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Theorem eucalglt 15222
 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 ‘(𝐸𝑋)) ≠ 0 → (2nd ‘(𝐸𝑋)) < (2nd𝑋)))
Distinct variable group:   𝑥,𝑦,𝑋
Allowed substitution hints:   𝐸(𝑥,𝑦)

Proof of Theorem eucalglt
StepHypRef Expression
1 eucalgval.1 . . . . . . . . 9 𝐸 = (𝑥 ∈ ℕ0, 𝑦 ∈ ℕ0 ↦ if(𝑦 = 0, ⟨𝑥, 𝑦⟩, ⟨𝑦, (𝑥 mod 𝑦)⟩))
21eucalgval 15219 . . . . . . . 8 (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩))
32adantr 481 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩))
4 simpr 477 . . . . . . . . 9 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) ≠ 0)
5 iftrue 4064 . . . . . . . . . . . . . 14 ((2nd𝑋) = 0 → if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) = 𝑋)
65eqeq2d 2631 . . . . . . . . . . . . 13 ((2nd𝑋) = 0 → ((𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) ↔ (𝐸𝑋) = 𝑋))
7 fveq2 6148 . . . . . . . . . . . . 13 ((𝐸𝑋) = 𝑋 → (2nd ‘(𝐸𝑋)) = (2nd𝑋))
86, 7syl6bi 243 . . . . . . . . . . . 12 ((2nd𝑋) = 0 → ((𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) → (2nd ‘(𝐸𝑋)) = (2nd𝑋)))
9 eqeq2 2632 . . . . . . . . . . . 12 ((2nd𝑋) = 0 → ((2nd ‘(𝐸𝑋)) = (2nd𝑋) ↔ (2nd ‘(𝐸𝑋)) = 0))
108, 9sylibd 229 . . . . . . . . . . 11 ((2nd𝑋) = 0 → ((𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) → (2nd ‘(𝐸𝑋)) = 0))
113, 10syl5com 31 . . . . . . . . . 10 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ((2nd𝑋) = 0 → (2nd ‘(𝐸𝑋)) = 0))
1211necon3ad 2803 . . . . . . . . 9 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ((2nd ‘(𝐸𝑋)) ≠ 0 → ¬ (2nd𝑋) = 0))
134, 12mpd 15 . . . . . . . 8 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ¬ (2nd𝑋) = 0)
1413iffalsed 4069 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) = ⟨(2nd𝑋), ( mod ‘𝑋)⟩)
153, 14eqtrd 2655 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (𝐸𝑋) = ⟨(2nd𝑋), ( mod ‘𝑋)⟩)
1615fveq2d 6152 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) = (2nd ‘⟨(2nd𝑋), ( mod ‘𝑋)⟩))
17 fvex 6158 . . . . . 6 (2nd𝑋) ∈ V
18 fvex 6158 . . . . . 6 ( mod ‘𝑋) ∈ V
1917, 18op2nd 7122 . . . . 5 (2nd ‘⟨(2nd𝑋), ( mod ‘𝑋)⟩) = ( mod ‘𝑋)
2016, 19syl6eq 2671 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) = ( mod ‘𝑋))
21 1st2nd2 7150 . . . . . . 7 (𝑋 ∈ (ℕ0 × ℕ0) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
2221adantr 481 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
2322fveq2d 6152 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ( mod ‘𝑋) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩))
24 df-ov 6607 . . . . 5 ((1st𝑋) mod (2nd𝑋)) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩)
2523, 24syl6eqr 2673 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ( mod ‘𝑋) = ((1st𝑋) mod (2nd𝑋)))
2620, 25eqtrd 2655 . . 3 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) = ((1st𝑋) mod (2nd𝑋)))
27 xp1st 7143 . . . . . 6 (𝑋 ∈ (ℕ0 × ℕ0) → (1st𝑋) ∈ ℕ0)
2827adantr 481 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (1st𝑋) ∈ ℕ0)
2928nn0red 11296 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (1st𝑋) ∈ ℝ)
30 xp2nd 7144 . . . . . . . . 9 (𝑋 ∈ (ℕ0 × ℕ0) → (2nd𝑋) ∈ ℕ0)
3130adantr 481 . . . . . . . 8 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd𝑋) ∈ ℕ0)
32 elnn0 11238 . . . . . . . 8 ((2nd𝑋) ∈ ℕ0 ↔ ((2nd𝑋) ∈ ℕ ∨ (2nd𝑋) = 0))
3331, 32sylib 208 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ((2nd𝑋) ∈ ℕ ∨ (2nd𝑋) = 0))
3433ord 392 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (¬ (2nd𝑋) ∈ ℕ → (2nd𝑋) = 0))
3513, 34mt3d 140 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd𝑋) ∈ ℕ)
3635nnrpd 11814 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd𝑋) ∈ ℝ+)
37 modlt 12619 . . . 4 (((1st𝑋) ∈ ℝ ∧ (2nd𝑋) ∈ ℝ+) → ((1st𝑋) mod (2nd𝑋)) < (2nd𝑋))
3829, 36, 37syl2anc 692 . . 3 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ((1st𝑋) mod (2nd𝑋)) < (2nd𝑋))
3926, 38eqbrtrd 4635 . 2 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) < (2nd𝑋))
4039ex 450 1 (𝑋 ∈ (ℕ0 × ℕ0) → ((2nd ‘(𝐸𝑋)) ≠ 0 → (2nd ‘(𝐸𝑋)) < (2nd𝑋)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ifcif 4058  ⟨cop 4154   class class class wbr 4613   × cxp 5072  ‘cfv 5847  (class class class)co 6604   ↦ cmpt2 6606  1st c1st 7111  2nd c2nd 7112  ℝcr 9879  0cc0 9880   < clt 10018  ℕcn 10964  ℕ0cn0 11236  ℝ+crp 11776   mod cmo 12608 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-sup 8292  df-inf 8293  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-rp 11777  df-fl 12533  df-mod 12609 This theorem is referenced by:  eucalgcvga  15223
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