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Theorem ruclem6 15636
 Description: Lemma for ruc 15644. Domain and range of the interval sequence. (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
ruclem6 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
Distinct variable groups:   𝑥,𝑚,𝑦,𝐹   𝑚,𝐺,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑚)   𝐶(𝑥,𝑦,𝑚)   𝐷(𝑥,𝑦,𝑚)

Proof of Theorem ruclem6
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ruc.5 . . . . . . 7 𝐺 = seq0(𝐷, 𝐶)
21fveq1i 6659 . . . . . 6 (𝐺‘0) = (seq0(𝐷, 𝐶)‘0)
3 0z 12031 . . . . . . 7 0 ∈ ℤ
4 seq1 13431 . . . . . . 7 (0 ∈ ℤ → (seq0(𝐷, 𝐶)‘0) = (𝐶‘0))
53, 4ax-mp 5 . . . . . 6 (seq0(𝐷, 𝐶)‘0) = (𝐶‘0)
62, 5eqtri 2781 . . . . 5 (𝐺‘0) = (𝐶‘0)
7 ruc.1 . . . . . 6 (𝜑𝐹:ℕ⟶ℝ)
8 ruc.2 . . . . . 6 (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
9 ruc.4 . . . . . 6 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
107, 8, 9, 1ruclem4 15635 . . . . 5 (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
116, 10syl5eqr 2807 . . . 4 (𝜑 → (𝐶‘0) = ⟨0, 1⟩)
12 0re 10681 . . . . 5 0 ∈ ℝ
13 1re 10679 . . . . 5 1 ∈ ℝ
14 opelxpi 5561 . . . . 5 ((0 ∈ ℝ ∧ 1 ∈ ℝ) → ⟨0, 1⟩ ∈ (ℝ × ℝ))
1512, 13, 14mp2an 691 . . . 4 ⟨0, 1⟩ ∈ (ℝ × ℝ)
1611, 15eqeltrdi 2860 . . 3 (𝜑 → (𝐶‘0) ∈ (ℝ × ℝ))
17 1st2nd2 7732 . . . . . 6 (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
1817ad2antrl 727 . . . . 5 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
1918oveq1d 7165 . . . 4 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (𝑧𝐷𝑤) = (⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤))
207adantr 484 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝐹:ℕ⟶ℝ)
218adantr 484 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
22 xp1st 7725 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
2322ad2antrl 727 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (1st𝑧) ∈ ℝ)
24 xp2nd 7726 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → (2nd𝑧) ∈ ℝ)
2524ad2antrl 727 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (2nd𝑧) ∈ ℝ)
26 simprr 772 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝑤 ∈ ℝ)
27 eqid 2758 . . . . . 6 (1st ‘(⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤)) = (1st ‘(⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤))
28 eqid 2758 . . . . . 6 (2nd ‘(⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤)) = (2nd ‘(⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤))
2920, 21, 23, 25, 26, 27, 28ruclem1 15632 . . . . 5 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → ((⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤) ∈ (ℝ × ℝ) ∧ (1st ‘(⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤)) = if((((1st𝑧) + (2nd𝑧)) / 2) < 𝑤, (1st𝑧), (((((1st𝑧) + (2nd𝑧)) / 2) + (2nd𝑧)) / 2)) ∧ (2nd ‘(⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤)) = if((((1st𝑧) + (2nd𝑧)) / 2) < 𝑤, (((1st𝑧) + (2nd𝑧)) / 2), (2nd𝑧))))
3029simp1d 1139 . . . 4 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤) ∈ (ℝ × ℝ))
3119, 30eqeltrd 2852 . . 3 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (𝑧𝐷𝑤) ∈ (ℝ × ℝ))
32 nn0uz 12320 . . 3 0 = (ℤ‘0)
33 0zd 12032 . . 3 (𝜑 → 0 ∈ ℤ)
34 0p1e1 11796 . . . . . . 7 (0 + 1) = 1
3534fveq2i 6661 . . . . . 6 (ℤ‘(0 + 1)) = (ℤ‘1)
36 nnuz 12321 . . . . . 6 ℕ = (ℤ‘1)
3735, 36eqtr4i 2784 . . . . 5 (ℤ‘(0 + 1)) = ℕ
3837eleq2i 2843 . . . 4 (𝑧 ∈ (ℤ‘(0 + 1)) ↔ 𝑧 ∈ ℕ)
399equncomi 4060 . . . . . . . 8 𝐶 = (𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})
4039fveq1i 6659 . . . . . . 7 (𝐶𝑧) = ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧)
41 nnne0 11708 . . . . . . . . 9 (𝑧 ∈ ℕ → 𝑧 ≠ 0)
4241necomd 3006 . . . . . . . 8 (𝑧 ∈ ℕ → 0 ≠ 𝑧)
43 fvunsn 6932 . . . . . . . 8 (0 ≠ 𝑧 → ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧) = (𝐹𝑧))
4442, 43syl 17 . . . . . . 7 (𝑧 ∈ ℕ → ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧) = (𝐹𝑧))
4540, 44syl5eq 2805 . . . . . 6 (𝑧 ∈ ℕ → (𝐶𝑧) = (𝐹𝑧))
4645adantl 485 . . . . 5 ((𝜑𝑧 ∈ ℕ) → (𝐶𝑧) = (𝐹𝑧))
477ffvelrnda 6842 . . . . 5 ((𝜑𝑧 ∈ ℕ) → (𝐹𝑧) ∈ ℝ)
4846, 47eqeltrd 2852 . . . 4 ((𝜑𝑧 ∈ ℕ) → (𝐶𝑧) ∈ ℝ)
4938, 48sylan2b 596 . . 3 ((𝜑𝑧 ∈ (ℤ‘(0 + 1))) → (𝐶𝑧) ∈ ℝ)
5016, 31, 32, 33, 49seqf2 13439 . 2 (𝜑 → seq0(𝐷, 𝐶):ℕ0⟶(ℝ × ℝ))
511feq1i 6489 . 2 (𝐺:ℕ0⟶(ℝ × ℝ) ↔ seq0(𝐷, 𝐶):ℕ0⟶(ℝ × ℝ))
5250, 51sylibr 237 1 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ⦋csb 3805   ∪ cun 3856  ifcif 4420  {csn 4522  ⟨cop 4528   class class class wbr 5032   × cxp 5522  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150   ∈ cmpo 7152  1st c1st 7691  2nd c2nd 7692  ℝcr 10574  0cc0 10575  1c1 10576   + caddc 10578   < clt 10713   / cdiv 11335  ℕcn 11674  2c2 11729  ℕ0cn0 11934  ℤcz 12020  ℤ≥cuz 12282  seqcseq 13418 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 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 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 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-div 11336  df-nn 11675  df-2 11737  df-n0 11935  df-z 12021  df-uz 12283  df-fz 12940  df-seq 13419 This theorem is referenced by:  ruclem8  15638  ruclem9  15639  ruclem10  15640  ruclem11  15641  ruclem12  15642
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