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Theorem ruclem8 15584
 Description: Lemma for ruc 15590. 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 6650 . . . . 5 (𝑘 = 0 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺‘0)))
2 2fveq3 6650 . . . . 5 (𝑘 = 0 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺‘0)))
31, 2breq12d 5043 . . . 4 (𝑘 = 0 → ((1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘)) ↔ (1st ‘(𝐺‘0)) < (2nd ‘(𝐺‘0))))
43imbi2d 344 . . 3 (𝑘 = 0 → ((𝜑 → (1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘))) ↔ (𝜑 → (1st ‘(𝐺‘0)) < (2nd ‘(𝐺‘0)))))
5 2fveq3 6650 . . . . 5 (𝑘 = 𝑛 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑛)))
6 2fveq3 6650 . . . . 5 (𝑘 = 𝑛 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑛)))
75, 6breq12d 5043 . . . 4 (𝑘 = 𝑛 → ((1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛))))
87imbi2d 344 . . 3 (𝑘 = 𝑛 → ((𝜑 → (1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘))) ↔ (𝜑 → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))))
9 2fveq3 6650 . . . . 5 (𝑘 = (𝑛 + 1) → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺‘(𝑛 + 1))))
10 2fveq3 6650 . . . . 5 (𝑘 = (𝑛 + 1) → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺‘(𝑛 + 1))))
119, 10breq12d 5043 . . . 4 (𝑘 = (𝑛 + 1) → ((1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘)) ↔ (1st ‘(𝐺‘(𝑛 + 1))) < (2nd ‘(𝐺‘(𝑛 + 1)))))
1211imbi2d 344 . . 3 (𝑘 = (𝑛 + 1) → ((𝜑 → (1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘))) ↔ (𝜑 → (1st ‘(𝐺‘(𝑛 + 1))) < (2nd ‘(𝐺‘(𝑛 + 1))))))
13 2fveq3 6650 . . . . 5 (𝑘 = 𝑁 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑁)))
14 2fveq3 6650 . . . . 5 (𝑘 = 𝑁 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑁)))
1513, 14breq12d 5043 . . . 4 (𝑘 = 𝑁 → ((1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑁)) < (2nd ‘(𝐺𝑁))))
1615imbi2d 344 . . 3 (𝑘 = 𝑁 → ((𝜑 → (1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘))) ↔ (𝜑 → (1st ‘(𝐺𝑁)) < (2nd ‘(𝐺𝑁)))))
17 0lt1 11153 . . . . 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 15581 . . . . . 6 (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
2423fveq2d 6649 . . . . 5 (𝜑 → (1st ‘(𝐺‘0)) = (1st ‘⟨0, 1⟩))
25 c0ex 10626 . . . . . 6 0 ∈ V
26 1ex 10628 . . . . . 6 1 ∈ V
2725, 26op1st 7681 . . . . 5 (1st ‘⟨0, 1⟩) = 0
2824, 27eqtrdi 2849 . . . 4 (𝜑 → (1st ‘(𝐺‘0)) = 0)
2923fveq2d 6649 . . . . 5 (𝜑 → (2nd ‘(𝐺‘0)) = (2nd ‘⟨0, 1⟩))
3025, 26op2nd 7682 . . . . 5 (2nd ‘⟨0, 1⟩) = 1
3129, 30eqtrdi 2849 . . . 4 (𝜑 → (2nd ‘(𝐺‘0)) = 1)
3218, 28, 313brtr4d 5062 . . 3 (𝜑 → (1st ‘(𝐺‘0)) < (2nd ‘(𝐺‘0)))
3319adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → 𝐹:ℕ⟶ℝ)
3420adantr 484 . . . . . . . . 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 15582 . . . . . . . . . . . 12 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
3635ffvelrnda 6828 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ (ℝ × ℝ))
3736adantrr 716 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (𝐺𝑛) ∈ (ℝ × ℝ))
38 xp1st 7705 . . . . . . . . . 10 ((𝐺𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
3937, 38syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (1st ‘(𝐺𝑛)) ∈ ℝ)
40 xp2nd 7706 . . . . . . . . . 10 ((𝐺𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
4137, 40syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
42 nn0p1nn 11926 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
43 ffvelrn 6826 . . . . . . . . . . 11 ((𝐹:ℕ⟶ℝ ∧ (𝑛 + 1) ∈ ℕ) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
4419, 42, 43syl2an 598 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ0) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
4544adantrr 716 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
46 eqid 2798 . . . . . . . . 9 (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) = (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
47 eqid 2798 . . . . . . . . 9 (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) = (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
48 simprr 772 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))
4933, 34, 39, 41, 45, 46, 47, 48ruclem2 15579 . . . . . . . 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 1140 . . . . . . 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 15583 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ0) → (𝐺‘(𝑛 + 1)) = ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))))
5251adantrr 716 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (𝐺‘(𝑛 + 1)) = ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))))
53 1st2nd2 7712 . . . . . . . . . . 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 7150 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))) = (⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
5652, 55eqtrd 2833 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (𝐺‘(𝑛 + 1)) = (⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
5756fveq2d 6649 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (1st ‘(𝐺‘(𝑛 + 1))) = (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
5856fveq2d 6649 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (2nd ‘(𝐺‘(𝑛 + 1))) = (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
5950, 57, 583brtr4d 5062 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (1st ‘(𝐺‘(𝑛 + 1))) < (2nd ‘(𝐺‘(𝑛 + 1))))
6059expr 460 . . . . 5 ((𝜑𝑛 ∈ ℕ0) → ((1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)) → (1st ‘(𝐺‘(𝑛 + 1))) < (2nd ‘(𝐺‘(𝑛 + 1)))))
6160expcom 417 . . . 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 12067 . 2 (𝑁 ∈ ℕ0 → (𝜑 → (1st ‘(𝐺𝑁)) < (2nd ‘(𝐺𝑁))))
6463impcom 411 1 ((𝜑𝑁 ∈ ℕ0) → (1st ‘(𝐺𝑁)) < (2nd ‘(𝐺𝑁)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ⦋csb 3828   ∪ cun 3879  ifcif 4425  {csn 4525  ⟨cop 4531   class class class wbr 5030   × cxp 5517  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  1st c1st 7671  2nd c2nd 7672  ℝcr 10527  0cc0 10528  1c1 10529   + caddc 10531   < clt 10666   ≤ cle 10667   / cdiv 11288  ℕcn 11627  2c2 11682  ℕ0cn0 11887  seqcseq 13366 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-cnex 10584  ax-resscn 10585  ax-1cn 10586  ax-icn 10587  ax-addcl 10588  ax-addrcl 10589  ax-mulcl 10590  ax-mulrcl 10591  ax-mulcom 10592  ax-addass 10593  ax-mulass 10594  ax-distr 10595  ax-i2m1 10596  ax-1ne0 10597  ax-1rid 10598  ax-rnegex 10599  ax-rrecex 10600  ax-cnre 10601  ax-pre-lttri 10602  ax-pre-lttrn 10603  ax-pre-ltadd 10604  ax-pre-mulgt0 10605 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-1st 7673  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-pnf 10668  df-mnf 10669  df-xr 10670  df-ltxr 10671  df-le 10672  df-sub 10863  df-neg 10864  df-div 11289  df-nn 11628  df-2 11690  df-n0 11888  df-z 11972  df-uz 12234  df-fz 12888  df-seq 13367 This theorem is referenced by:  ruclem9  15585  ruclem10  15586  ruclem12  15588
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