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Theorem ruclem6 16117
Description: Lemma for ruc 16125. Domain and codomain 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 6843 . . . . . 6 (𝐺‘0) = (seq0(𝐷, 𝐶)‘0)
3 0z 12510 . . . . . . 7 0 ∈ ℤ
4 seq1 13919 . . . . . . 7 (0 ∈ ℤ → (seq0(𝐷, 𝐶)‘0) = (𝐶‘0))
53, 4ax-mp 5 . . . . . 6 (seq0(𝐷, 𝐶)‘0) = (𝐶‘0)
62, 5eqtri 2764 . . . . 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 16116 . . . . 5 (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
116, 10eqtr3id 2790 . . . 4 (𝜑 → (𝐶‘0) = ⟨0, 1⟩)
12 0re 11157 . . . . 5 0 ∈ ℝ
13 1re 11155 . . . . 5 1 ∈ ℝ
14 opelxpi 5670 . . . . 5 ((0 ∈ ℝ ∧ 1 ∈ ℝ) → ⟨0, 1⟩ ∈ (ℝ × ℝ))
1512, 13, 14mp2an 690 . . . 4 ⟨0, 1⟩ ∈ (ℝ × ℝ)
1611, 15eqeltrdi 2846 . . 3 (𝜑 → (𝐶‘0) ∈ (ℝ × ℝ))
17 1st2nd2 7960 . . . . . 6 (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
1817ad2antrl 726 . . . . 5 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
1918oveq1d 7372 . . . 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 7953 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
2322ad2antrl 726 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (1st𝑧) ∈ ℝ)
24 xp2nd 7954 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → (2nd𝑧) ∈ ℝ)
2524ad2antrl 726 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (2nd𝑧) ∈ ℝ)
26 simprr 771 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝑤 ∈ ℝ)
27 eqid 2736 . . . . . 6 (1st ‘(⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤)) = (1st ‘(⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤))
28 eqid 2736 . . . . . 6 (2nd ‘(⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤)) = (2nd ‘(⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤))
2920, 21, 23, 25, 26, 27, 28ruclem1 16113 . . . . 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 1142 . . . 4 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤) ∈ (ℝ × ℝ))
3119, 30eqeltrd 2838 . . 3 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (𝑧𝐷𝑤) ∈ (ℝ × ℝ))
32 nn0uz 12805 . . 3 0 = (ℤ‘0)
33 0zd 12511 . . 3 (𝜑 → 0 ∈ ℤ)
34 0p1e1 12275 . . . . . . 7 (0 + 1) = 1
3534fveq2i 6845 . . . . . 6 (ℤ‘(0 + 1)) = (ℤ‘1)
36 nnuz 12806 . . . . . 6 ℕ = (ℤ‘1)
3735, 36eqtr4i 2767 . . . . 5 (ℤ‘(0 + 1)) = ℕ
3837eleq2i 2829 . . . 4 (𝑧 ∈ (ℤ‘(0 + 1)) ↔ 𝑧 ∈ ℕ)
399equncomi 4115 . . . . . . . 8 𝐶 = (𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})
4039fveq1i 6843 . . . . . . 7 (𝐶𝑧) = ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧)
41 nnne0 12187 . . . . . . . . 9 (𝑧 ∈ ℕ → 𝑧 ≠ 0)
4241necomd 2999 . . . . . . . 8 (𝑧 ∈ ℕ → 0 ≠ 𝑧)
43 fvunsn 7125 . . . . . . . 8 (0 ≠ 𝑧 → ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧) = (𝐹𝑧))
4442, 43syl 17 . . . . . . 7 (𝑧 ∈ ℕ → ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧) = (𝐹𝑧))
4540, 44eqtrid 2788 . . . . . 6 (𝑧 ∈ ℕ → (𝐶𝑧) = (𝐹𝑧))
4645adantl 482 . . . . 5 ((𝜑𝑧 ∈ ℕ) → (𝐶𝑧) = (𝐹𝑧))
477ffvelcdmda 7035 . . . . 5 ((𝜑𝑧 ∈ ℕ) → (𝐹𝑧) ∈ ℝ)
4846, 47eqeltrd 2838 . . . 4 ((𝜑𝑧 ∈ ℕ) → (𝐶𝑧) ∈ ℝ)
4938, 48sylan2b 594 . . 3 ((𝜑𝑧 ∈ (ℤ‘(0 + 1))) → (𝐶𝑧) ∈ ℝ)
5016, 31, 32, 33, 49seqf2 13927 . 2 (𝜑 → seq0(𝐷, 𝐶):ℕ0⟶(ℝ × ℝ))
511feq1i 6659 . 2 (𝐺:ℕ0⟶(ℝ × ℝ) ↔ seq0(𝐷, 𝐶):ℕ0⟶(ℝ × ℝ))
5250, 51sylibr 233 1 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2943  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   / cdiv 11812  cn 12153  2c2 12208  0cn0 12413  cz 12499  cuz 12763  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:  ruclem8  16119  ruclem9  16120  ruclem10  16121  ruclem11  16122  ruclem12  16123
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