Theorem eucalglt 16512

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 16509	. . . . . . . 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 4485	. . . . . . . . . . . . . 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 6834	. . . . . . . . . . . . 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 2945	. . . . . . . . 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 4490	. . . . . . 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 6838	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘(𝐸‘𝑋)) = (2nd ‘⟨(2nd ‘𝑋), ( mod ‘𝑋)⟩))
17		fvex 6847	. . . . . 6 ⊢ (2nd ‘𝑋) ∈ V
18		fvex 6847	. . . . . 6 ⊢ ( mod ‘𝑋) ∈ V
19	17, 18	op2nd 7942	. . . . 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 7972	. . . . . . 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 6838	. . . . 5 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → ( mod ‘𝑋) = ( mod ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
24		df-ov 7361	. . . . 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 7965	. . . . . 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 12463	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (1st ‘𝑋) ∈ ℝ)
30		xp2nd 7966	. . . . . . . . 9 ⊢ (𝑋 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑋) ∈ ℕ0)
31	30	adantr 480	. . . . . . . 8 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘𝑋) ∈ ℕ0)
32		elnn0 12403	. . . . . . . 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 12947	. . . 4 ⊢ ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸‘𝑋)) ≠ 0) → (2nd ‘𝑋) ∈ ℝ+)
37		modlt 13800	. . . 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 5120	. 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 1541   ∈ wcel 2113   ≠ wne 2932  ifcif 4479  ⟨cop 4586   class class class wbr 5098   × cxp 5622  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  1^st c1st 7931  2^nd c2nd 7932  ℝcr 11025  0cc0 11026   < clt 11166  ℕcn 12145  ℕ₀cn0 12401  ℝ⁺crp 12905   mod cmo 13789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9345  df-inf 9346  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-n0 12402  df-z 12489  df-uz 12752  df-rp 12906  df-fl 13712  df-mod 13790

This theorem is referenced by:  eucalgcvga  16513