 0)
3231oveq2d 5790 . . 3 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ((1st ‘(𝑅𝑁)) gcd (2nd ‘(𝑅𝑁))) = ((1st ‘(𝑅𝑁)) gcd 0))
33 xp1st 6063 . . . 4 ((𝑅𝑁) ∈ (ℕ0 × ℕ0) → (1st ‘(𝑅𝑁)) ∈ ℕ0)
34 nn0gcdid0 11680 . . . 4 ((1st ‘(𝑅𝑁)) ∈ ℕ0 → ((1st ‘(𝑅𝑁)) gcd 0) = (1st ‘(𝑅𝑁)))
3512, 33, 343syl 17 . . 3 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ((1st ‘(𝑅𝑁)) gcd 0) = (1st ‘(𝑅𝑁)))
3617, 32, 353eqtrrd 2177 . 2 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (1st ‘(𝑅𝑁)) = ( gcd ‘(𝑅𝑁)))
377eucalginv 11748 . . . . . 6 (𝑧 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸𝑧)) = ( gcd ‘𝑧))
388ffvelrni 5554 . . . . . . 7 (𝑧 ∈ (ℕ0 × ℕ0) → (𝐸𝑧) ∈ (ℕ0 × ℕ0))
39 fvres 5445 . . . . . . 7 ((𝐸𝑧) ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) = ( gcd ‘(𝐸𝑧)))
4038, 39syl 14 . . . . . 6 (𝑧 ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) = ( gcd ‘(𝐸𝑧)))
41 fvres 5445 . . . . . 6 (𝑧 ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘𝑧) = ( gcd ‘𝑧))
4237, 40, 413eqtr4d 2182 . . . . 5 (𝑧 ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝐸𝑧)) = (( gcd ↾ (ℕ0 × ℕ0))‘𝑧))
432, 8, 42alginv 11739 . . . 4 ((𝐴 ∈ (ℕ0 × ℕ0) ∧ 𝑁 ∈ ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅𝑁)) = (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘0)))
446, 43sylancom 416 . . 3 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅𝑁)) = (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘0)))
45 fvres 5445 . . . 4 ((𝑅𝑁) ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅𝑁)) = ( gcd ‘(𝑅𝑁)))
4612, 45syl 14 . . 3 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅𝑁)) = ( gcd ‘(𝑅𝑁)))
47 0nn0 9004 . . . . 5 0 ∈ ℕ0
48 ffvelrn 5553 . . . . 5 ((𝑅:ℕ0⟶(ℕ0 × ℕ0) ∧ 0 ∈ ℕ0) → (𝑅‘0) ∈ (ℕ0 × ℕ0))
4910, 47, 48sylancl 409 . . . 4 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑅‘0) ∈ (ℕ0 × ℕ0))
50 fvres 5445 . . . 4 ((𝑅‘0) ∈ (ℕ0 × ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘0)) = ( gcd ‘(𝑅‘0)))
5149, 50syl 14 . . 3 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (( gcd ↾ (ℕ0 × ℕ0))‘(𝑅‘0)) = ( gcd ‘(𝑅‘0)))
5244, 46, 513eqtr3d 2180 . 2 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ( gcd ‘(𝑅𝑁)) = ( gcd ‘(𝑅‘0)))
531, 2, 3, 6, 9ialgr0 11736 . . . . 5 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑅‘0) = 𝐴)
5453, 4syl6eq 2188 . . . 4 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑅‘0) = ⟨𝑀, 𝑁⟩)
5554fveq2d 5425 . . 3 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ( gcd ‘(𝑅‘0)) = ( gcd ‘⟨𝑀, 𝑁⟩))
56 df-ov 5777 . . 3 (𝑀 gcd 𝑁) = ( gcd ‘⟨𝑀, 𝑁⟩)
5755, 56syl6eqr 2190 . 2 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ( gcd ‘(𝑅‘0)) = (𝑀 gcd 𝑁))
5836, 52, 573eqtrd 2176 1 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (1st ‘(𝑅𝑁)) = (𝑀 gcd 𝑁))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  ifcif 3474  {csn 3527  cop 3530   × cxp 4537  cres 4541  ccom 4543  wf 5119  cfv 5123  (class class class)co 5774  cmpo 5776  1st c1st 6036  2nd c2nd 6037  0cc0 7632  0cn0 8989  cz 9066  cuz 9338   mod cmo 10107  seqcseq 10230   gcd cgcd 11646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751  ax-caucvg 7752
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-n0 8990  df-z 9067  df-uz 9339  df-q 9424  df-rp 9454  df-fz 9803  df-fzo 9932  df-fl 10055  df-mod 10108  df-seqfrec 10231  df-exp 10305  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783  df-dvds 11505  df-gcd 11647
This theorem is referenced by: (None)
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