Theorem eucalginv 16646

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 16644	. . 3 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸‘𝑋) = if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩))
3	2	fveq2d 6885	. 2 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸‘𝑋)) = ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)))
4		1st2nd2 8021	. . . . . . . . 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 6885	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( mod ‘𝑋) = ( mod ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
7		df-ov 7413	. . . . . . 7 ⊢ ((1st ‘𝑋) mod (2nd ‘𝑋)) = ( mod ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
8	6, 7	eqtr4di 2816	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( mod ‘𝑋) = ((1st ‘𝑋) mod (2nd ‘𝑋)))
9	8	oveq2d 7426	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((2nd ‘𝑋) gcd ( mod ‘𝑋)) = ((2nd ‘𝑋) gcd ((1st ‘𝑋) mod (2nd ‘𝑋))))
10		nnz 12616	. . . . . 6 ⊢ ((2nd ‘𝑋) ∈ ℕ → (2nd ‘𝑋) ∈ ℤ)
11		xp1st 8014	. . . . . . . . . 10 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (1st ‘𝑋) ∈ ℕ0)
12	11	adantr 485	. . . . . . . . 9 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → (1st ‘𝑋) ∈ ℕ0)
13	12	nn0zd 12620	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → (1st ‘𝑋) ∈ ℤ)
14		zmodcl 13929	. . . . . . . 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 12620	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((1st ‘𝑋) mod (2nd ‘𝑋)) ∈ ℤ)
17		gcdcom 16575	. . . . . 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 16594	. . . . . 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 2802	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ((2nd ‘𝑋) gcd ( mod ‘𝑋)) = ((1st ‘𝑋) gcd (2nd ‘𝑋)))
22		nnne0 12274	. . . . . . . . 9 ⊢ ((2nd ‘𝑋) ∈ ℕ → (2nd ‘𝑋) ≠ 0)
23	22	adantl 486	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → (2nd ‘𝑋) ≠ 0)
24	23	neneqd 2963	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ¬ (2nd ‘𝑋) = 0)
25	24	iffalsed 4498	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) = ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)
26	25	fveq2d 6885	. . . . 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 7413	. . . . 5 ⊢ ((2nd ‘𝑋) gcd ( mod ‘𝑋)) = ( gcd ‘⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)
28	26, 27	eqtr4di 2816	. . . 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 6885	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( gcd ‘𝑋) = ( gcd ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
30		df-ov 7413	. . . . 5 ⊢ ((1st ‘𝑋) gcd (2nd ‘𝑋)) = ( gcd ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
31	29, 30	eqtr4di 2816	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( gcd ‘𝑋) = ((1st ‘𝑋) gcd (2nd ‘𝑋)))
32	21, 28, 31	3eqtr4d 2808	. . 3 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘𝑋) ∈ ℕ) → ( gcd ‘if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)) = ( gcd ‘𝑋))
33		iftrue 4493	. . . . 5 ⊢ ((2nd ‘𝑋) = 0 → if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) = 𝑋)
34	33	fveq2d 6885	. . . 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 8015	. . . 4 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑋) ∈ ℕ0)
37		elnn0 12510	. . . 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 2798	1 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → ( gcd ‘(𝐸‘𝑋)) = ( gcd ‘𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ifcif 4487  ⟨cop 4595   × cxp 5659  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1^st c1st 7980  2^nd c2nd 7981  0cc0 11104  ℕcn 12237  ℕ₀cn0 12508  ℤcz 12595   mod cmo 13907   gcd cgcd 16556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11160  ax-resscn 11161  ax-1cn 11162  ax-icn 11163  ax-addcl 11164  ax-addrcl 11165  ax-mulcl 11166  ax-mulrcl 11167  ax-mulcom 11168  ax-addass 11169  ax-mulass 11170  ax-distr 11171  ax-i2m1 11172  ax-1ne0 11173  ax-1rid 11174  ax-rnegex 11175  ax-rrecex 11176  ax-cnre 11177  ax-pre-lttri 11178  ax-pre-lttrn 11179  ax-pre-ltadd 11180  ax-pre-mulgt0 11181  ax-pre-sup 11182

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9398  df-inf 9399  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11447  df-neg 11448  df-div 11876  df-nn 12238  df-2 12307  df-3 12308  df-n0 12509  df-z 12596  df-uz 12867  df-rp 13021  df-fl 13830  df-mod 13908  df-seq 14043  df-exp 14103  df-cj 15155  df-re 15156  df-im 15157  df-sqrt 15291  df-abs 15292  df-dvds 16315  df-gcd 16557

This theorem is used by:  eucalg  16649