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Theorem eucalglt 15919
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 15916 . . . . . . . 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 485 . . . . . . . . 9 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) ≠ 0)
5 iftrue 4476 . . . . . . . . . . . . . 14 ((2nd𝑋) = 0 → if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) = 𝑋)
65eqeq2d 2837 . . . . . . . . . . . . 13 ((2nd𝑋) = 0 → ((𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) ↔ (𝐸𝑋) = 𝑋))
7 fveq2 6667 . . . . . . . . . . . . 13 ((𝐸𝑋) = 𝑋 → (2nd ‘(𝐸𝑋)) = (2nd𝑋))
86, 7syl6bi 254 . . . . . . . . . . . 12 ((2nd𝑋) = 0 → ((𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) → (2nd ‘(𝐸𝑋)) = (2nd𝑋)))
9 eqeq2 2838 . . . . . . . . . . . 12 ((2nd𝑋) = 0 → ((2nd ‘(𝐸𝑋)) = (2nd𝑋) ↔ (2nd ‘(𝐸𝑋)) = 0))
108, 9sylibd 240 . . . . . . . . . . 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 3034 . . . . . . . . 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 4481 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) = ⟨(2nd𝑋), ( mod ‘𝑋)⟩)
153, 14eqtrd 2861 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (𝐸𝑋) = ⟨(2nd𝑋), ( mod ‘𝑋)⟩)
1615fveq2d 6671 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) = (2nd ‘⟨(2nd𝑋), ( mod ‘𝑋)⟩))
17 fvex 6680 . . . . . 6 (2nd𝑋) ∈ V
18 fvex 6680 . . . . . 6 ( mod ‘𝑋) ∈ V
1917, 18op2nd 7689 . . . . 5 (2nd ‘⟨(2nd𝑋), ( mod ‘𝑋)⟩) = ( mod ‘𝑋)
2016, 19syl6eq 2877 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) = ( mod ‘𝑋))
21 1st2nd2 7719 . . . . . . 7 (𝑋 ∈ (ℕ0 × ℕ0) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
2221adantr 481 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
2322fveq2d 6671 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ( mod ‘𝑋) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩))
24 df-ov 7151 . . . . 5 ((1st𝑋) mod (2nd𝑋)) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩)
2523, 24syl6eqr 2879 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ( mod ‘𝑋) = ((1st𝑋) mod (2nd𝑋)))
2620, 25eqtrd 2861 . . 3 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) = ((1st𝑋) mod (2nd𝑋)))
27 xp1st 7712 . . . . . 6 (𝑋 ∈ (ℕ0 × ℕ0) → (1st𝑋) ∈ ℕ0)
2827adantr 481 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (1st𝑋) ∈ ℕ0)
2928nn0red 11945 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (1st𝑋) ∈ ℝ)
30 xp2nd 7713 . . . . . . . . 9 (𝑋 ∈ (ℕ0 × ℕ0) → (2nd𝑋) ∈ ℕ0)
3130adantr 481 . . . . . . . 8 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd𝑋) ∈ ℕ0)
32 elnn0 11888 . . . . . . . 8 ((2nd𝑋) ∈ ℕ0 ↔ ((2nd𝑋) ∈ ℕ ∨ (2nd𝑋) = 0))
3331, 32sylib 219 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ((2nd𝑋) ∈ ℕ ∨ (2nd𝑋) = 0))
3433ord 860 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (¬ (2nd𝑋) ∈ ℕ → (2nd𝑋) = 0))
3513, 34mt3d 150 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd𝑋) ∈ ℕ)
3635nnrpd 12419 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd𝑋) ∈ ℝ+)
37 modlt 13238 . . . 4 (((1st𝑋) ∈ ℝ ∧ (2nd𝑋) ∈ ℝ+) → ((1st𝑋) mod (2nd𝑋)) < (2nd𝑋))
3829, 36, 37syl2anc 584 . . 3 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ((1st𝑋) mod (2nd𝑋)) < (2nd𝑋))
3926, 38eqbrtrd 5085 . 2 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) < (2nd𝑋))
4039ex 413 1 (𝑋 ∈ (ℕ0 × ℕ0) → ((2nd ‘(𝐸𝑋)) ≠ 0 → (2nd ‘(𝐸𝑋)) < (2nd𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 843   = wceq 1530  wcel 2107  wne 3021  ifcif 4470  cop 4570   class class class wbr 5063   × cxp 5552  cfv 6352  (class class class)co 7148  cmpo 7150  1st c1st 7678  2nd c2nd 7679  cr 10525  0cc0 10526   < clt 10664  cn 11627  0cn0 11886  +crp 12379   mod cmo 13227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-sup 8895  df-inf 8896  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-n0 11887  df-z 11971  df-uz 12233  df-rp 12380  df-fl 13152  df-mod 13228
This theorem is referenced by:  eucalgcvga  15920
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