MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eucalginv Structured version   Visualization version   GIF version

Theorem eucalginv 15906
Description: The invariant of the step function 𝐸 for Euclid's Algorithm is the gcd operator applied to the state. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 29-May-2014.)
Hypothesis
Ref Expression
eucalgval.1 𝐸 = (𝑥 ∈ ℕ0, 𝑦 ∈ ℕ0 ↦ if(𝑦 = 0, ⟨𝑥, 𝑦⟩, ⟨𝑦, (𝑥 mod 𝑦)⟩))
Assertion
Ref Expression
eucalginv (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸𝑋)) = ( gcd ‘𝑋))
Distinct variable group:   𝑥,𝑦,𝑋
Allowed substitution hints:   𝐸(𝑥,𝑦)

Proof of Theorem eucalginv
StepHypRef Expression
1 eucalgval.1 . . . 4 𝐸 = (𝑥 ∈ ℕ0, 𝑦 ∈ ℕ0 ↦ if(𝑦 = 0, ⟨𝑥, 𝑦⟩, ⟨𝑦, (𝑥 mod 𝑦)⟩))
21eucalgval 15904 . . 3 (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩))
32fveq2d 6650 . 2 (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸𝑋)) = ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)))
4 1st2nd2 7706 . . . . . . . . 9 (𝑋 ∈ (ℕ0 × ℕ0) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
54adantr 483 . . . . . . . 8 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
65fveq2d 6650 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( mod ‘𝑋) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩))
7 df-ov 7136 . . . . . . 7 ((1st𝑋) mod (2nd𝑋)) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩)
86, 7syl6eqr 2873 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( mod ‘𝑋) = ((1st𝑋) mod (2nd𝑋)))
98oveq2d 7149 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ((2nd𝑋) gcd ( mod ‘𝑋)) = ((2nd𝑋) gcd ((1st𝑋) mod (2nd𝑋))))
10 nnz 11983 . . . . . 6 ((2nd𝑋) ∈ ℕ → (2nd𝑋) ∈ ℤ)
11 xp1st 7699 . . . . . . . . . 10 (𝑋 ∈ (ℕ0 × ℕ0) → (1st𝑋) ∈ ℕ0)
1211adantr 483 . . . . . . . . 9 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → (1st𝑋) ∈ ℕ0)
1312nn0zd 12064 . . . . . . . 8 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → (1st𝑋) ∈ ℤ)
14 zmodcl 13243 . . . . . . . 8 (((1st𝑋) ∈ ℤ ∧ (2nd𝑋) ∈ ℕ) → ((1st𝑋) mod (2nd𝑋)) ∈ ℕ0)
1513, 14sylancom 590 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ((1st𝑋) mod (2nd𝑋)) ∈ ℕ0)
1615nn0zd 12064 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ((1st𝑋) mod (2nd𝑋)) ∈ ℤ)
17 gcdcom 15840 . . . . . 6 (((2nd𝑋) ∈ ℤ ∧ ((1st𝑋) mod (2nd𝑋)) ∈ ℤ) → ((2nd𝑋) gcd ((1st𝑋) mod (2nd𝑋))) = (((1st𝑋) mod (2nd𝑋)) gcd (2nd𝑋)))
1810, 16, 17syl2an2 684 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ((2nd𝑋) gcd ((1st𝑋) mod (2nd𝑋))) = (((1st𝑋) mod (2nd𝑋)) gcd (2nd𝑋)))
19 modgcd 15858 . . . . . 6 (((1st𝑋) ∈ ℤ ∧ (2nd𝑋) ∈ ℕ) → (((1st𝑋) mod (2nd𝑋)) gcd (2nd𝑋)) = ((1st𝑋) gcd (2nd𝑋)))
2013, 19sylancom 590 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → (((1st𝑋) mod (2nd𝑋)) gcd (2nd𝑋)) = ((1st𝑋) gcd (2nd𝑋)))
219, 18, 203eqtrd 2859 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ((2nd𝑋) gcd ( mod ‘𝑋)) = ((1st𝑋) gcd (2nd𝑋)))
22 nnne0 11650 . . . . . . . . 9 ((2nd𝑋) ∈ ℕ → (2nd𝑋) ≠ 0)
2322adantl 484 . . . . . . . 8 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → (2nd𝑋) ≠ 0)
2423neneqd 3011 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ¬ (2nd𝑋) = 0)
2524iffalsed 4454 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) = ⟨(2nd𝑋), ( mod ‘𝑋)⟩)
2625fveq2d 6650 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘⟨(2nd𝑋), ( mod ‘𝑋)⟩))
27 df-ov 7136 . . . . 5 ((2nd𝑋) gcd ( mod ‘𝑋)) = ( gcd ‘⟨(2nd𝑋), ( mod ‘𝑋)⟩)
2826, 27syl6eqr 2873 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)) = ((2nd𝑋) gcd ( mod ‘𝑋)))
295fveq2d 6650 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( gcd ‘𝑋) = ( gcd ‘⟨(1st𝑋), (2nd𝑋)⟩))
30 df-ov 7136 . . . . 5 ((1st𝑋) gcd (2nd𝑋)) = ( gcd ‘⟨(1st𝑋), (2nd𝑋)⟩)
3129, 30syl6eqr 2873 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( gcd ‘𝑋) = ((1st𝑋) gcd (2nd𝑋)))
3221, 28, 313eqtr4d 2865 . . 3 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
33 iftrue 4449 . . . . 5 ((2nd𝑋) = 0 → if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) = 𝑋)
3433fveq2d 6650 . . . 4 ((2nd𝑋) = 0 → ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
3534adantl 484 . . 3 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) = 0) → ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
36 xp2nd 7700 . . . 4 (𝑋 ∈ (ℕ0 × ℕ0) → (2nd𝑋) ∈ ℕ0)
37 elnn0 11878 . . . 4 ((2nd𝑋) ∈ ℕ0 ↔ ((2nd𝑋) ∈ ℕ ∨ (2nd𝑋) = 0))
3836, 37sylib 220 . . 3 (𝑋 ∈ (ℕ0 × ℕ0) → ((2nd𝑋) ∈ ℕ ∨ (2nd𝑋) = 0))
3932, 35, 38mpjaodan 955 . 2 (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
403, 39eqtrd 2855 1 (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸𝑋)) = ( gcd ‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wo 843   = wceq 1537  wcel 2114  wne 3006  ifcif 4443  cop 4549   × cxp 5529  cfv 6331  (class class class)co 7133  cmpo 7135  1st c1st 7665  2nd c2nd 7666  0cc0 10515  cn 11616  0cn0 11876  cz 11960   mod cmo 13221   gcd cgcd 15821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592  ax-pre-sup 10593
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-er 8267  df-en 8488  df-dom 8489  df-sdom 8490  df-sup 8884  df-inf 8885  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-div 11276  df-nn 11617  df-2 11679  df-3 11680  df-n0 11877  df-z 11961  df-uz 12223  df-rp 12369  df-fl 13146  df-mod 13222  df-seq 13354  df-exp 13415  df-cj 14438  df-re 14439  df-im 14440  df-sqrt 14574  df-abs 14575  df-dvds 15588  df-gcd 15822
This theorem is referenced by:  eucalg  15909
  Copyright terms: Public domain W3C validator