Theorem eucalglt 12565

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 12562	. . . . . . 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 3607	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝑋) = 0 → if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) = 𝑋)
6	5	eqeq2d 2241	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑋) = 0 → ((𝐸‘𝑋) = if((2nd ‘𝑋) = 0, 𝑋, ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩) ↔ (𝐸‘𝑋) = 𝑋))
7		fveq2 5623	. . . . . . . . . . . 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 2239	. . . . . . . . . . 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 2442	. . . . . . . 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 3612	. . . . . 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 2262	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (𝐸‘𝑋) = ⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩)
16	15	fveq2d 5627	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘(𝐸‘𝑋)) = (2nd ‘⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩))
17		xp2nd 6302	. . . . . 6 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑋) ∈ ℕ0)
18	17	adantr 276	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘𝑋) ∈ ℕ0)
19		1st2nd2 6311	. . . . . . . . 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 5627	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ( mod ‘𝑋) = ( mod ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
22		df-ov 5997	. . . . . . 7 ⊢ ((1st ‘𝑋) mod (2nd ‘𝑋)) = ( mod ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
23	21, 22	eqtr4di 2280	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ( mod ‘𝑋) = ((1st ‘𝑋) mod (2nd ‘𝑋)))
24		xp1st 6301	. . . . . . . . 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 9555	. . . . . . 7 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (1st ‘𝑋) ∈ ℤ)
27	13	neqned 2407	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘𝑋) ≠ 0)
28		elnnne0 9371	. . . . . . . 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 10554	. . . . . 6 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ((1st ‘𝑋) mod (2nd ‘𝑋)) ∈ ℕ0)
31	23, 30	eqeltrd 2306	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ( mod ‘𝑋) ∈ ℕ0)
32		op2ndg 6287	. . . . 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 2266	. . 3 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘(𝐸‘𝑋)) = ((1st ‘𝑋) mod (2nd ‘𝑋)))
35		zq 9809	. . . . 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 9555	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘𝑋) ∈ ℤ)
38		zq 9809	. . . . 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 9142	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → 0 < (2nd ‘𝑋))
41		modqlt 10542	. . . 4 ⊢ (((1st ‘𝑋) ∈ ℚ ∧ (2nd ‘𝑋) ∈ ℚ ∧ 0 < (2nd ‘𝑋)) → ((1st ‘𝑋) mod (2nd ‘𝑋)) < (2nd ‘𝑋))
42	36, 39, 40, 41	syl3anc 1271	. . 3 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ((1st ‘𝑋) mod (2nd ‘𝑋)) < (2nd ‘𝑋))
43	34, 42	eqbrtrd 4104	. 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 1395   ∈ wcel 2200   ≠ wne 2400  ifcif 3602  ⟨cop 3669   class class class wbr 4082   × cxp 4714  ‘cfv 5314  (class class class)co 5994   ∈ cmpo 5996  1^st c1st 6274  2^nd c2nd 6275  0cc0 7987   < clt 8169  ℕcn 9098  ℕ₀cn0 9357  ℤcz 9434  ℚcq 9802   mod cmo 10531

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105  ax-arch 8106

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-po 4384  df-iso 4385  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-n0 9358  df-z 9435  df-q 9803  df-rp 9838  df-fl 10477  df-mod 10532

This theorem is referenced by:  eucalgcvga  12566