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Theorem eucalginv 15629
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 15627 . . 3 (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩))
32fveq2d 6413 . 2 (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸𝑋)) = ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)))
4 1st2nd2 7438 . . . . . . . . 9 (𝑋 ∈ (ℕ0 × ℕ0) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
54adantr 473 . . . . . . . 8 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
65fveq2d 6413 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( mod ‘𝑋) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩))
7 df-ov 6879 . . . . . . 7 ((1st𝑋) mod (2nd𝑋)) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩)
86, 7syl6eqr 2849 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( mod ‘𝑋) = ((1st𝑋) mod (2nd𝑋)))
98oveq2d 6892 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ((2nd𝑋) gcd ( mod ‘𝑋)) = ((2nd𝑋) gcd ((1st𝑋) mod (2nd𝑋))))
10 nnz 11685 . . . . . . 7 ((2nd𝑋) ∈ ℕ → (2nd𝑋) ∈ ℤ)
1110adantl 474 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → (2nd𝑋) ∈ ℤ)
12 xp1st 7431 . . . . . . . . . 10 (𝑋 ∈ (ℕ0 × ℕ0) → (1st𝑋) ∈ ℕ0)
1312adantr 473 . . . . . . . . 9 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → (1st𝑋) ∈ ℕ0)
1413nn0zd 11766 . . . . . . . 8 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → (1st𝑋) ∈ ℤ)
15 zmodcl 12941 . . . . . . . 8 (((1st𝑋) ∈ ℤ ∧ (2nd𝑋) ∈ ℕ) → ((1st𝑋) mod (2nd𝑋)) ∈ ℕ0)
1614, 15sylancom 583 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ((1st𝑋) mod (2nd𝑋)) ∈ ℕ0)
1716nn0zd 11766 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ((1st𝑋) mod (2nd𝑋)) ∈ ℤ)
18 gcdcom 15567 . . . . . 6 (((2nd𝑋) ∈ ℤ ∧ ((1st𝑋) mod (2nd𝑋)) ∈ ℤ) → ((2nd𝑋) gcd ((1st𝑋) mod (2nd𝑋))) = (((1st𝑋) mod (2nd𝑋)) gcd (2nd𝑋)))
1911, 17, 18syl2anc 580 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ((2nd𝑋) gcd ((1st𝑋) mod (2nd𝑋))) = (((1st𝑋) mod (2nd𝑋)) gcd (2nd𝑋)))
20 modgcd 15585 . . . . . 6 (((1st𝑋) ∈ ℤ ∧ (2nd𝑋) ∈ ℕ) → (((1st𝑋) mod (2nd𝑋)) gcd (2nd𝑋)) = ((1st𝑋) gcd (2nd𝑋)))
2114, 20sylancom 583 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → (((1st𝑋) mod (2nd𝑋)) gcd (2nd𝑋)) = ((1st𝑋) gcd (2nd𝑋)))
229, 19, 213eqtrd 2835 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ((2nd𝑋) gcd ( mod ‘𝑋)) = ((1st𝑋) gcd (2nd𝑋)))
23 nnne0 11347 . . . . . . . . 9 ((2nd𝑋) ∈ ℕ → (2nd𝑋) ≠ 0)
2423adantl 474 . . . . . . . 8 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → (2nd𝑋) ≠ 0)
2524neneqd 2974 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ¬ (2nd𝑋) = 0)
2625iffalsed 4286 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) = ⟨(2nd𝑋), ( mod ‘𝑋)⟩)
2726fveq2d 6413 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘⟨(2nd𝑋), ( mod ‘𝑋)⟩))
28 df-ov 6879 . . . . 5 ((2nd𝑋) gcd ( mod ‘𝑋)) = ( gcd ‘⟨(2nd𝑋), ( mod ‘𝑋)⟩)
2927, 28syl6eqr 2849 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)) = ((2nd𝑋) gcd ( mod ‘𝑋)))
305fveq2d 6413 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( gcd ‘𝑋) = ( gcd ‘⟨(1st𝑋), (2nd𝑋)⟩))
31 df-ov 6879 . . . . 5 ((1st𝑋) gcd (2nd𝑋)) = ( gcd ‘⟨(1st𝑋), (2nd𝑋)⟩)
3230, 31syl6eqr 2849 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( gcd ‘𝑋) = ((1st𝑋) gcd (2nd𝑋)))
3322, 29, 323eqtr4d 2841 . . 3 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
34 iftrue 4281 . . . . 5 ((2nd𝑋) = 0 → if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) = 𝑋)
3534fveq2d 6413 . . . 4 ((2nd𝑋) = 0 → ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
3635adantl 474 . . 3 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) = 0) → ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
37 xp2nd 7432 . . . 4 (𝑋 ∈ (ℕ0 × ℕ0) → (2nd𝑋) ∈ ℕ0)
38 elnn0 11578 . . . 4 ((2nd𝑋) ∈ ℕ0 ↔ ((2nd𝑋) ∈ ℕ ∨ (2nd𝑋) = 0))
3937, 38sylib 210 . . 3 (𝑋 ∈ (ℕ0 × ℕ0) → ((2nd𝑋) ∈ ℕ ∨ (2nd𝑋) = 0))
4033, 36, 39mpjaodan 982 . 2 (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
413, 40eqtrd 2831 1 (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸𝑋)) = ( gcd ‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wo 874   = wceq 1653  wcel 2157  wne 2969  ifcif 4275  cop 4372   × cxp 5308  cfv 6099  (class class class)co 6876  cmpt2 6878  1st c1st 7397  2nd c2nd 7398  0cc0 10222  cn 11310  0cn0 11576  cz 11662   mod cmo 12919   gcd cgcd 15548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-cnex 10278  ax-resscn 10279  ax-1cn 10280  ax-icn 10281  ax-addcl 10282  ax-addrcl 10283  ax-mulcl 10284  ax-mulrcl 10285  ax-mulcom 10286  ax-addass 10287  ax-mulass 10288  ax-distr 10289  ax-i2m1 10290  ax-1ne0 10291  ax-1rid 10292  ax-rnegex 10293  ax-rrecex 10294  ax-cnre 10295  ax-pre-lttri 10296  ax-pre-lttrn 10297  ax-pre-ltadd 10298  ax-pre-mulgt0 10299  ax-pre-sup 10300
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-nel 3073  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-om 7298  df-1st 7399  df-2nd 7400  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-er 7980  df-en 8194  df-dom 8195  df-sdom 8196  df-sup 8588  df-inf 8589  df-pnf 10363  df-mnf 10364  df-xr 10365  df-ltxr 10366  df-le 10367  df-sub 10556  df-neg 10557  df-div 10975  df-nn 11311  df-2 11372  df-3 11373  df-n0 11577  df-z 11663  df-uz 11927  df-rp 12071  df-fl 12844  df-mod 12920  df-seq 13052  df-exp 13111  df-cj 14177  df-re 14178  df-im 14179  df-sqrt 14313  df-abs 14314  df-dvds 15317  df-gcd 15549
This theorem is referenced by:  eucalg  15632
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