Theorem eucalglt 16522

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 16519	. . . . . . . 8 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸‘𝑋) = if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩))
3	2	adantr 482	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (𝐸‘𝑋) = if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩))
4		simpr 486	. . . . . . . . 9 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘(𝐸‘𝑋)) ≠ 0)
5		iftrue 4535	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑋) = 0 → if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) = 𝑋)
6	5	eqeq2d 2744	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝑋) = 0 → ((𝐸‘𝑋) = if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) ↔ (𝐸‘𝑋) = 𝑋))
7		fveq2 6892	. . . . . . . . . . . . 13 ⊢ ((𝐸‘𝑋) = 𝑋 → (2nd ‘(𝐸‘𝑋)) = (2nd ‘𝑋))
8	6, 7	syl6bi 253	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑋) = 0 → ((𝐸‘𝑋) = if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) → (2nd ‘(𝐸‘𝑋)) = (2nd ‘𝑋)))
9		eqeq2 2745	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑋) = 0 → ((2nd ‘(𝐸‘𝑋)) = (2nd ‘𝑋) ↔ (2nd ‘(𝐸‘𝑋)) = 0))
10	8, 9	sylibd 238	. . . . . . . . . . 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 2954	. . . . . . . . 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 4540	. . . . . . 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 2773	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (𝐸‘𝑋) = ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)
16	15	fveq2d 6896	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘(𝐸‘𝑋)) = (2nd ‘⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩))
17		fvex 6905	. . . . . 6 ⊢ (2nd ‘𝑋) ∈ V
18		fvex 6905	. . . . . 6 ⊢ ( mod ‘𝑋) ∈ V
19	17, 18	op2nd 7984	. . . . 5 ⊢ (2nd ‘⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) = ( mod ‘𝑋)
20	16, 19	eqtrdi 2789	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘(𝐸‘𝑋)) = ( mod ‘𝑋))
21		1st2nd2 8014	. . . . . . 7 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
22	21	adantr 482	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
23	22	fveq2d 6896	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ( mod ‘𝑋) = ( mod ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
24		df-ov 7412	. . . . 5 ⊢ ((1st ‘𝑋) mod (2nd ‘𝑋)) = ( mod ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
25	23, 24	eqtr4di 2791	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ( mod ‘𝑋) = ((1st ‘𝑋) mod (2nd ‘𝑋)))
26	20, 25	eqtrd 2773	. . 3 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘(𝐸‘𝑋)) = ((1st ‘𝑋) mod (2nd ‘𝑋)))
27		xp1st 8007	. . . . . 6 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (1st ‘𝑋) ∈ ℕ0)
28	27	adantr 482	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (1st ‘𝑋) ∈ ℕ0)
29	28	nn0red 12533	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (1st ‘𝑋) ∈ ℝ)
30		xp2nd 8008	. . . . . . . . 9 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑋) ∈ ℕ0)
31	30	adantr 482	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘𝑋) ∈ ℕ0)
32		elnn0 12474	. . . . . . . 8 ⊢ ((2nd ‘𝑋) ∈ ℕ0 ↔ ((2nd ‘𝑋) ∈ ℕ ∨ (2nd ‘𝑋) = 0))
33	31, 32	sylib 217	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ((2nd ‘𝑋) ∈ ℕ ∨ (2nd ‘𝑋) = 0))
34	33	ord 863	. . . . . 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 13014	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘𝑋) ∈ ℝ+)
37		modlt 13845	. . . 4 ⊢ (((1st ‘𝑋) ∈ ℝ ∧ (2nd ‘𝑋) ∈ ℝ+) → ((1st ‘𝑋) mod (2nd ‘𝑋)) < (2nd ‘𝑋))
38	29, 36, 37	syl2anc 585	. . 3 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ((1st ‘𝑋) mod (2nd ‘𝑋)) < (2nd ‘𝑋))
39	26, 38	eqbrtrd 5171	. 2 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘(𝐸‘𝑋)) < (2nd ‘𝑋))
40	39	ex 414	1 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → ((2nd ‘(𝐸‘𝑋)) ≠ 0 → (2nd ‘(𝐸‘𝑋)) < (2nd ‘𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ifcif 4529  ⟨cop 4635   class class class wbr 5149   × cxp 5675  ‘cfv 6544  (class class class)co 7409   ∈ cmpo 7411  1^st c1st 7973  2^nd c2nd 7974  ℝcr 11109  0cc0 11110   < clt 11248  ℕcn 12212  ℕ₀cn0 12472  ℝ⁺crp 12974   mod cmo 13834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-rp 12975  df-fl 13757  df-mod 13835

This theorem is referenced by:  eucalgcvga  16523