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Theorem ruclem8 16273
Description: Lemma for ruc 16279. 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 6911 . . . . 5 (𝑘 = 0 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺‘0)))
2 2fveq3 6911 . . . . 5 (𝑘 = 0 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺‘0)))
31, 2breq12d 5156 . . . 4 (𝑘 = 0 → ((1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘)) ↔ (1st ‘(𝐺‘0)) < (2nd ‘(𝐺‘0))))
43imbi2d 340 . . 3 (𝑘 = 0 → ((𝜑 → (1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘))) ↔ (𝜑 → (1st ‘(𝐺‘0)) < (2nd ‘(𝐺‘0)))))
5 2fveq3 6911 . . . . 5 (𝑘 = 𝑛 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑛)))
6 2fveq3 6911 . . . . 5 (𝑘 = 𝑛 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑛)))
75, 6breq12d 5156 . . . 4 (𝑘 = 𝑛 → ((1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛))))
87imbi2d 340 . . 3 (𝑘 = 𝑛 → ((𝜑 → (1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘))) ↔ (𝜑 → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))))
9 2fveq3 6911 . . . . 5 (𝑘 = (𝑛 + 1) → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺‘(𝑛 + 1))))
10 2fveq3 6911 . . . . 5 (𝑘 = (𝑛 + 1) → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺‘(𝑛 + 1))))
119, 10breq12d 5156 . . . 4 (𝑘 = (𝑛 + 1) → ((1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘)) ↔ (1st ‘(𝐺‘(𝑛 + 1))) < (2nd ‘(𝐺‘(𝑛 + 1)))))
1211imbi2d 340 . . 3 (𝑘 = (𝑛 + 1) → ((𝜑 → (1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘))) ↔ (𝜑 → (1st ‘(𝐺‘(𝑛 + 1))) < (2nd ‘(𝐺‘(𝑛 + 1))))))
13 2fveq3 6911 . . . . 5 (𝑘 = 𝑁 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑁)))
14 2fveq3 6911 . . . . 5 (𝑘 = 𝑁 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑁)))
1513, 14breq12d 5156 . . . 4 (𝑘 = 𝑁 → ((1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑁)) < (2nd ‘(𝐺𝑁))))
1615imbi2d 340 . . 3 (𝑘 = 𝑁 → ((𝜑 → (1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘))) ↔ (𝜑 → (1st ‘(𝐺𝑁)) < (2nd ‘(𝐺𝑁)))))
17 0lt1 11785 . . . . 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 16270 . . . . . 6 (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
2423fveq2d 6910 . . . . 5 (𝜑 → (1st ‘(𝐺‘0)) = (1st ‘⟨0, 1⟩))
25 c0ex 11255 . . . . . 6 0 ∈ V
26 1ex 11257 . . . . . 6 1 ∈ V
2725, 26op1st 8022 . . . . 5 (1st ‘⟨0, 1⟩) = 0
2824, 27eqtrdi 2793 . . . 4 (𝜑 → (1st ‘(𝐺‘0)) = 0)
2923fveq2d 6910 . . . . 5 (𝜑 → (2nd ‘(𝐺‘0)) = (2nd ‘⟨0, 1⟩))
3025, 26op2nd 8023 . . . . 5 (2nd ‘⟨0, 1⟩) = 1
3129, 30eqtrdi 2793 . . . 4 (𝜑 → (2nd ‘(𝐺‘0)) = 1)
3218, 28, 313brtr4d 5175 . . 3 (𝜑 → (1st ‘(𝐺‘0)) < (2nd ‘(𝐺‘0)))
3319adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → 𝐹:ℕ⟶ℝ)
3420adantr 480 . . . . . . . . 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 16271 . . . . . . . . . . . 12 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
3635ffvelcdmda 7104 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ (ℝ × ℝ))
3736adantrr 717 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (𝐺𝑛) ∈ (ℝ × ℝ))
38 xp1st 8046 . . . . . . . . . 10 ((𝐺𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
3937, 38syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (1st ‘(𝐺𝑛)) ∈ ℝ)
40 xp2nd 8047 . . . . . . . . . 10 ((𝐺𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
4137, 40syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
42 nn0p1nn 12565 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
43 ffvelcdm 7101 . . . . . . . . . . 11 ((𝐹:ℕ⟶ℝ ∧ (𝑛 + 1) ∈ ℕ) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
4419, 42, 43syl2an 596 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ0) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
4544adantrr 717 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
46 eqid 2737 . . . . . . . . 9 (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) = (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
47 eqid 2737 . . . . . . . . 9 (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) = (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
48 simprr 773 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))
4933, 34, 39, 41, 45, 46, 47, 48ruclem2 16268 . . . . . . . 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 1144 . . . . . . 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 16272 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ0) → (𝐺‘(𝑛 + 1)) = ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))))
5251adantrr 717 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (𝐺‘(𝑛 + 1)) = ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))))
53 1st2nd2 8053 . . . . . . . . . . 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 7446 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))) = (⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
5652, 55eqtrd 2777 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (𝐺‘(𝑛 + 1)) = (⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
5756fveq2d 6910 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (1st ‘(𝐺‘(𝑛 + 1))) = (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
5856fveq2d 6910 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (2nd ‘(𝐺‘(𝑛 + 1))) = (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
5950, 57, 583brtr4d 5175 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (1st ‘(𝐺‘(𝑛 + 1))) < (2nd ‘(𝐺‘(𝑛 + 1))))
6059expr 456 . . . . 5 ((𝜑𝑛 ∈ ℕ0) → ((1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)) → (1st ‘(𝐺‘(𝑛 + 1))) < (2nd ‘(𝐺‘(𝑛 + 1)))))
6160expcom 413 . . . 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 12713 . 2 (𝑁 ∈ ℕ0 → (𝜑 → (1st ‘(𝐺𝑁)) < (2nd ‘(𝐺𝑁))))
6463impcom 407 1 ((𝜑𝑁 ∈ ℕ0) → (1st ‘(𝐺𝑁)) < (2nd ‘(𝐺𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  csb 3899  cun 3949  ifcif 4525  {csn 4626  cop 4632   class class class wbr 5143   × cxp 5683  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   < clt 11295  cle 11296   / cdiv 11920  cn 12266  2c2 12321  0cn0 12526  seqcseq 14042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-seq 14043
This theorem is referenced by:  ruclem9  16274  ruclem10  16275  ruclem12  16277
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