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Theorem eucalginv 12074
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 12072 . . 3 (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩))
32fveq2d 5534 . 2 (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸𝑋)) = ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)))
4 1st2nd2 6194 . . . . . . . . 9 (𝑋 ∈ (ℕ0 × ℕ0) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
54adantr 276 . . . . . . . 8 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
65fveq2d 5534 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( mod ‘𝑋) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩))
7 df-ov 5894 . . . . . . 7 ((1st𝑋) mod (2nd𝑋)) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩)
86, 7eqtr4di 2240 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( mod ‘𝑋) = ((1st𝑋) mod (2nd𝑋)))
98oveq2d 5907 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ((2nd𝑋) gcd ( mod ‘𝑋)) = ((2nd𝑋) gcd ((1st𝑋) mod (2nd𝑋))))
10 nnz 9290 . . . . . . 7 ((2nd𝑋) ∈ ℕ → (2nd𝑋) ∈ ℤ)
1110adantl 277 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → (2nd𝑋) ∈ ℤ)
12 xp1st 6184 . . . . . . . . . 10 (𝑋 ∈ (ℕ0 × ℕ0) → (1st𝑋) ∈ ℕ0)
1312adantr 276 . . . . . . . . 9 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → (1st𝑋) ∈ ℕ0)
1413nn0zd 9391 . . . . . . . 8 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → (1st𝑋) ∈ ℤ)
15 zmodcl 10362 . . . . . . . 8 (((1st𝑋) ∈ ℤ ∧ (2nd𝑋) ∈ ℕ) → ((1st𝑋) mod (2nd𝑋)) ∈ ℕ0)
1614, 15sylancom 420 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ((1st𝑋) mod (2nd𝑋)) ∈ ℕ0)
1716nn0zd 9391 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ((1st𝑋) mod (2nd𝑋)) ∈ ℤ)
18 gcdcom 11992 . . . . . 6 (((2nd𝑋) ∈ ℤ ∧ ((1st𝑋) mod (2nd𝑋)) ∈ ℤ) → ((2nd𝑋) gcd ((1st𝑋) mod (2nd𝑋))) = (((1st𝑋) mod (2nd𝑋)) gcd (2nd𝑋)))
1911, 17, 18syl2anc 411 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ((2nd𝑋) gcd ((1st𝑋) mod (2nd𝑋))) = (((1st𝑋) mod (2nd𝑋)) gcd (2nd𝑋)))
20 modgcd 12010 . . . . . 6 (((1st𝑋) ∈ ℤ ∧ (2nd𝑋) ∈ ℕ) → (((1st𝑋) mod (2nd𝑋)) gcd (2nd𝑋)) = ((1st𝑋) gcd (2nd𝑋)))
2114, 20sylancom 420 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → (((1st𝑋) mod (2nd𝑋)) gcd (2nd𝑋)) = ((1st𝑋) gcd (2nd𝑋)))
229, 19, 213eqtrd 2226 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ((2nd𝑋) gcd ( mod ‘𝑋)) = ((1st𝑋) gcd (2nd𝑋)))
23 nnne0 8965 . . . . . . . . 9 ((2nd𝑋) ∈ ℕ → (2nd𝑋) ≠ 0)
2423adantl 277 . . . . . . . 8 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → (2nd𝑋) ≠ 0)
2524neneqd 2381 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ¬ (2nd𝑋) = 0)
2625iffalsed 3559 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) = ⟨(2nd𝑋), ( mod ‘𝑋)⟩)
2726fveq2d 5534 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘⟨(2nd𝑋), ( mod ‘𝑋)⟩))
28 df-ov 5894 . . . . 5 ((2nd𝑋) gcd ( mod ‘𝑋)) = ( gcd ‘⟨(2nd𝑋), ( mod ‘𝑋)⟩)
2927, 28eqtr4di 2240 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)) = ((2nd𝑋) gcd ( mod ‘𝑋)))
305fveq2d 5534 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( gcd ‘𝑋) = ( gcd ‘⟨(1st𝑋), (2nd𝑋)⟩))
31 df-ov 5894 . . . . 5 ((1st𝑋) gcd (2nd𝑋)) = ( gcd ‘⟨(1st𝑋), (2nd𝑋)⟩)
3230, 31eqtr4di 2240 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( gcd ‘𝑋) = ((1st𝑋) gcd (2nd𝑋)))
3322, 29, 323eqtr4d 2232 . . 3 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) ∈ ℕ) → ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
34 iftrue 3554 . . . . 5 ((2nd𝑋) = 0 → if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) = 𝑋)
3534fveq2d 5534 . . . 4 ((2nd𝑋) = 0 → ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
3635adantl 277 . . 3 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd𝑋) = 0) → ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
37 xp2nd 6185 . . . 4 (𝑋 ∈ (ℕ0 × ℕ0) → (2nd𝑋) ∈ ℕ0)
38 elnn0 9196 . . . 4 ((2nd𝑋) ∈ ℕ0 ↔ ((2nd𝑋) ∈ ℕ ∨ (2nd𝑋) = 0))
3937, 38sylib 122 . . 3 (𝑋 ∈ (ℕ0 × ℕ0) → ((2nd𝑋) ∈ ℕ ∨ (2nd𝑋) = 0))
4033, 36, 39mpjaodan 799 . 2 (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
413, 40eqtrd 2222 1 (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸𝑋)) = ( gcd ‘𝑋))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 709   = wceq 1364  wcel 2160  wne 2360  ifcif 3549  cop 3610   × cxp 4639  cfv 5231  (class class class)co 5891  cmpo 5893  1st c1st 6157  2nd c2nd 6158  0cc0 7829  cn 8937  0cn0 9194  cz 9271   mod cmo 10340   gcd cgcd 11961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-mulrcl 7928  ax-addcom 7929  ax-mulcom 7930  ax-addass 7931  ax-mulass 7932  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-1rid 7936  ax-0id 7937  ax-rnegex 7938  ax-precex 7939  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-apti 7944  ax-pre-ltadd 7945  ax-pre-mulgt0 7946  ax-pre-mulext 7947  ax-arch 7948  ax-caucvg 7949
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-frec 6410  df-sup 7001  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-reap 8550  df-ap 8557  df-div 8648  df-inn 8938  df-2 8996  df-3 8997  df-4 8998  df-n0 9195  df-z 9272  df-uz 9547  df-q 9638  df-rp 9672  df-fz 10027  df-fzo 10161  df-fl 10288  df-mod 10341  df-seqfrec 10464  df-exp 10538  df-cj 10869  df-re 10870  df-im 10871  df-rsqrt 11025  df-abs 11026  df-dvds 11813  df-gcd 11962
This theorem is referenced by:  eucalg  12077
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