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Theorem ruclem10 15376
 Description: Lemma for ruc 15380. 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 15372 . . . 4 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
6 ruclem10.6 . . . 4 (𝜑𝑀 ∈ ℕ0)
75, 6ffvelrnd 6626 . . 3 (𝜑 → (𝐺𝑀) ∈ (ℝ × ℝ))
8 xp1st 7479 . . 3 ((𝐺𝑀) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑀)) ∈ ℝ)
97, 8syl 17 . 2 (𝜑 → (1st ‘(𝐺𝑀)) ∈ ℝ)
10 ruclem10.7 . . . . 5 (𝜑𝑁 ∈ ℕ0)
1110, 6ifcld 4352 . . . 4 (𝜑 → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℕ0)
125, 11ffvelrnd 6626 . . 3 (𝜑 → (𝐺‘if(𝑀𝑁, 𝑁, 𝑀)) ∈ (ℝ × ℝ))
13 xp1st 7479 . . 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 6626 . . 3 (𝜑 → (𝐺𝑁) ∈ (ℝ × ℝ))
16 xp2nd 7480 . . 3 ((𝐺𝑁) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑁)) ∈ ℝ)
1715, 16syl 17 . 2 (𝜑 → (2nd ‘(𝐺𝑁)) ∈ ℝ)
186nn0red 11707 . . . . . 6 (𝜑𝑀 ∈ ℝ)
1910nn0red 11707 . . . . . 6 (𝜑𝑁 ∈ ℝ)
20 max1 12332 . . . . . 6 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀))
2118, 19, 20syl2anc 579 . . . . 5 (𝜑𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀))
226nn0zd 11836 . . . . . 6 (𝜑𝑀 ∈ ℤ)
2311nn0zd 11836 . . . . . 6 (𝜑 → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℤ)
24 eluz 12010 . . . . . 6 ((𝑀 ∈ ℤ ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℤ) → (if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑀) ↔ 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
2522, 23, 24syl2anc 579 . . . . 5 (𝜑 → (if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑀) ↔ 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
2621, 25mpbird 249 . . . 4 (𝜑 → if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑀))
271, 2, 3, 4, 6, 26ruclem9 15375 . . 3 (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ∧ (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ≤ (2nd ‘(𝐺𝑀))))
2827simpld 490 . 2 (𝜑 → (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))))
29 xp2nd 7480 . . . 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 15374 . . . 4 ((𝜑 ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℕ0) → (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))))
3211, 31mpdan 677 . . 3 (𝜑 → (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))))
33 max2 12334 . . . . . . 7 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀))
3418, 19, 33syl2anc 579 . . . . . 6 (𝜑𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀))
3510nn0zd 11836 . . . . . . 7 (𝜑𝑁 ∈ ℤ)
36 eluz 12010 . . . . . . 7 ((𝑁 ∈ ℤ ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℤ) → (if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑁) ↔ 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
3735, 23, 36syl2anc 579 . . . . . 6 (𝜑 → (if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑁) ↔ 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
3834, 37mpbird 249 . . . . 5 (𝜑 → if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑁))
391, 2, 3, 4, 10, 38ruclem9 15375 . . . 4 (𝜑 → ((1st ‘(𝐺𝑁)) ≤ (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ∧ (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ≤ (2nd ‘(𝐺𝑁))))
4039simprd 491 . . 3 (𝜑 → (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ≤ (2nd ‘(𝐺𝑁)))
4114, 30, 17, 32, 40ltletrd 10538 . 2 (𝜑 → (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺𝑁)))
429, 14, 17, 28, 41lelttrd 10536 1 (𝜑 → (1st ‘(𝐺𝑀)) < (2nd ‘(𝐺𝑁)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   = wceq 1601   ∈ wcel 2107  ⦋csb 3751   ∪ cun 3790  ifcif 4307  {csn 4398  ⟨cop 4404   class class class wbr 4888   × cxp 5355  ⟶wf 6133  ‘cfv 6137  (class class class)co 6924   ↦ cmpt2 6926  1st c1st 7445  2nd c2nd 7446  ℝcr 10273  0cc0 10274  1c1 10275   + caddc 10277   < clt 10413   ≤ cle 10414   / cdiv 11034  ℕcn 11378  2c2 11434  ℕ0cn0 11646  ℤcz 11732  ℤ≥cuz 11996  seqcseq 13123 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-er 8028  df-en 8244  df-dom 8245  df-sdom 8246  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-div 11035  df-nn 11379  df-2 11442  df-n0 11647  df-z 11733  df-uz 11997  df-fz 12648  df-seq 13124 This theorem is referenced by:  ruclem11  15377  ruclem12  15378
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