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Theorem eucalglt 15136
 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 15133 . . . . . . . 8 (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩))
32adantr 480 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩))
4 simpr 476 . . . . . . . . 9 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) ≠ 0)
5 iftrue 4042 . . . . . . . . . . . . . 14 ((2nd𝑋) = 0 → if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) = 𝑋)
65eqeq2d 2620 . . . . . . . . . . . . 13 ((2nd𝑋) = 0 → ((𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) ↔ (𝐸𝑋) = 𝑋))
7 fveq2 6103 . . . . . . . . . . . . 13 ((𝐸𝑋) = 𝑋 → (2nd ‘(𝐸𝑋)) = (2nd𝑋))
86, 7syl6bi 242 . . . . . . . . . . . 12 ((2nd𝑋) = 0 → ((𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) → (2nd ‘(𝐸𝑋)) = (2nd𝑋)))
9 eqeq2 2621 . . . . . . . . . . . 12 ((2nd𝑋) = 0 → ((2nd ‘(𝐸𝑋)) = (2nd𝑋) ↔ (2nd ‘(𝐸𝑋)) = 0))
108, 9sylibd 228 . . . . . . . . . . 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 2795 . . . . . . . . 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 4047 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) = ⟨(2nd𝑋), ( mod ‘𝑋)⟩)
153, 14eqtrd 2644 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (𝐸𝑋) = ⟨(2nd𝑋), ( mod ‘𝑋)⟩)
1615fveq2d 6107 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) = (2nd ‘⟨(2nd𝑋), ( mod ‘𝑋)⟩))
17 fvex 6113 . . . . . 6 (2nd𝑋) ∈ V
18 fvex 6113 . . . . . 6 ( mod ‘𝑋) ∈ V
1917, 18op2nd 7068 . . . . 5 (2nd ‘⟨(2nd𝑋), ( mod ‘𝑋)⟩) = ( mod ‘𝑋)
2016, 19syl6eq 2660 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) = ( mod ‘𝑋))
21 1st2nd2 7096 . . . . . . 7 (𝑋 ∈ (ℕ0 × ℕ0) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
2221adantr 480 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
2322fveq2d 6107 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ( mod ‘𝑋) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩))
24 df-ov 6552 . . . . 5 ((1st𝑋) mod (2nd𝑋)) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩)
2523, 24syl6eqr 2662 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ( mod ‘𝑋) = ((1st𝑋) mod (2nd𝑋)))
2620, 25eqtrd 2644 . . 3 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) = ((1st𝑋) mod (2nd𝑋)))
27 xp1st 7089 . . . . . 6 (𝑋 ∈ (ℕ0 × ℕ0) → (1st𝑋) ∈ ℕ0)
2827adantr 480 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (1st𝑋) ∈ ℕ0)
2928nn0red 11229 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (1st𝑋) ∈ ℝ)
30 xp2nd 7090 . . . . . . . . 9 (𝑋 ∈ (ℕ0 × ℕ0) → (2nd𝑋) ∈ ℕ0)
3130adantr 480 . . . . . . . 8 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd𝑋) ∈ ℕ0)
32 elnn0 11171 . . . . . . . 8 ((2nd𝑋) ∈ ℕ0 ↔ ((2nd𝑋) ∈ ℕ ∨ (2nd𝑋) = 0))
3331, 32sylib 207 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ((2nd𝑋) ∈ ℕ ∨ (2nd𝑋) = 0))
3433ord 391 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (¬ (2nd𝑋) ∈ ℕ → (2nd𝑋) = 0))
3513, 34mt3d 139 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd𝑋) ∈ ℕ)
3635nnrpd 11746 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd𝑋) ∈ ℝ+)
37 modlt 12541 . . . 4 (((1st𝑋) ∈ ℝ ∧ (2nd𝑋) ∈ ℝ+) → ((1st𝑋) mod (2nd𝑋)) < (2nd𝑋))
3829, 36, 37syl2anc 691 . . 3 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ((1st𝑋) mod (2nd𝑋)) < (2nd𝑋))
3926, 38eqbrtrd 4605 . 2 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) < (2nd𝑋))
4039ex 449 1 (𝑋 ∈ (ℕ0 × ℕ0) → ((2nd ‘(𝐸𝑋)) ≠ 0 → (2nd ‘(𝐸𝑋)) < (2nd𝑋)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ifcif 4036  ⟨cop 4131   class class class wbr 4583   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  ℝcr 9814  0cc0 9815   < clt 9953  ℕcn 10897  ℕ0cn0 11169  ℝ+crp 11708   mod cmo 12530 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fl 12455  df-mod 12531 This theorem is referenced by:  eucalgcvga  15137
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