Theorem eucalginv 12010

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 12008	. . 3 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸‘𝑋) = if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩))
3	2	fveq2d 5500	. 2 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸‘𝑋)) = ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)))
4		1st2nd2 6154	. . . . . . . . 9 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
5	4	adantr 274	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
6	5	fveq2d 5500	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( mod ‘𝑋) = ( mod ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
7		df-ov 5856	. . . . . . 7 ⊢ ((1st ‘𝑋) mod (2nd ‘𝑋)) = ( mod ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
8	6, 7	eqtr4di 2221	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( mod ‘𝑋) = ((1st ‘𝑋) mod (2nd ‘𝑋)))
9	8	oveq2d 5869	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((2nd ‘𝑋) gcd ( mod ‘𝑋)) = ((2nd ‘𝑋) gcd ((1st ‘𝑋) mod (2nd ‘𝑋))))
10		nnz 9231	. . . . . . 7 ⊢ ((2nd ‘𝑋) ∈ ℕ → (2nd ‘𝑋) ∈ ℤ)
11	10	adantl 275	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → (2nd ‘𝑋) ∈ ℤ)
12		xp1st 6144	. . . . . . . . . 10 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (1st ‘𝑋) ∈ ℕ0)
13	12	adantr 274	. . . . . . . . 9 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → (1st ‘𝑋) ∈ ℕ0)
14	13	nn0zd 9332	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → (1st ‘𝑋) ∈ ℤ)
15		zmodcl 10300	. . . . . . . 8 ⊢ (((1st ‘𝑋) ∈ ℤ ∧ (2nd ‘𝑋) ∈ ℕ) → ((1st ‘𝑋) mod (2nd ‘𝑋)) ∈ ℕ0)
16	14, 15	sylancom 418	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((1st ‘𝑋) mod (2nd ‘𝑋)) ∈ ℕ0)
17	16	nn0zd 9332	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((1st ‘𝑋) mod (2nd ‘𝑋)) ∈ ℤ)
18		gcdcom 11928	. . . . . 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)))
19	11, 17, 18	syl2anc 409	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((2nd ‘𝑋) gcd ((1st ‘𝑋) mod (2nd ‘𝑋))) = (((1st ‘𝑋) mod (2nd ‘𝑋)) gcd (2nd ‘𝑋)))
20		modgcd 11946	. . . . . 6 ⊢ (((1st ‘𝑋) ∈ ℤ ∧ (2nd ‘𝑋) ∈ ℕ) → (((1st ‘𝑋) mod (2nd ‘𝑋)) gcd (2nd ‘𝑋)) = ((1st ‘𝑋) gcd (2nd ‘𝑋)))
21	14, 20	sylancom 418	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → (((1st ‘𝑋) mod (2nd ‘𝑋)) gcd (2nd ‘𝑋)) = ((1st ‘𝑋) gcd (2nd ‘𝑋)))
22	9, 19, 21	3eqtrd 2207	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((2nd ‘𝑋) gcd ( mod ‘𝑋)) = ((1st ‘𝑋) gcd (2nd ‘𝑋)))
23		nnne0 8906	. . . . . . . . 9 ⊢ ((2nd ‘𝑋) ∈ ℕ → (2nd ‘𝑋) ≠ 0)
24	23	adantl 275	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → (2nd ‘𝑋) ≠ 0)
25	24	neneqd 2361	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ¬ (2nd ‘𝑋) = 0)
26	25	iffalsed 3536	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) = ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)
27	26	fveq2d 5500	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩))
28		df-ov 5856	. . . . 5 ⊢ ((2nd ‘𝑋) gcd ( mod ‘𝑋)) = ( gcd ‘⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)
29	27, 28	eqtr4di 2221	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)) = ((2nd ‘𝑋) gcd ( mod ‘𝑋)))
30	5	fveq2d 5500	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( gcd ‘𝑋) = ( gcd ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
31		df-ov 5856	. . . . 5 ⊢ ((1st ‘𝑋) gcd (2nd ‘𝑋)) = ( gcd ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
32	30, 31	eqtr4di 2221	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( gcd ‘𝑋) = ((1st ‘𝑋) gcd (2nd ‘𝑋)))
33	22, 29, 32	3eqtr4d 2213	. . 3 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
34		iftrue 3531	. . . . 5 ⊢ ((2nd ‘𝑋) = 0 → if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) = 𝑋)
35	34	fveq2d 5500	. . . 4 ⊢ ((2nd ‘𝑋) = 0 → ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
36	35	adantl 275	. . 3 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) = 0) → ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
37		xp2nd 6145	. . . 4 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑋) ∈ ℕ0)
38		elnn0 9137	. . . 4 ⊢ ((2nd ‘𝑋) ∈ ℕ0 ↔ ((2nd ‘𝑋) ∈ ℕ ∨ (2nd ‘𝑋) = 0))
39	37, 38	sylib 121	. . 3 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → ((2nd ‘𝑋) ∈ ℕ ∨ (2nd ‘𝑋) = 0))
40	33, 36, 39	mpjaodan 793	. 2 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
41	3, 40	eqtrd 2203	1 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸‘𝑋)) = ( gcd ‘𝑋))

Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 703   = wceq 1348   ∈ wcel 2141   ≠ wne 2340  ifcif 3526  ⟨cop 3586   × cxp 4609  ‘cfv 5198  (class class class)co 5853   ∈ cmpo 5855  1^st c1st 6117  2^nd c2nd 6118  0cc0 7774  ℕcn 8878  ℕ₀cn0 9135  ℤcz 9212   mod cmo 10278   gcd cgcd 11897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-gcd 11898

This theorem is referenced by:  eucalg  12013