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Theorem ruclem10 15012
Description: Lemma for ruc 15016. Every first component of the 𝐺 sequence is less than every second component. That is, the sequences form a chain a1 < a2 <... < b2 < b1, where ai are the first components and bi are the second components. (Contributed by Mario Carneiro, 28-May-2014.)
Hypotheses
Ref Expression
ruc.1 (𝜑𝐹:ℕ⟶ℝ)
ruc.2 (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
ruc.4 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
ruc.5 𝐺 = seq0(𝐷, 𝐶)
ruclem10.6 (𝜑𝑀 ∈ ℕ0)
ruclem10.7 (𝜑𝑁 ∈ ℕ0)
Assertion
Ref Expression
ruclem10 (𝜑 → (1st ‘(𝐺𝑀)) < (2nd ‘(𝐺𝑁)))
Distinct variable groups:   𝑥,𝑚,𝑦,𝐹   𝑚,𝐺,𝑥,𝑦   𝑚,𝑀,𝑥,𝑦   𝑚,𝑁,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑚)   𝐶(𝑥,𝑦,𝑚)   𝐷(𝑥,𝑦,𝑚)

Proof of Theorem ruclem10
StepHypRef Expression
1 ruc.1 . . . . 5 (𝜑𝐹:ℕ⟶ℝ)
2 ruc.2 . . . . 5 (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
3 ruc.4 . . . . 5 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
4 ruc.5 . . . . 5 𝐺 = seq0(𝐷, 𝐶)
51, 2, 3, 4ruclem6 15008 . . . 4 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
6 ruclem10.6 . . . 4 (𝜑𝑀 ∈ ℕ0)
75, 6ffvelrnd 6400 . . 3 (𝜑 → (𝐺𝑀) ∈ (ℝ × ℝ))
8 xp1st 7242 . . 3 ((𝐺𝑀) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑀)) ∈ ℝ)
97, 8syl 17 . 2 (𝜑 → (1st ‘(𝐺𝑀)) ∈ ℝ)
10 ruclem10.7 . . . . 5 (𝜑𝑁 ∈ ℕ0)
1110, 6ifcld 4164 . . . 4 (𝜑 → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℕ0)
125, 11ffvelrnd 6400 . . 3 (𝜑 → (𝐺‘if(𝑀𝑁, 𝑁, 𝑀)) ∈ (ℝ × ℝ))
13 xp1st 7242 . . 3 ((𝐺‘if(𝑀𝑁, 𝑁, 𝑀)) ∈ (ℝ × ℝ) → (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ∈ ℝ)
1412, 13syl 17 . 2 (𝜑 → (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ∈ ℝ)
155, 10ffvelrnd 6400 . . 3 (𝜑 → (𝐺𝑁) ∈ (ℝ × ℝ))
16 xp2nd 7243 . . 3 ((𝐺𝑁) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑁)) ∈ ℝ)
1715, 16syl 17 . 2 (𝜑 → (2nd ‘(𝐺𝑁)) ∈ ℝ)
186nn0red 11390 . . . . . 6 (𝜑𝑀 ∈ ℝ)
1910nn0red 11390 . . . . . 6 (𝜑𝑁 ∈ ℝ)
20 max1 12054 . . . . . 6 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀))
2118, 19, 20syl2anc 694 . . . . 5 (𝜑𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀))
226nn0zd 11518 . . . . . 6 (𝜑𝑀 ∈ ℤ)
2311nn0zd 11518 . . . . . 6 (𝜑 → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℤ)
24 eluz 11739 . . . . . 6 ((𝑀 ∈ ℤ ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℤ) → (if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑀) ↔ 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
2522, 23, 24syl2anc 694 . . . . 5 (𝜑 → (if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑀) ↔ 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
2621, 25mpbird 247 . . . 4 (𝜑 → if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑀))
271, 2, 3, 4, 6, 26ruclem9 15011 . . 3 (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ∧ (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ≤ (2nd ‘(𝐺𝑀))))
2827simpld 474 . 2 (𝜑 → (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))))
29 xp2nd 7243 . . . 4 ((𝐺‘if(𝑀𝑁, 𝑁, 𝑀)) ∈ (ℝ × ℝ) → (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ∈ ℝ)
3012, 29syl 17 . . 3 (𝜑 → (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ∈ ℝ)
311, 2, 3, 4ruclem8 15010 . . . 4 ((𝜑 ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℕ0) → (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))))
3211, 31mpdan 703 . . 3 (𝜑 → (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))))
33 max2 12056 . . . . . . 7 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀))
3418, 19, 33syl2anc 694 . . . . . 6 (𝜑𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀))
3510nn0zd 11518 . . . . . . 7 (𝜑𝑁 ∈ ℤ)
36 eluz 11739 . . . . . . 7 ((𝑁 ∈ ℤ ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℤ) → (if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑁) ↔ 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
3735, 23, 36syl2anc 694 . . . . . 6 (𝜑 → (if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑁) ↔ 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
3834, 37mpbird 247 . . . . 5 (𝜑 → if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑁))
391, 2, 3, 4, 10, 38ruclem9 15011 . . . 4 (𝜑 → ((1st ‘(𝐺𝑁)) ≤ (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ∧ (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ≤ (2nd ‘(𝐺𝑁))))
4039simprd 478 . . 3 (𝜑 → (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ≤ (2nd ‘(𝐺𝑁)))
4114, 30, 17, 32, 40ltletrd 10235 . 2 (𝜑 → (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺𝑁)))
429, 14, 17, 28, 41lelttrd 10233 1 (𝜑 → (1st ‘(𝐺𝑀)) < (2nd ‘(𝐺𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  csb 3566  cun 3605  ifcif 4119  {csn 4210  cop 4216   class class class wbr 4685   × cxp 5141  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  cle 10113   / cdiv 10722  cn 11058  2c2 11108  0cn0 11330  cz 11415  cuz 11725  seqcseq 12841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-seq 12842
This theorem is referenced by:  ruclem11  15013  ruclem12  15014
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