Theorem eucalginv 16217

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 ‘(𝐸‘𝑋)) = ( gcd ‘𝑋))

Distinct variable group:   𝑥,𝑦,𝑋

Allowed substitution hints:   𝐸(𝑥,𝑦)

Proof of Theorem eucalginv

Step	Hyp	Ref	Expression
1		eucalgval.1	. . . 4 ⊢ 𝐸 = (𝑥 ∈ ℕ0, 𝑦 ∈ ℕ0 ↦ if(𝑦 = 0, ⟨𝑥, 𝑦⟩, ⟨𝑦, (𝑥 mod 𝑦)⟩))
2	1	eucalgval 16215	. . 3 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸‘𝑋) = if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩))
3	2	fveq2d 6760	. 2 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸‘𝑋)) = ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)))
4		1st2nd2 7843	. . . . . . . . 9 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
5	4	adantr 480	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
6	5	fveq2d 6760	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( mod ‘𝑋) = ( mod ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
7		df-ov 7258	. . . . . . 7 ⊢ ((1st ‘𝑋) mod (2nd ‘𝑋)) = ( mod ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
8	6, 7	eqtr4di 2797	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( mod ‘𝑋) = ((1st ‘𝑋) mod (2nd ‘𝑋)))
9	8	oveq2d 7271	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((2nd ‘𝑋) gcd ( mod ‘𝑋)) = ((2nd ‘𝑋) gcd ((1st ‘𝑋) mod (2nd ‘𝑋))))
10		nnz 12272	. . . . . 6 ⊢ ((2nd ‘𝑋) ∈ ℕ → (2nd ‘𝑋) ∈ ℤ)
11		xp1st 7836	. . . . . . . . . 10 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (1st ‘𝑋) ∈ ℕ0)
12	11	adantr 480	. . . . . . . . 9 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → (1st ‘𝑋) ∈ ℕ0)
13	12	nn0zd 12353	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → (1st ‘𝑋) ∈ ℤ)
14		zmodcl 13539	. . . . . . . 8 ⊢ (((1st ‘𝑋) ∈ ℤ ∧ (2nd ‘𝑋) ∈ ℕ) → ((1st ‘𝑋) mod (2nd ‘𝑋)) ∈ ℕ0)
15	13, 14	sylancom 587	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((1st ‘𝑋) mod (2nd ‘𝑋)) ∈ ℕ0)
16	15	nn0zd 12353	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((1st ‘𝑋) mod (2nd ‘𝑋)) ∈ ℤ)
17		gcdcom 16148	. . . . . 6 ⊢ (((2nd ‘𝑋) ∈ ℤ ∧ ((1st ‘𝑋) mod (2nd ‘𝑋)) ∈ ℤ) → ((2nd ‘𝑋) gcd ((1st ‘𝑋) mod (2nd ‘𝑋))) = (((1st ‘𝑋) mod (2nd ‘𝑋)) gcd (2nd ‘𝑋)))
18	10, 16, 17	syl2an2 682	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((2nd ‘𝑋) gcd ((1st ‘𝑋) mod (2nd ‘𝑋))) = (((1st ‘𝑋) mod (2nd ‘𝑋)) gcd (2nd ‘𝑋)))
19		modgcd 16168	. . . . . 6 ⊢ (((1st ‘𝑋) ∈ ℤ ∧ (2nd ‘𝑋) ∈ ℕ) → (((1st ‘𝑋) mod (2nd ‘𝑋)) gcd (2nd ‘𝑋)) = ((1st ‘𝑋) gcd (2nd ‘𝑋)))
20	13, 19	sylancom 587	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → (((1st ‘𝑋) mod (2nd ‘𝑋)) gcd (2nd ‘𝑋)) = ((1st ‘𝑋) gcd (2nd ‘𝑋)))
21	9, 18, 20	3eqtrd 2782	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((2nd ‘𝑋) gcd ( mod ‘𝑋)) = ((1st ‘𝑋) gcd (2nd ‘𝑋)))
22		nnne0 11937	. . . . . . . . 9 ⊢ ((2nd ‘𝑋) ∈ ℕ → (2nd ‘𝑋) ≠ 0)
23	22	adantl 481	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → (2nd ‘𝑋) ≠ 0)
24	23	neneqd 2947	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ¬ (2nd ‘𝑋) = 0)
25	24	iffalsed 4467	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) = ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)
26	25	fveq2d 6760	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩))
27		df-ov 7258	. . . . 5 ⊢ ((2nd ‘𝑋) gcd ( mod ‘𝑋)) = ( gcd ‘⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)
28	26, 27	eqtr4di 2797	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)) = ((2nd ‘𝑋) gcd ( mod ‘𝑋)))
29	5	fveq2d 6760	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( gcd ‘𝑋) = ( gcd ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
30		df-ov 7258	. . . . 5 ⊢ ((1st ‘𝑋) gcd (2nd ‘𝑋)) = ( gcd ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
31	29, 30	eqtr4di 2797	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( gcd ‘𝑋) = ((1st ‘𝑋) gcd (2nd ‘𝑋)))
32	21, 28, 31	3eqtr4d 2788	. . 3 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
33		iftrue 4462	. . . . 5 ⊢ ((2nd ‘𝑋) = 0 → if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) = 𝑋)
34	33	fveq2d 6760	. . . 4 ⊢ ((2nd ‘𝑋) = 0 → ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
35	34	adantl 481	. . 3 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) = 0) → ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
36		xp2nd 7837	. . . 4 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑋) ∈ ℕ0)
37		elnn0 12165	. . . 4 ⊢ ((2nd ‘𝑋) ∈ ℕ0 ↔ ((2nd ‘𝑋) ∈ ℕ ∨ (2nd ‘𝑋) = 0))
38	36, 37	sylib 217	. . 3 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → ((2nd ‘𝑋) ∈ ℕ ∨ (2nd ‘𝑋) = 0))
39	32, 35, 38	mpjaodan 955	. 2 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
40	3, 39	eqtrd 2778	1 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸‘𝑋)) = ( gcd ‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ifcif 4456  ⟨cop 4564   × cxp 5578  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  1^st c1st 7802  2^nd c2nd 7803  0cc0 10802  ℕcn 11903  ℕ₀cn0 12163  ℤcz 12249   mod cmo 13517   gcd cgcd 16129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-sup 9131  df-inf 9132  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-rp 12660  df-fl 13440  df-mod 13518  df-seq 13650  df-exp 13711  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-dvds 15892  df-gcd 16130

This theorem is referenced by:  eucalg  16220