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Theorem ruclem6 15576
Description: Lemma for ruc 15584. 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 6664 . . . . . 6 (𝐺‘0) = (seq0(𝐷, 𝐶)‘0)
3 0z 11980 . . . . . . 7 0 ∈ ℤ
4 seq1 13370 . . . . . . 7 (0 ∈ ℤ → (seq0(𝐷, 𝐶)‘0) = (𝐶‘0))
53, 4ax-mp 5 . . . . . 6 (seq0(𝐷, 𝐶)‘0) = (𝐶‘0)
62, 5eqtri 2841 . . . . 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 15575 . . . . 5 (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
116, 10syl5eqr 2867 . . . 4 (𝜑 → (𝐶‘0) = ⟨0, 1⟩)
12 0re 10631 . . . . 5 0 ∈ ℝ
13 1re 10629 . . . . 5 1 ∈ ℝ
14 opelxpi 5585 . . . . 5 ((0 ∈ ℝ ∧ 1 ∈ ℝ) → ⟨0, 1⟩ ∈ (ℝ × ℝ))
1512, 13, 14mp2an 688 . . . 4 ⟨0, 1⟩ ∈ (ℝ × ℝ)
1611, 15syl6eqel 2918 . . 3 (𝜑 → (𝐶‘0) ∈ (ℝ × ℝ))
17 1st2nd2 7717 . . . . . 6 (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
1817ad2antrl 724 . . . . 5 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
1918oveq1d 7160 . . . 4 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (𝑧𝐷𝑤) = (⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤))
207adantr 481 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝐹:ℕ⟶ℝ)
218adantr 481 . . . . . 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 7710 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
2322ad2antrl 724 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (1st𝑧) ∈ ℝ)
24 xp2nd 7711 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → (2nd𝑧) ∈ ℝ)
2524ad2antrl 724 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (2nd𝑧) ∈ ℝ)
26 simprr 769 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝑤 ∈ ℝ)
27 eqid 2818 . . . . . 6 (1st ‘(⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤)) = (1st ‘(⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤))
28 eqid 2818 . . . . . 6 (2nd ‘(⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤)) = (2nd ‘(⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤))
2920, 21, 23, 25, 26, 27, 28ruclem1 15572 . . . . 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 1134 . . . 4 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤) ∈ (ℝ × ℝ))
3119, 30eqeltrd 2910 . . 3 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (𝑧𝐷𝑤) ∈ (ℝ × ℝ))
32 nn0uz 12268 . . 3 0 = (ℤ‘0)
33 0zd 11981 . . 3 (𝜑 → 0 ∈ ℤ)
34 0p1e1 11747 . . . . . . 7 (0 + 1) = 1
3534fveq2i 6666 . . . . . 6 (ℤ‘(0 + 1)) = (ℤ‘1)
36 nnuz 12269 . . . . . 6 ℕ = (ℤ‘1)
3735, 36eqtr4i 2844 . . . . 5 (ℤ‘(0 + 1)) = ℕ
3837eleq2i 2901 . . . 4 (𝑧 ∈ (ℤ‘(0 + 1)) ↔ 𝑧 ∈ ℕ)
399equncomi 4128 . . . . . . . 8 𝐶 = (𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})
4039fveq1i 6664 . . . . . . 7 (𝐶𝑧) = ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧)
41 nnne0 11659 . . . . . . . . 9 (𝑧 ∈ ℕ → 𝑧 ≠ 0)
4241necomd 3068 . . . . . . . 8 (𝑧 ∈ ℕ → 0 ≠ 𝑧)
43 fvunsn 6933 . . . . . . . 8 (0 ≠ 𝑧 → ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧) = (𝐹𝑧))
4442, 43syl 17 . . . . . . 7 (𝑧 ∈ ℕ → ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧) = (𝐹𝑧))
4540, 44syl5eq 2865 . . . . . 6 (𝑧 ∈ ℕ → (𝐶𝑧) = (𝐹𝑧))
4645adantl 482 . . . . 5 ((𝜑𝑧 ∈ ℕ) → (𝐶𝑧) = (𝐹𝑧))
477ffvelrnda 6843 . . . . 5 ((𝜑𝑧 ∈ ℕ) → (𝐹𝑧) ∈ ℝ)
4846, 47eqeltrd 2910 . . . 4 ((𝜑𝑧 ∈ ℕ) → (𝐶𝑧) ∈ ℝ)
4938, 48sylan2b 593 . . 3 ((𝜑𝑧 ∈ (ℤ‘(0 + 1))) → (𝐶𝑧) ∈ ℝ)
5016, 31, 32, 33, 49seqf2 13377 . 2 (𝜑 → seq0(𝐷, 𝐶):ℕ0⟶(ℝ × ℝ))
511feq1i 6498 . 2 (𝐺:ℕ0⟶(ℝ × ℝ) ↔ seq0(𝐷, 𝐶):ℕ0⟶(ℝ × ℝ))
5250, 51sylibr 235 1 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wne 3013  csb 3880  cun 3931  ifcif 4463  {csn 4557  cop 4563   class class class wbr 5057   × cxp 5546  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  1st c1st 7676  2nd c2nd 7677  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   < clt 10663   / cdiv 11285  cn 11626  2c2 11680  0cn0 11885  cz 11969  cuz 12231  seqcseq 13357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-seq 13358
This theorem is referenced by:  ruclem8  15578  ruclem9  15579  ruclem10  15580  ruclem11  15581  ruclem12  15582
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