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Theorem ruclem10 15581
 Description: Lemma for ruc 15585. 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 15577 . . . 4 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
6 ruclem10.6 . . . 4 (𝜑𝑀 ∈ ℕ0)
75, 6ffvelrnd 6833 . . 3 (𝜑 → (𝐺𝑀) ∈ (ℝ × ℝ))
8 xp1st 7704 . . 3 ((𝐺𝑀) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑀)) ∈ ℝ)
97, 8syl 17 . 2 (𝜑 → (1st ‘(𝐺𝑀)) ∈ ℝ)
10 ruclem10.7 . . . . 5 (𝜑𝑁 ∈ ℕ0)
1110, 6ifcld 4493 . . . 4 (𝜑 → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℕ0)
125, 11ffvelrnd 6833 . . 3 (𝜑 → (𝐺‘if(𝑀𝑁, 𝑁, 𝑀)) ∈ (ℝ × ℝ))
13 xp1st 7704 . . 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 6833 . . 3 (𝜑 → (𝐺𝑁) ∈ (ℝ × ℝ))
16 xp2nd 7705 . . 3 ((𝐺𝑁) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑁)) ∈ ℝ)
1715, 16syl 17 . 2 (𝜑 → (2nd ‘(𝐺𝑁)) ∈ ℝ)
186nn0red 11942 . . . . . 6 (𝜑𝑀 ∈ ℝ)
1910nn0red 11942 . . . . . 6 (𝜑𝑁 ∈ ℝ)
20 max1 12564 . . . . . 6 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀))
2118, 19, 20syl2anc 587 . . . . 5 (𝜑𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀))
226nn0zd 12071 . . . . . 6 (𝜑𝑀 ∈ ℤ)
2311nn0zd 12071 . . . . . 6 (𝜑 → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℤ)
24 eluz 12243 . . . . . 6 ((𝑀 ∈ ℤ ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℤ) → (if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑀) ↔ 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
2522, 23, 24syl2anc 587 . . . . 5 (𝜑 → (if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑀) ↔ 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
2621, 25mpbird 260 . . . 4 (𝜑 → if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑀))
271, 2, 3, 4, 6, 26ruclem9 15580 . . 3 (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ∧ (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ≤ (2nd ‘(𝐺𝑀))))
2827simpld 498 . 2 (𝜑 → (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))))
29 xp2nd 7705 . . . 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 15579 . . . 4 ((𝜑 ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℕ0) → (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))))
3211, 31mpdan 686 . . 3 (𝜑 → (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))))
33 max2 12566 . . . . . . 7 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀))
3418, 19, 33syl2anc 587 . . . . . 6 (𝜑𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀))
3510nn0zd 12071 . . . . . . 7 (𝜑𝑁 ∈ ℤ)
36 eluz 12243 . . . . . . 7 ((𝑁 ∈ ℤ ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℤ) → (if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑁) ↔ 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
3735, 23, 36syl2anc 587 . . . . . 6 (𝜑 → (if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑁) ↔ 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
3834, 37mpbird 260 . . . . 5 (𝜑 → if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑁))
391, 2, 3, 4, 10, 38ruclem9 15580 . . . 4 (𝜑 → ((1st ‘(𝐺𝑁)) ≤ (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ∧ (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ≤ (2nd ‘(𝐺𝑁))))
4039simprd 499 . . 3 (𝜑 → (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ≤ (2nd ‘(𝐺𝑁)))
4114, 30, 17, 32, 40ltletrd 10785 . 2 (𝜑 → (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺𝑁)))
429, 14, 17, 28, 41lelttrd 10783 1 (𝜑 → (1st ‘(𝐺𝑀)) < (2nd ‘(𝐺𝑁)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2115  ⦋csb 3865   ∪ cun 3916  ifcif 4448  {csn 4548  ⟨cop 4554   class class class wbr 5047   × cxp 5534  ⟶wf 6332  ‘cfv 6336  (class class class)co 7138   ∈ cmpo 7140  1st c1st 7670  2nd c2nd 7671  ℝcr 10521  0cc0 10522  1c1 10523   + caddc 10525   < clt 10660   ≤ cle 10661   / cdiv 11282  ℕcn 11623  2c2 11678  ℕ0cn0 11883  ℤcz 11967  ℤ≥cuz 12229  seqcseq 13362 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-cnex 10578  ax-resscn 10579  ax-1cn 10580  ax-icn 10581  ax-addcl 10582  ax-addrcl 10583  ax-mulcl 10584  ax-mulrcl 10585  ax-mulcom 10586  ax-addass 10587  ax-mulass 10588  ax-distr 10589  ax-i2m1 10590  ax-1ne0 10591  ax-1rid 10592  ax-rnegex 10593  ax-rrecex 10594  ax-cnre 10595  ax-pre-lttri 10596  ax-pre-lttrn 10597  ax-pre-ltadd 10598  ax-pre-mulgt0 10599 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  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 7096  df-ov 7141  df-oprab 7142  df-mpo 7143  df-om 7564  df-1st 7672  df-2nd 7673  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10662  df-mnf 10663  df-xr 10664  df-ltxr 10665  df-le 10666  df-sub 10857  df-neg 10858  df-div 11283  df-nn 11624  df-2 11686  df-n0 11884  df-z 11968  df-uz 12230  df-fz 12884  df-seq 13363 This theorem is referenced by:  ruclem11  15582  ruclem12  15583
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