Theorem eucalglt 16609

Description: The second member of the state decreases with each iteration of the step function 𝐸 for Euclid's Algorithm. (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
eucalglt	⊢ (𝑋 ∈ (ℕ0 × ℕ0) → ((2nd ‘(𝐸‘𝑋)) ≠ 0 → (2nd ‘(𝐸‘𝑋)) < (2nd ‘𝑋)))

Distinct variable group:   𝑥,𝑦,𝑋

Allowed substitution hints:   𝐸(𝑥,𝑦)

Proof of Theorem eucalglt

Step	Hyp	Ref	Expression
1		eucalgval.1	. . . . . . . . 9 ⊢ 𝐸 = (𝑥 ∈ ℕ0, 𝑦 ∈ ℕ0 ↦ if(𝑦 = 0, ⟨𝑥, 𝑦⟩, ⟨𝑦, (𝑥 mod 𝑦)⟩))
2	1	eucalgval 16606	. . . . . . . 8 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸‘𝑋) = if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩))
3	2	adantr 480	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (𝐸‘𝑋) = if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩))
4		simpr 484	. . . . . . . . 9 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘(𝐸‘𝑋)) ≠ 0)
5		iftrue 4511	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑋) = 0 → if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) = 𝑋)
6	5	eqeq2d 2747	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝑋) = 0 → ((𝐸‘𝑋) = if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) ↔ (𝐸‘𝑋) = 𝑋))
7		fveq2 6881	. . . . . . . . . . . . 13 ⊢ ((𝐸‘𝑋) = 𝑋 → (2nd ‘(𝐸‘𝑋)) = (2nd ‘𝑋))
8	6, 7	biimtrdi 253	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑋) = 0 → ((𝐸‘𝑋) = if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) → (2nd ‘(𝐸‘𝑋)) = (2nd ‘𝑋)))
9		eqeq2 2748	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑋) = 0 → ((2nd ‘(𝐸‘𝑋)) = (2nd ‘𝑋) ↔ (2nd ‘(𝐸‘𝑋)) = 0))
10	8, 9	sylibd 239	. . . . . . . . . . 11 ⊢ ((2nd ‘𝑋) = 0 → ((𝐸‘𝑋) = if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) → (2nd ‘(𝐸‘𝑋)) = 0))
11	3, 10	syl5com 31	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ((2nd ‘𝑋) = 0 → (2nd ‘(𝐸‘𝑋)) = 0))
12	11	necon3ad 2946	. . . . . . . . 9 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ((2nd ‘(𝐸‘𝑋)) ≠ 0 → ¬ (2nd ‘𝑋) = 0))
13	4, 12	mpd 15	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ¬ (2nd ‘𝑋) = 0)
14	13	iffalsed 4516	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) = ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)
15	3, 14	eqtrd 2771	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (𝐸‘𝑋) = ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)
16	15	fveq2d 6885	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘(𝐸‘𝑋)) = (2nd ‘⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩))
17		fvex 6894	. . . . . 6 ⊢ (2nd ‘𝑋) ∈ V
18		fvex 6894	. . . . . 6 ⊢ ( mod ‘𝑋) ∈ V
19	17, 18	op2nd 8002	. . . . 5 ⊢ (2nd ‘⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) = ( mod ‘𝑋)
20	16, 19	eqtrdi 2787	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘(𝐸‘𝑋)) = ( mod ‘𝑋))
21		1st2nd2 8032	. . . . . . 7 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
22	21	adantr 480	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
23	22	fveq2d 6885	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ( mod ‘𝑋) = ( mod ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
24		df-ov 7413	. . . . 5 ⊢ ((1st ‘𝑋) mod (2nd ‘𝑋)) = ( mod ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
25	23, 24	eqtr4di 2789	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ( mod ‘𝑋) = ((1st ‘𝑋) mod (2nd ‘𝑋)))
26	20, 25	eqtrd 2771	. . 3 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘(𝐸‘𝑋)) = ((1st ‘𝑋) mod (2nd ‘𝑋)))
27		xp1st 8025	. . . . . 6 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (1st ‘𝑋) ∈ ℕ0)
28	27	adantr 480	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (1st ‘𝑋) ∈ ℕ0)
29	28	nn0red 12568	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (1st ‘𝑋) ∈ ℝ)
30		xp2nd 8026	. . . . . . . . 9 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑋) ∈ ℕ0)
31	30	adantr 480	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘𝑋) ∈ ℕ0)
32		elnn0 12508	. . . . . . . 8 ⊢ ((2nd ‘𝑋) ∈ ℕ0 ↔ ((2nd ‘𝑋) ∈ ℕ ∨ (2nd ‘𝑋) = 0))
33	31, 32	sylib 218	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ((2nd ‘𝑋) ∈ ℕ ∨ (2nd ‘𝑋) = 0))
34	33	ord 864	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (¬ (2nd ‘𝑋) ∈ ℕ → (2nd ‘𝑋) = 0))
35	13, 34	mt3d 148	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘𝑋) ∈ ℕ)
36	35	nnrpd 13054	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘𝑋) ∈ ℝ+)
37		modlt 13902	. . . 4 ⊢ (((1st ‘𝑋) ∈ ℝ ∧ (2nd ‘𝑋) ∈ ℝ+) → ((1st ‘𝑋) mod (2nd ‘𝑋)) < (2nd ‘𝑋))
38	29, 36, 37	syl2anc 584	. . 3 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ((1st ‘𝑋) mod (2nd ‘𝑋)) < (2nd ‘𝑋))
39	26, 38	eqbrtrd 5146	. 2 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘(𝐸‘𝑋)) < (2nd ‘𝑋))
40	39	ex 412	1 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → ((2nd ‘(𝐸‘𝑋)) ≠ 0 → (2nd ‘(𝐸‘𝑋)) < (2nd ‘𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ifcif 4505  ⟨cop 4612   class class class wbr 5124   × cxp 5657  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1^st c1st 7991  2^nd c2nd 7992  ℝcr 11133  0cc0 11134   < clt 11274  ℕcn 12245  ℕ₀cn0 12506  ℝ⁺crp 13013   mod cmo 13891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211  ax-pre-sup 11212

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  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 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-sup 9459  df-inf 9460  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-div 11900  df-nn 12246  df-n0 12507  df-z 12594  df-uz 12858  df-rp 13014  df-fl 13814  df-mod 13892

This theorem is referenced by:  eucalgcvga  16610