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Theorem ruclem8 15578
Description: Lemma for ruc 15584. The intervals of the 𝐺 sequence are all nonempty. (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)
Assertion
Ref Expression
ruclem8 ((𝜑𝑁 ∈ ℕ0) → (1st ‘(𝐺𝑁)) < (2nd ‘(𝐺𝑁)))
Distinct variable groups:   𝑥,𝑚,𝑦,𝐹   𝑚,𝐺,𝑥,𝑦   𝑚,𝑁,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑚)   𝐶(𝑥,𝑦,𝑚)   𝐷(𝑥,𝑦,𝑚)

Proof of Theorem ruclem8
Dummy variables 𝑛 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2fveq3 6668 . . . . 5 (𝑘 = 0 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺‘0)))
2 2fveq3 6668 . . . . 5 (𝑘 = 0 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺‘0)))
31, 2breq12d 5070 . . . 4 (𝑘 = 0 → ((1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘)) ↔ (1st ‘(𝐺‘0)) < (2nd ‘(𝐺‘0))))
43imbi2d 342 . . 3 (𝑘 = 0 → ((𝜑 → (1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘))) ↔ (𝜑 → (1st ‘(𝐺‘0)) < (2nd ‘(𝐺‘0)))))
5 2fveq3 6668 . . . . 5 (𝑘 = 𝑛 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑛)))
6 2fveq3 6668 . . . . 5 (𝑘 = 𝑛 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑛)))
75, 6breq12d 5070 . . . 4 (𝑘 = 𝑛 → ((1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛))))
87imbi2d 342 . . 3 (𝑘 = 𝑛 → ((𝜑 → (1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘))) ↔ (𝜑 → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))))
9 2fveq3 6668 . . . . 5 (𝑘 = (𝑛 + 1) → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺‘(𝑛 + 1))))
10 2fveq3 6668 . . . . 5 (𝑘 = (𝑛 + 1) → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺‘(𝑛 + 1))))
119, 10breq12d 5070 . . . 4 (𝑘 = (𝑛 + 1) → ((1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘)) ↔ (1st ‘(𝐺‘(𝑛 + 1))) < (2nd ‘(𝐺‘(𝑛 + 1)))))
1211imbi2d 342 . . 3 (𝑘 = (𝑛 + 1) → ((𝜑 → (1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘))) ↔ (𝜑 → (1st ‘(𝐺‘(𝑛 + 1))) < (2nd ‘(𝐺‘(𝑛 + 1))))))
13 2fveq3 6668 . . . . 5 (𝑘 = 𝑁 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑁)))
14 2fveq3 6668 . . . . 5 (𝑘 = 𝑁 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑁)))
1513, 14breq12d 5070 . . . 4 (𝑘 = 𝑁 → ((1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑁)) < (2nd ‘(𝐺𝑁))))
1615imbi2d 342 . . 3 (𝑘 = 𝑁 → ((𝜑 → (1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘))) ↔ (𝜑 → (1st ‘(𝐺𝑁)) < (2nd ‘(𝐺𝑁)))))
17 0lt1 11150 . . . . 5 0 < 1
1817a1i 11 . . . 4 (𝜑 → 0 < 1)
19 ruc.1 . . . . . . 7 (𝜑𝐹:ℕ⟶ℝ)
20 ruc.2 . . . . . . 7 (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
21 ruc.4 . . . . . . 7 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
22 ruc.5 . . . . . . 7 𝐺 = seq0(𝐷, 𝐶)
2319, 20, 21, 22ruclem4 15575 . . . . . 6 (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
2423fveq2d 6667 . . . . 5 (𝜑 → (1st ‘(𝐺‘0)) = (1st ‘⟨0, 1⟩))
25 c0ex 10623 . . . . . 6 0 ∈ V
26 1ex 10625 . . . . . 6 1 ∈ V
2725, 26op1st 7686 . . . . 5 (1st ‘⟨0, 1⟩) = 0
2824, 27syl6eq 2869 . . . 4 (𝜑 → (1st ‘(𝐺‘0)) = 0)
2923fveq2d 6667 . . . . 5 (𝜑 → (2nd ‘(𝐺‘0)) = (2nd ‘⟨0, 1⟩))
3025, 26op2nd 7687 . . . . 5 (2nd ‘⟨0, 1⟩) = 1
3129, 30syl6eq 2869 . . . 4 (𝜑 → (2nd ‘(𝐺‘0)) = 1)
3218, 28, 313brtr4d 5089 . . 3 (𝜑 → (1st ‘(𝐺‘0)) < (2nd ‘(𝐺‘0)))
3319adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → 𝐹:ℕ⟶ℝ)
3420adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
3519, 20, 21, 22ruclem6 15576 . . . . . . . . . . . 12 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
3635ffvelrnda 6843 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ (ℝ × ℝ))
3736adantrr 713 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (𝐺𝑛) ∈ (ℝ × ℝ))
38 xp1st 7710 . . . . . . . . . 10 ((𝐺𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
3937, 38syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (1st ‘(𝐺𝑛)) ∈ ℝ)
40 xp2nd 7711 . . . . . . . . . 10 ((𝐺𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
4137, 40syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
42 nn0p1nn 11924 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
43 ffvelrn 6841 . . . . . . . . . . 11 ((𝐹:ℕ⟶ℝ ∧ (𝑛 + 1) ∈ ℕ) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
4419, 42, 43syl2an 595 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ0) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
4544adantrr 713 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
46 eqid 2818 . . . . . . . . 9 (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) = (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
47 eqid 2818 . . . . . . . . 9 (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) = (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
48 simprr 769 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))
4933, 34, 39, 41, 45, 46, 47, 48ruclem2 15573 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → ((1st ‘(𝐺𝑛)) ≤ (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) ∧ (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) < (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) ∧ (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) ≤ (2nd ‘(𝐺𝑛))))
5049simp2d 1135 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) < (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
5119, 20, 21, 22ruclem7 15577 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ0) → (𝐺‘(𝑛 + 1)) = ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))))
5251adantrr 713 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (𝐺‘(𝑛 + 1)) = ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))))
53 1st2nd2 7717 . . . . . . . . . . 11 ((𝐺𝑛) ∈ (ℝ × ℝ) → (𝐺𝑛) = ⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩)
5437, 53syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (𝐺𝑛) = ⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩)
5554oveq1d 7160 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))) = (⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
5652, 55eqtrd 2853 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (𝐺‘(𝑛 + 1)) = (⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
5756fveq2d 6667 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (1st ‘(𝐺‘(𝑛 + 1))) = (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
5856fveq2d 6667 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (2nd ‘(𝐺‘(𝑛 + 1))) = (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
5950, 57, 583brtr4d 5089 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (1st ‘(𝐺‘(𝑛 + 1))) < (2nd ‘(𝐺‘(𝑛 + 1))))
6059expr 457 . . . . 5 ((𝜑𝑛 ∈ ℕ0) → ((1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)) → (1st ‘(𝐺‘(𝑛 + 1))) < (2nd ‘(𝐺‘(𝑛 + 1)))))
6160expcom 414 . . . 4 (𝑛 ∈ ℕ0 → (𝜑 → ((1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)) → (1st ‘(𝐺‘(𝑛 + 1))) < (2nd ‘(𝐺‘(𝑛 + 1))))))
6261a2d 29 . . 3 (𝑛 ∈ ℕ0 → ((𝜑 → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛))) → (𝜑 → (1st ‘(𝐺‘(𝑛 + 1))) < (2nd ‘(𝐺‘(𝑛 + 1))))))
634, 8, 12, 16, 32, 62nn0ind 12065 . 2 (𝑁 ∈ ℕ0 → (𝜑 → (1st ‘(𝐺𝑁)) < (2nd ‘(𝐺𝑁))))
6463impcom 408 1 ((𝜑𝑁 ∈ ℕ0) → (1st ‘(𝐺𝑁)) < (2nd ‘(𝐺𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  csb 3880  cun 3931  ifcif 4463  {csn 4557  cop 4563   class class class wbr 5057   × cxp 5546  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  1st c1st 7676  2nd c2nd 7677  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   < clt 10663  cle 10664   / cdiv 11285  cn 11626  2c2 11680  0cn0 11885  seqcseq 13357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-seq 13358
This theorem is referenced by:  ruclem9  15579  ruclem10  15580  ruclem12  15582
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