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Theorem eucalglt 15633
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 15630 . . . . . . . 8 (𝑋 ∈ (ℕ0 × ℕ0) → (𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩))
32adantr 473 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩))
4 simpr 478 . . . . . . . . 9 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) ≠ 0)
5 iftrue 4283 . . . . . . . . . . . . . 14 ((2nd𝑋) = 0 → if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) = 𝑋)
65eqeq2d 2809 . . . . . . . . . . . . 13 ((2nd𝑋) = 0 → ((𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) ↔ (𝐸𝑋) = 𝑋))
7 fveq2 6411 . . . . . . . . . . . . 13 ((𝐸𝑋) = 𝑋 → (2nd ‘(𝐸𝑋)) = (2nd𝑋))
86, 7syl6bi 245 . . . . . . . . . . . 12 ((2nd𝑋) = 0 → ((𝐸𝑋) = if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) → (2nd ‘(𝐸𝑋)) = (2nd𝑋)))
9 eqeq2 2810 . . . . . . . . . . . 12 ((2nd𝑋) = 0 → ((2nd ‘(𝐸𝑋)) = (2nd𝑋) ↔ (2nd ‘(𝐸𝑋)) = 0))
108, 9sylibd 231 . . . . . . . . . . 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 2984 . . . . . . . . 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 4288 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → if((2nd𝑋) = 0, 𝑋, ⟨(2nd𝑋), ( mod ‘𝑋)⟩) = ⟨(2nd𝑋), ( mod ‘𝑋)⟩)
153, 14eqtrd 2833 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (𝐸𝑋) = ⟨(2nd𝑋), ( mod ‘𝑋)⟩)
1615fveq2d 6415 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) = (2nd ‘⟨(2nd𝑋), ( mod ‘𝑋)⟩))
17 fvex 6424 . . . . . 6 (2nd𝑋) ∈ V
18 fvex 6424 . . . . . 6 ( mod ‘𝑋) ∈ V
1917, 18op2nd 7410 . . . . 5 (2nd ‘⟨(2nd𝑋), ( mod ‘𝑋)⟩) = ( mod ‘𝑋)
2016, 19syl6eq 2849 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) = ( mod ‘𝑋))
21 1st2nd2 7440 . . . . . . 7 (𝑋 ∈ (ℕ0 × ℕ0) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
2221adantr 473 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
2322fveq2d 6415 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ( mod ‘𝑋) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩))
24 df-ov 6881 . . . . 5 ((1st𝑋) mod (2nd𝑋)) = ( mod ‘⟨(1st𝑋), (2nd𝑋)⟩)
2523, 24syl6eqr 2851 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ( mod ‘𝑋) = ((1st𝑋) mod (2nd𝑋)))
2620, 25eqtrd 2833 . . 3 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) = ((1st𝑋) mod (2nd𝑋)))
27 xp1st 7433 . . . . . 6 (𝑋 ∈ (ℕ0 × ℕ0) → (1st𝑋) ∈ ℕ0)
2827adantr 473 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (1st𝑋) ∈ ℕ0)
2928nn0red 11641 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (1st𝑋) ∈ ℝ)
30 xp2nd 7434 . . . . . . . . 9 (𝑋 ∈ (ℕ0 × ℕ0) → (2nd𝑋) ∈ ℕ0)
3130adantr 473 . . . . . . . 8 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd𝑋) ∈ ℕ0)
32 elnn0 11582 . . . . . . . 8 ((2nd𝑋) ∈ ℕ0 ↔ ((2nd𝑋) ∈ ℕ ∨ (2nd𝑋) = 0))
3331, 32sylib 210 . . . . . . 7 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ((2nd𝑋) ∈ ℕ ∨ (2nd𝑋) = 0))
3433ord 891 . . . . . 6 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (¬ (2nd𝑋) ∈ ℕ → (2nd𝑋) = 0))
3513, 34mt3d 143 . . . . 5 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd𝑋) ∈ ℕ)
3635nnrpd 12115 . . . 4 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd𝑋) ∈ ℝ+)
37 modlt 12934 . . . 4 (((1st𝑋) ∈ ℝ ∧ (2nd𝑋) ∈ ℝ+) → ((1st𝑋) mod (2nd𝑋)) < (2nd𝑋))
3829, 36, 37syl2anc 580 . . 3 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → ((1st𝑋) mod (2nd𝑋)) < (2nd𝑋))
3926, 38eqbrtrd 4865 . 2 ((𝑋 ∈ (ℕ0 × ℕ0) ∧ (2nd ‘(𝐸𝑋)) ≠ 0) → (2nd ‘(𝐸𝑋)) < (2nd𝑋))
4039ex 402 1 (𝑋 ∈ (ℕ0 × ℕ0) → ((2nd ‘(𝐸𝑋)) ≠ 0 → (2nd ‘(𝐸𝑋)) < (2nd𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385  wo 874   = wceq 1653  wcel 2157  wne 2971  ifcif 4277  cop 4374   class class class wbr 4843   × cxp 5310  cfv 6101  (class class class)co 6878  cmpt2 6880  1st c1st 7399  2nd c2nd 7400  cr 10223  0cc0 10224   < clt 10363  cn 11312  0cn0 11580  +crp 12074   mod cmo 12923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-pre-sup 10302
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-sup 8590  df-inf 8591  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-div 10977  df-nn 11313  df-n0 11581  df-z 11667  df-uz 11931  df-rp 12075  df-fl 12848  df-mod 12924
This theorem is referenced by:  eucalgcvga  15634
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