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Theorem ruclem8 16119
Description: Lemma for ruc 16125. 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 6847 . . . . 5 (𝑘 = 0 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺‘0)))
2 2fveq3 6847 . . . . 5 (𝑘 = 0 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺‘0)))
31, 2breq12d 5118 . . . 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 6847 . . . . 5 (𝑘 = 𝑛 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑛)))
6 2fveq3 6847 . . . . 5 (𝑘 = 𝑛 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑛)))
75, 6breq12d 5118 . . . 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 6847 . . . . 5 (𝑘 = (𝑛 + 1) → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺‘(𝑛 + 1))))
10 2fveq3 6847 . . . . 5 (𝑘 = (𝑛 + 1) → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺‘(𝑛 + 1))))
119, 10breq12d 5118 . . . 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 6847 . . . . 5 (𝑘 = 𝑁 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑁)))
14 2fveq3 6847 . . . . 5 (𝑘 = 𝑁 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑁)))
1513, 14breq12d 5118 . . . 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 11677 . . . . 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 16116 . . . . . 6 (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
2423fveq2d 6846 . . . . 5 (𝜑 → (1st ‘(𝐺‘0)) = (1st ‘⟨0, 1⟩))
25 c0ex 11149 . . . . . 6 0 ∈ V
26 1ex 11151 . . . . . 6 1 ∈ V
2725, 26op1st 7929 . . . . 5 (1st ‘⟨0, 1⟩) = 0
2824, 27eqtrdi 2792 . . . 4 (𝜑 → (1st ‘(𝐺‘0)) = 0)
2923fveq2d 6846 . . . . 5 (𝜑 → (2nd ‘(𝐺‘0)) = (2nd ‘⟨0, 1⟩))
3025, 26op2nd 7930 . . . . 5 (2nd ‘⟨0, 1⟩) = 1
3129, 30eqtrdi 2792 . . . 4 (𝜑 → (2nd ‘(𝐺‘0)) = 1)
3218, 28, 313brtr4d 5137 . . 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 16117 . . . . . . . . . . . 12 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
3635ffvelcdmda 7035 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ (ℝ × ℝ))
3736adantrr 715 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (𝐺𝑛) ∈ (ℝ × ℝ))
38 xp1st 7953 . . . . . . . . . 10 ((𝐺𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
3937, 38syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (1st ‘(𝐺𝑛)) ∈ ℝ)
40 xp2nd 7954 . . . . . . . . . 10 ((𝐺𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
4137, 40syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
42 nn0p1nn 12452 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
43 ffvelcdm 7032 . . . . . . . . . . 11 ((𝐹:ℕ⟶ℝ ∧ (𝑛 + 1) ∈ ℕ) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
4419, 42, 43syl2an 596 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ0) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
4544adantrr 715 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
46 eqid 2736 . . . . . . . . 9 (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) = (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
47 eqid 2736 . . . . . . . . 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 16114 . . . . . . . 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 1143 . . . . . . 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 16118 . . . . . . . . . 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 7960 . . . . . . . . . . 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 7372 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))) = (⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
5652, 55eqtrd 2776 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (𝐺‘(𝑛 + 1)) = (⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
5756fveq2d 6846 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (1st ‘(𝐺‘(𝑛 + 1))) = (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
5856fveq2d 6846 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))) → (2nd ‘(𝐺‘(𝑛 + 1))) = (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
5950, 57, 583brtr4d 5137 . . . . . 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 12598 . 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 1541  wcel 2106  csb 3855  cun 3908  ifcif 4486  {csn 4586  cop 4592   class class class wbr 5105   × cxp 5631  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   < clt 11189  cle 11190   / cdiv 11812  cn 12153  2c2 12208  0cn0 12413  seqcseq 13906
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-seq 13907
This theorem is referenced by:  ruclem9  16120  ruclem10  16121  ruclem12  16123
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