Theorem eucalglt 12634

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	. . . . . . . 8 ⊢ 𝐸 = (𝑥 ∈ ℕ0, 𝑦 ∈ ℕ0 ↦ if(𝑦 = 0, ⟨𝑥, 𝑦⟩, ⟨𝑦, (𝑥 mod 𝑦)⟩))
2	1	eucalgval 12631	. . . . . . 7 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸‘𝑋) = if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩))
3	2	adantr 276	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (𝐸‘𝑋) = if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩))
4		simpr 110	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘(𝐸‘𝑋)) ≠ 0)
5		iftrue 3610	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝑋) = 0 → if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) = 𝑋)
6	5	eqeq2d 2243	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑋) = 0 → ((𝐸‘𝑋) = if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) ↔ (𝐸‘𝑋) = 𝑋))
7		fveq2 5639	. . . . . . . . . . . 12 ⊢ ((𝐸‘𝑋) = 𝑋 → (2nd ‘(𝐸‘𝑋)) = (2nd ‘𝑋))
8	6, 7	biimtrdi 163	. . . . . . . . . . 11 ⊢ ((2nd ‘𝑋) = 0 → ((𝐸‘𝑋) = if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) → (2nd ‘(𝐸‘𝑋)) = (2nd ‘𝑋)))
9		eqeq2 2241	. . . . . . . . . . 11 ⊢ ((2nd ‘𝑋) = 0 → ((2nd ‘(𝐸‘𝑋)) = (2nd ‘𝑋) ↔ (2nd ‘(𝐸‘𝑋)) = 0))
10	8, 9	sylibd 149	. . . . . . . . . 10 ⊢ ((2nd ‘𝑋) = 0 → ((𝐸‘𝑋) = if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) → (2nd ‘(𝐸‘𝑋)) = 0))
11	3, 10	syl5com 29	. . . . . . . . 9 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ((2nd ‘𝑋) = 0 → (2nd ‘(𝐸‘𝑋)) = 0))
12	11	necon3ad 2444	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ((2nd ‘(𝐸‘𝑋)) ≠ 0 → ¬ (2nd ‘𝑋) = 0))
13	4, 12	mpd 13	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ¬ (2nd ‘𝑋) = 0)
14	13	iffalsed 3615	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) = ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)
15	3, 14	eqtrd 2264	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (𝐸‘𝑋) = ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)
16	15	fveq2d 5643	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘(𝐸‘𝑋)) = (2nd ‘⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩))
17		xp2nd 6329	. . . . . 6 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑋) ∈ ℕ0)
18	17	adantr 276	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘𝑋) ∈ ℕ0)
19		1st2nd2 6338	. . . . . . . . 9 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
20	19	adantr 276	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
21	20	fveq2d 5643	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ( mod ‘𝑋) = ( mod ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
22		df-ov 6021	. . . . . . 7 ⊢ ((1st ‘𝑋) mod (2nd ‘𝑋)) = ( mod ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
23	21, 22	eqtr4di 2282	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ( mod ‘𝑋) = ((1st ‘𝑋) mod (2nd ‘𝑋)))
24		xp1st 6328	. . . . . . . . 9 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (1st ‘𝑋) ∈ ℕ0)
25	24	adantr 276	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (1st ‘𝑋) ∈ ℕ0)
26	25	nn0zd 9600	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (1st ‘𝑋) ∈ ℤ)
27	13	neqned 2409	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘𝑋) ≠ 0)
28		elnnne0 9416	. . . . . . . 8 ⊢ ((2nd ‘𝑋) ∈ ℕ ↔ ((2nd ‘𝑋) ∈ ℕ0 ∧ (2nd ‘𝑋) ≠ 0))
29	18, 27, 28	sylanbrc 417	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘𝑋) ∈ ℕ)
30	26, 29	zmodcld 10608	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ((1st ‘𝑋) mod (2nd ‘𝑋)) ∈ ℕ0)
31	23, 30	eqeltrd 2308	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ( mod ‘𝑋) ∈ ℕ0)
32		op2ndg 6314	. . . . 5 ⊢ (((2nd ‘𝑋) ∈ ℕ0 ∧ ( mod ‘𝑋) ∈ ℕ0) → (2nd ‘⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) = ( mod ‘𝑋))
33	18, 31, 32	syl2anc 411	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) = ( mod ‘𝑋))
34	16, 33, 23	3eqtrd 2268	. . 3 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘(𝐸‘𝑋)) = ((1st ‘𝑋) mod (2nd ‘𝑋)))
35		zq 9860	. . . . 5 ⊢ ((1st ‘𝑋) ∈ ℤ → (1st ‘𝑋) ∈ ℚ)
36	26, 35	syl 14	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (1st ‘𝑋) ∈ ℚ)
37	18	nn0zd 9600	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘𝑋) ∈ ℤ)
38		zq 9860	. . . . 5 ⊢ ((2nd ‘𝑋) ∈ ℤ → (2nd ‘𝑋) ∈ ℚ)
39	37, 38	syl 14	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘𝑋) ∈ ℚ)
40	29	nngt0d 9187	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → 0 < (2nd ‘𝑋))
41		modqlt 10596	. . . 4 ⊢ (((1st ‘𝑋) ∈ ℚ ∧ (2nd ‘𝑋) ∈ ℚ ∧ 0 < (2nd ‘𝑋)) → ((1st ‘𝑋) mod (2nd ‘𝑋)) < (2nd ‘𝑋))
42	36, 39, 40, 41	syl3anc 1273	. . 3 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ((1st ‘𝑋) mod (2nd ‘𝑋)) < (2nd ‘𝑋))
43	34, 42	eqbrtrd 4110	. 2 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘(𝐸‘𝑋)) < (2nd ‘𝑋))
44	43	ex 115	1 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → ((2nd ‘(𝐸‘𝑋)) ≠ 0 → (2nd ‘(𝐸‘𝑋)) < (2nd ‘𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202   ≠ wne 2402  ifcif 3605  ⟨cop 3672   class class class wbr 4088   × cxp 4723  ‘cfv 5326  (class class class)co 6018   ∈ cmpo 6020  1^st c1st 6301  2^nd c2nd 6302  0cc0 8032   < clt 8214  ℕcn 9143  ℕ₀cn0 9402  ℤcz 9479  ℚcq 9853   mod cmo 10585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-n0 9403  df-z 9480  df-q 9854  df-rp 9889  df-fl 10531  df-mod 10586

This theorem is referenced by:  eucalgcvga  12635