Theorem eucalginv 16759

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 16757	. . 3 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸‘𝑋) = if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩))
3	2	fveq2d 6889	. 2 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸‘𝑋)) = ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)))
4		1st2nd2 8040	. . . . . . . . 9 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
5	4	adantr 486	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
6	5	fveq2d 6889	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( mod ‘𝑋) = ( mod ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
7		df-ov 7423	. . . . . . 7 ⊢ ((1st ‘𝑋) mod (2nd ‘𝑋)) = ( mod ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
8	6, 7	eqtr4di 2813	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( mod ‘𝑋) = ((1st ‘𝑋) mod (2nd ‘𝑋)))
9	8	oveq2d 7436	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((2nd ‘𝑋) gcd ( mod ‘𝑋)) = ((2nd ‘𝑋) gcd ((1st ‘𝑋) mod (2nd ‘𝑋))))
10		nnz 12714	. . . . . 6 ⊢ ((2nd ‘𝑋) ∈ ℕ → (2nd ‘𝑋) ∈ ℤ)
11		xp1st 8033	. . . . . . . . . 10 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (1st ‘𝑋) ∈ ℕ0)
12	11	adantr 486	. . . . . . . . 9 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → (1st ‘𝑋) ∈ ℕ0)
13	12	nn0zd 12718	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → (1st ‘𝑋) ∈ ℤ)
14		zmodcl 14031	. . . . . . . 8 ⊢ (((1st ‘𝑋) ∈ ℤ ∧ (2nd ‘𝑋) ∈ ℕ) → ((1st ‘𝑋) mod (2nd ‘𝑋)) ∈ ℕ0)
15	13, 14	sylancom 600	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((1st ‘𝑋) mod (2nd ‘𝑋)) ∈ ℕ0)
16	15	nn0zd 12718	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((1st ‘𝑋) mod (2nd ‘𝑋)) ∈ ℤ)
17		gcdcom 16685	. . . . . 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 699	. . . . 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 16705	. . . . . 6 ⊢ (((1st ‘𝑋) ∈ ℤ ∧ (2nd ‘𝑋) ∈ ℕ) → (((1st ‘𝑋) mod (2nd ‘𝑋)) gcd (2nd ‘𝑋)) = ((1st ‘𝑋) gcd (2nd ‘𝑋)))
20	13, 19	sylancom 600	. . . . 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 2799	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((2nd ‘𝑋) gcd ( mod ‘𝑋)) = ((1st ‘𝑋) gcd (2nd ‘𝑋)))
22		nnne0 12372	. . . . . . . . 9 ⊢ ((2nd ‘𝑋) ∈ ℕ → (2nd ‘𝑋) ≠ 0)
23	22	adantl 487	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → (2nd ‘𝑋) ≠ 0)
24	23	neneqd 2960	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ¬ (2nd ‘𝑋) = 0)
25	24	iffalsed 4492	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) = ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)
26	25	fveq2d 6889	. . . . 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 7423	. . . . 5 ⊢ ((2nd ‘𝑋) gcd ( mod ‘𝑋)) = ( gcd ‘⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)
28	26, 27	eqtr4di 2813	. . . 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 6889	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( gcd ‘𝑋) = ( gcd ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
30		df-ov 7423	. . . . 5 ⊢ ((1st ‘𝑋) gcd (2nd ‘𝑋)) = ( gcd ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
31	29, 30	eqtr4di 2813	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( gcd ‘𝑋) = ((1st ‘𝑋) gcd (2nd ‘𝑋)))
32	21, 28, 31	3eqtr4d 2805	. . 3 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
33		iftrue 4487	. . . . 5 ⊢ ((2nd ‘𝑋) = 0 → if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) = 𝑋)
34	33	fveq2d 6889	. . . 4 ⊢ ((2nd ‘𝑋) = 0 → ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
35	34	adantl 487	. . 3 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) = 0) → ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
36		xp2nd 8034	. . . 4 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑋) ∈ ℕ0)
37		elnn0 12608	. . . 4 ⊢ ((2nd ‘𝑋) ∈ ℕ0 ↔ ((2nd ‘𝑋) ∈ ℕ ∨ (2nd ‘𝑋) = 0))
38	36, 37	sylib 221	. . 3 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → ((2nd ‘𝑋) ∈ ℕ ∨ (2nd ‘𝑋) = 0))
39	32, 35, 38	mpjaodan 973	. 2 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
40	3, 39	eqtrd 2795	1 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸‘𝑋)) = ( gcd ‘𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ifcif 4481  ⟨cop 4589   × cxp 5648  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000  0cc0 11200  ℕcn 12335  ℕ₀cn0 12606  ℤcz 12693   mod cmo 14009   gcd cgcd 16664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-sup 9434  df-inf 9435  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-n0 12607  df-z 12694  df-uz 12966  df-rp 13121  df-fl 13932  df-mod 14010  df-seq 14145  df-exp 14205  df-cj 15266  df-re 15267  df-im 15268  df-sqrt 15402  df-abs 15403  df-dvds 16423  df-gcd 16665

This theorem is used by:  eucalg  16762