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Theorem ruclem8 16234
Description: Lemma for ruc 16240. 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 6898 . . . . 5 (𝑘 = 0 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺‘0)))
2 2fveq3 6898 . . . . 5 (𝑘 = 0 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺‘0)))
31, 2breq12d 5158 . . . 4 (𝑘 = 0 → ((1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘)) ↔ (1st ‘(𝐺‘0)) < (2nd ‘(𝐺‘0))))
43imbi2d 339 . . 3 (𝑘 = 0 → ((𝜑 → (1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘))) ↔ (𝜑 → (1st ‘(𝐺‘0)) < (2nd ‘(𝐺‘0)))))
5 2fveq3 6898 . . . . 5 (𝑘 = 𝑛 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑛)))
6 2fveq3 6898 . . . . 5 (𝑘 = 𝑛 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑛)))
75, 6breq12d 5158 . . . 4 (𝑘 = 𝑛 → ((1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛))))
87imbi2d 339 . . 3 (𝑘 = 𝑛 → ((𝜑 → (1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘))) ↔ (𝜑 → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))))
9 2fveq3 6898 . . . . 5 (𝑘 = (𝑛 + 1) → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺‘(𝑛 + 1))))
10 2fveq3 6898 . . . . 5 (𝑘 = (𝑛 + 1) → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺‘(𝑛 + 1))))
119, 10breq12d 5158 . . . 4 (𝑘 = (𝑛 + 1) → ((1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘)) ↔ (1st ‘(𝐺‘(𝑛 + 1))) < (2nd ‘(𝐺‘(𝑛 + 1)))))
1211imbi2d 339 . . 3 (𝑘 = (𝑛 + 1) → ((𝜑 → (1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘))) ↔ (𝜑 → (1st ‘(𝐺‘(𝑛 + 1))) < (2nd ‘(𝐺‘(𝑛 + 1))))))
13 2fveq3 6898 . . . . 5 (𝑘 = 𝑁 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑁)))
14 2fveq3 6898 . . . . 5 (𝑘 = 𝑁 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑁)))
1513, 14breq12d 5158 . . . 4 (𝑘 = 𝑁 → ((1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑁)) < (2nd ‘(𝐺𝑁))))
1615imbi2d 339 . . 3 (𝑘 = 𝑁 → ((𝜑 → (1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑘))) ↔ (𝜑 → (1st ‘(𝐺𝑁)) < (2nd ‘(𝐺𝑁)))))
17 0lt1 11777 . . . . 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 16231 . . . . . 6 (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
2423fveq2d 6897 . . . . 5 (𝜑 → (1st ‘(𝐺‘0)) = (1st ‘⟨0, 1⟩))
25 c0ex 11249 . . . . . 6 0 ∈ V
26 1ex 11251 . . . . . 6 1 ∈ V
2725, 26op1st 8003 . . . . 5 (1st ‘⟨0, 1⟩) = 0
2824, 27eqtrdi 2782 . . . 4 (𝜑 → (1st ‘(𝐺‘0)) = 0)
2923fveq2d 6897 . . . . 5 (𝜑 → (2nd ‘(𝐺‘0)) = (2nd ‘⟨0, 1⟩))
3025, 26op2nd 8004 . . . . 5 (2nd ‘⟨0, 1⟩) = 1
3129, 30eqtrdi 2782 . . . 4 (𝜑 → (2nd ‘(𝐺‘0)) = 1)
3218, 28, 313brtr4d 5177 . . 3 (𝜑 → (1st ‘(𝐺‘0)) < (2nd ‘(𝐺‘0)))
3319adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → 𝐹:ℕ⟶ℝ)
3420adantr 479 . . . . . . . . 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 16232 . . . . . . . . . . . 12 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
3635ffvelcdmda 7090 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ (ℝ × ℝ))
3736adantrr 715 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (𝐺𝑛) ∈ (ℝ × ℝ))
38 xp1st 8027 . . . . . . . . . 10 ((𝐺𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
3937, 38syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (1st ‘(𝐺𝑛)) ∈ ℝ)
40 xp2nd 8028 . . . . . . . . . 10 ((𝐺𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
4137, 40syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
42 nn0p1nn 12557 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
43 ffvelcdm 7087 . . . . . . . . . . 11 ((𝐹:ℕ⟶ℝ ∧ (𝑛 + 1) ∈ ℕ) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
4419, 42, 43syl2an 594 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ0) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
4544adantrr 715 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
46 eqid 2726 . . . . . . . . 9 (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) = (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
47 eqid 2726 . . . . . . . . 9 (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) = (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
48 simprr 771 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))
4933, 34, 39, 41, 45, 46, 47, 48ruclem2 16229 . . . . . . . 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 16233 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ0) → (𝐺‘(𝑛 + 1)) = ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))))
5251adantrr 715 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (𝐺‘(𝑛 + 1)) = ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))))
53 1st2nd2 8034 . . . . . . . . . . 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 7431 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))) = (⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
5652, 55eqtrd 2766 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (𝐺‘(𝑛 + 1)) = (⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
5756fveq2d 6897 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (1st ‘(𝐺‘(𝑛 + 1))) = (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
5856fveq2d 6897 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (2nd ‘(𝐺‘(𝑛 + 1))) = (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
5950, 57, 583brtr4d 5177 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (1st ‘(𝐺‘(𝑛 + 1))) < (2nd ‘(𝐺‘(𝑛 + 1))))
6059expr 455 . . . . 5 ((𝜑𝑛 ∈ ℕ0) → ((1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)) → (1st ‘(𝐺‘(𝑛 + 1))) < (2nd ‘(𝐺‘(𝑛 + 1)))))
6160expcom 412 . . . 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 12703 . 2 (𝑁 ∈ ℕ0 → (𝜑 → (1st ‘(𝐺𝑁)) < (2nd ‘(𝐺𝑁))))
6463impcom 406 1 ((𝜑𝑁 ∈ ℕ0) → (1st ‘(𝐺𝑁)) < (2nd ‘(𝐺𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  csb 3891  cun 3944  ifcif 4523  {csn 4623  cop 4629   class class class wbr 5145   × cxp 5672  wf 6542  cfv 6546  (class class class)co 7416  cmpo 7418  1st c1st 7993  2nd c2nd 7994  cr 11148  0cc0 11149  1c1 11150   + caddc 11152   < clt 11289  cle 11290   / cdiv 11912  cn 12258  2c2 12313  0cn0 12518  seqcseq 14015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7995  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-er 8726  df-en 8967  df-dom 8968  df-sdom 8969  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-div 11913  df-nn 12259  df-2 12321  df-n0 12519  df-z 12605  df-uz 12869  df-fz 13533  df-seq 14016
This theorem is referenced by:  ruclem9  16235  ruclem10  16236  ruclem12  16238
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