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Theorem eucalglt 15974
 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 15971 . . . . . . . 8 (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩))
32adantr 485 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩))
4 simpr 489 . . . . . . . . 9 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) ≠ 0)
5 iftrue 4427 . . . . . . . . . . . . . 14 ((2nd𝑋) = 0 → if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) = 𝑋)
65eqeq2d 2770 . . . . . . . . . . . . 13 ((2nd𝑋) = 0 → ((𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) ↔ (𝐸𝑋) = 𝑋))
7 fveq2 6659 . . . . . . . . . . . . 13 ((𝐸𝑋) = 𝑋 → (2nd ‘(𝐸𝑋)) = (2nd𝑋))
86, 7syl6bi 256 . . . . . . . . . . . 12 ((2nd𝑋) = 0 → ((𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) → (2nd ‘(𝐸𝑋)) = (2nd𝑋)))
9 eqeq2 2771 . . . . . . . . . . . 12 ((2nd𝑋) = 0 → ((2nd ‘(𝐸𝑋)) = (2nd𝑋) ↔ (2nd ‘(𝐸𝑋)) = 0))
108, 9sylibd 242 . . . . . . . . . . 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 2965 . . . . . . . . 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 4432 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) = ⟨(2nd𝑋), ( mod ‘𝑋)⟩)
153, 14eqtrd 2794 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (𝐸𝑋) = ⟨(2nd𝑋), ( mod ‘𝑋)⟩)
1615fveq2d 6663 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) = (2nd ‘⟨(2nd𝑋), ( mod ‘𝑋)⟩))
17 fvex 6672 . . . . . 6 (2nd𝑋) ∈ V
18 fvex 6672 . . . . . 6 ( mod ‘𝑋) ∈ V
1917, 18op2nd 7703 . . . . 5 (2nd ‘⟨(2nd𝑋), ( mod ‘𝑋)⟩) = ( mod ‘𝑋)
2016, 19eqtrdi 2810 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) = ( mod ‘𝑋))
21 1st2nd2 7733 . . . . . . 7 (𝑋 ∈ (ℕ0 × ℕ0) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
2221adantr 485 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
2322fveq2d 6663 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ( mod ‘𝑋) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩))
24 df-ov 7154 . . . . 5 ((1st𝑋) mod (2nd𝑋)) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩)
2523, 24eqtr4di 2812 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ( mod ‘𝑋) = ((1st𝑋) mod (2nd𝑋)))
2620, 25eqtrd 2794 . . 3 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) = ((1st𝑋) mod (2nd𝑋)))
27 xp1st 7726 . . . . . 6 (𝑋 ∈ (ℕ0 × ℕ0) → (1st𝑋) ∈ ℕ0)
2827adantr 485 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (1st𝑋) ∈ ℕ0)
2928nn0red 11988 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (1st𝑋) ∈ ℝ)
30 xp2nd 7727 . . . . . . . . 9 (𝑋 ∈ (ℕ0 × ℕ0) → (2nd𝑋) ∈ ℕ0)
3130adantr 485 . . . . . . . 8 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd𝑋) ∈ ℕ0)
32 elnn0 11929 . . . . . . . 8 ((2nd𝑋) ∈ ℕ0 ↔ ((2nd𝑋) ∈ ℕ ∨ (2nd𝑋) = 0))
3331, 32sylib 221 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ((2nd𝑋) ∈ ℕ ∨ (2nd𝑋) = 0))
3433ord 862 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (¬ (2nd𝑋) ∈ ℕ → (2nd𝑋) = 0))
3513, 34mt3d 150 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd𝑋) ∈ ℕ)
3635nnrpd 12463 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd𝑋) ∈ ℝ+)
37 modlt 13290 . . . 4 (((1st𝑋) ∈ ℝ ∧ (2nd𝑋) ∈ ℝ+) → ((1st𝑋) mod (2nd𝑋)) < (2nd𝑋))
3829, 36, 37syl2anc 588 . . 3 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ((1st𝑋) mod (2nd𝑋)) < (2nd𝑋))
3926, 38eqbrtrd 5055 . 2 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) < (2nd𝑋))
4039ex 417 1 (𝑋 ∈ (ℕ0 × ℕ0) → ((2nd ‘(𝐸𝑋)) ≠ 0 → (2nd ‘(𝐸𝑋)) < (2nd𝑋)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 845   = wceq 1539   ∈ wcel 2112   ≠ wne 2952  ifcif 4421  ⟨cop 4529   class class class wbr 5033   × cxp 5523  ‘cfv 6336  (class class class)co 7151   ∈ cmpo 7153  1st c1st 7692  2nd c2nd 7693  ℝcr 10567  0cc0 10568   < clt 10706  ℕcn 11667  ℕ0cn0 11927  ℝ+crp 12423   mod cmo 13279 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-cnex 10624  ax-resscn 10625  ax-1cn 10626  ax-icn 10627  ax-addcl 10628  ax-addrcl 10629  ax-mulcl 10630  ax-mulrcl 10631  ax-mulcom 10632  ax-addass 10633  ax-mulass 10634  ax-distr 10635  ax-i2m1 10636  ax-1ne0 10637  ax-1rid 10638  ax-rnegex 10639  ax-rrecex 10640  ax-cnre 10641  ax-pre-lttri 10642  ax-pre-lttrn 10643  ax-pre-ltadd 10644  ax-pre-mulgt0 10645  ax-pre-sup 10646 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7581  df-1st 7694  df-2nd 7695  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-er 8300  df-en 8529  df-dom 8530  df-sdom 8531  df-sup 8932  df-inf 8933  df-pnf 10708  df-mnf 10709  df-xr 10710  df-ltxr 10711  df-le 10712  df-sub 10903  df-neg 10904  df-div 11329  df-nn 11668  df-n0 11928  df-z 12014  df-uz 12276  df-rp 12424  df-fl 13204  df-mod 13280 This theorem is referenced by:  eucalgcvga  15975
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