Theorem eucalginv 16642

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 16640	. . 3 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸‘𝑋) = if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩))
3	2	fveq2d 6886	. 2 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸‘𝑋)) = ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)))
4		1st2nd2 8025	. . . . . . . . 9 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
5	4	adantr 485	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
6	5	fveq2d 6886	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( mod ‘𝑋) = ( mod ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
7		df-ov 7414	. . . . . . 7 ⊢ ((1st ‘𝑋) mod (2nd ‘𝑋)) = ( mod ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
8	6, 7	eqtr4di 2822	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( mod ‘𝑋) = ((1st ‘𝑋) mod (2nd ‘𝑋)))
9	8	oveq2d 7427	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((2nd ‘𝑋) gcd ( mod ‘𝑋)) = ((2nd ‘𝑋) gcd ((1st ‘𝑋) mod (2nd ‘𝑋))))
10		nnz 12612	. . . . . 6 ⊢ ((2nd ‘𝑋) ∈ ℕ → (2nd ‘𝑋) ∈ ℤ)
11		xp1st 8018	. . . . . . . . . 10 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (1st ‘𝑋) ∈ ℕ0)
12	11	adantr 485	. . . . . . . . 9 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → (1st ‘𝑋) ∈ ℕ0)
13	12	nn0zd 12616	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → (1st ‘𝑋) ∈ ℤ)
14		zmodcl 13924	. . . . . . . 8 ⊢ (((1st ‘𝑋) ∈ ℤ ∧ (2nd ‘𝑋) ∈ ℕ) → ((1st ‘𝑋) mod (2nd ‘𝑋)) ∈ ℕ0)
15	13, 14	sylancom 599	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((1st ‘𝑋) mod (2nd ‘𝑋)) ∈ ℕ0)
16	15	nn0zd 12616	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((1st ‘𝑋) mod (2nd ‘𝑋)) ∈ ℤ)
17		gcdcom 16571	. . . . . 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 698	. . . . 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 16590	. . . . . 6 ⊢ (((1st ‘𝑋) ∈ ℤ ∧ (2nd ‘𝑋) ∈ ℕ) → (((1st ‘𝑋) mod (2nd ‘𝑋)) gcd (2nd ‘𝑋)) = ((1st ‘𝑋) gcd (2nd ‘𝑋)))
20	13, 19	sylancom 599	. . . . 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 2808	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((2nd ‘𝑋) gcd ( mod ‘𝑋)) = ((1st ‘𝑋) gcd (2nd ‘𝑋)))
22		nnne0 12270	. . . . . . . . 9 ⊢ ((2nd ‘𝑋) ∈ ℕ → (2nd ‘𝑋) ≠ 0)
23	22	adantl 486	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → (2nd ‘𝑋) ≠ 0)
24	23	neneqd 2969	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ¬ (2nd ‘𝑋) = 0)
25	24	iffalsed 4503	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) = ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)
26	25	fveq2d 6886	. . . . 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 7414	. . . . 5 ⊢ ((2nd ‘𝑋) gcd ( mod ‘𝑋)) = ( gcd ‘⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)
28	26, 27	eqtr4di 2822	. . . 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 6886	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( gcd ‘𝑋) = ( gcd ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
30		df-ov 7414	. . . . 5 ⊢ ((1st ‘𝑋) gcd (2nd ‘𝑋)) = ( gcd ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
31	29, 30	eqtr4di 2822	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( gcd ‘𝑋) = ((1st ‘𝑋) gcd (2nd ‘𝑋)))
32	21, 28, 31	3eqtr4d 2814	. . 3 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
33		iftrue 4498	. . . . 5 ⊢ ((2nd ‘𝑋) = 0 → if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) = 𝑋)
34	33	fveq2d 6886	. . . 4 ⊢ ((2nd ‘𝑋) = 0 → ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
35	34	adantl 486	. . 3 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) = 0) → ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
36		xp2nd 8019	. . . 4 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑋) ∈ ℕ0)
37		elnn0 12506	. . . 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 2804	1 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸‘𝑋)) = ( gcd ‘𝑋))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ifcif 4492  ⟨cop 4600   × cxp 5660  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  1^st c1st 7984  2^nd c2nd 7985  0cc0 11100  ℕcn 12233  ℕ₀cn0 12504  ℤcz 12591   mod cmo 13902   gcd cgcd 16552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177  ax-pre-sup 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-sup 9402  df-inf 9403  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-div 11872  df-nn 12234  df-2 12303  df-3 12304  df-n0 12505  df-z 12592  df-uz 12863  df-rp 13017  df-fl 13825  df-mod 13903  df-seq 14038  df-exp 14098  df-cj 15150  df-re 15151  df-im 15152  df-sqrt 15286  df-abs 15287  df-dvds 16311  df-gcd 16553

This theorem is referenced by:  eucalg  16645