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Theorem ruclem6 14749
Description: Lemma for ruc 14757. 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 6089 . . . . . 6 (𝐺‘0) = (seq0(𝐷, 𝐶)‘0)
3 0z 11221 . . . . . . 7 0 ∈ ℤ
4 seq1 12631 . . . . . . 7 (0 ∈ ℤ → (seq0(𝐷, 𝐶)‘0) = (𝐶‘0))
53, 4ax-mp 5 . . . . . 6 (seq0(𝐷, 𝐶)‘0) = (𝐶‘0)
62, 5eqtri 2631 . . . . 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 14748 . . . . 5 (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
116, 10syl5eqr 2657 . . . 4 (𝜑 → (𝐶‘0) = ⟨0, 1⟩)
12 0re 9896 . . . . 5 0 ∈ ℝ
13 1re 9895 . . . . 5 1 ∈ ℝ
14 opelxpi 5062 . . . . 5 ((0 ∈ ℝ ∧ 1 ∈ ℝ) → ⟨0, 1⟩ ∈ (ℝ × ℝ))
1512, 13, 14mp2an 703 . . . 4 ⟨0, 1⟩ ∈ (ℝ × ℝ)
1611, 15syl6eqel 2695 . . 3 (𝜑 → (𝐶‘0) ∈ (ℝ × ℝ))
17 1st2nd2 7073 . . . . . 6 (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
1817ad2antrl 759 . . . . 5 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
1918oveq1d 6542 . . . 4 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (𝑧𝐷𝑤) = (⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤))
207adantr 479 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝐹:ℕ⟶ℝ)
218adantr 479 . . . . . 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 7066 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
2322ad2antrl 759 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (1st𝑧) ∈ ℝ)
24 xp2nd 7067 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → (2nd𝑧) ∈ ℝ)
2524ad2antrl 759 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (2nd𝑧) ∈ ℝ)
26 simprr 791 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → 𝑤 ∈ ℝ)
27 eqid 2609 . . . . . 6 (1st ‘(⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤)) = (1st ‘(⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤))
28 eqid 2609 . . . . . 6 (2nd ‘(⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤)) = (2nd ‘(⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤))
2920, 21, 23, 25, 26, 27, 28ruclem1 14745 . . . . 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 1065 . . . 4 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (⟨(1st𝑧), (2nd𝑧)⟩𝐷𝑤) ∈ (ℝ × ℝ))
3119, 30eqeltrd 2687 . . 3 ((𝜑 ∧ (𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ ℝ)) → (𝑧𝐷𝑤) ∈ (ℝ × ℝ))
32 nn0uz 11554 . . 3 0 = (ℤ‘0)
33 0zd 11222 . . 3 (𝜑 → 0 ∈ ℤ)
34 0p1e1 10979 . . . . . . 7 (0 + 1) = 1
3534fveq2i 6091 . . . . . 6 (ℤ‘(0 + 1)) = (ℤ‘1)
36 nnuz 11555 . . . . . 6 ℕ = (ℤ‘1)
3735, 36eqtr4i 2634 . . . . 5 (ℤ‘(0 + 1)) = ℕ
3837eleq2i 2679 . . . 4 (𝑧 ∈ (ℤ‘(0 + 1)) ↔ 𝑧 ∈ ℕ)
399equncomi 3720 . . . . . . . 8 𝐶 = (𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})
4039fveq1i 6089 . . . . . . 7 (𝐶𝑧) = ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧)
41 nnne0 10900 . . . . . . . . 9 (𝑧 ∈ ℕ → 𝑧 ≠ 0)
4241necomd 2836 . . . . . . . 8 (𝑧 ∈ ℕ → 0 ≠ 𝑧)
43 fvunsn 6328 . . . . . . . 8 (0 ≠ 𝑧 → ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧) = (𝐹𝑧))
4442, 43syl 17 . . . . . . 7 (𝑧 ∈ ℕ → ((𝐹 ∪ {⟨0, ⟨0, 1⟩⟩})‘𝑧) = (𝐹𝑧))
4540, 44syl5eq 2655 . . . . . 6 (𝑧 ∈ ℕ → (𝐶𝑧) = (𝐹𝑧))
4645adantl 480 . . . . 5 ((𝜑𝑧 ∈ ℕ) → (𝐶𝑧) = (𝐹𝑧))
477ffvelrnda 6252 . . . . 5 ((𝜑𝑧 ∈ ℕ) → (𝐹𝑧) ∈ ℝ)
4846, 47eqeltrd 2687 . . . 4 ((𝜑𝑧 ∈ ℕ) → (𝐶𝑧) ∈ ℝ)
4938, 48sylan2b 490 . . 3 ((𝜑𝑧 ∈ (ℤ‘(0 + 1))) → (𝐶𝑧) ∈ ℝ)
5016, 31, 32, 33, 49seqf2 12637 . 2 (𝜑 → seq0(𝐷, 𝐶):ℕ0⟶(ℝ × ℝ))
511feq1i 5935 . 2 (𝐺:ℕ0⟶(ℝ × ℝ) ↔ seq0(𝐷, 𝐶):ℕ0⟶(ℝ × ℝ))
5250, 51sylibr 222 1 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wne 2779  csb 3498  cun 3537  ifcif 4035  {csn 4124  cop 4130   class class class wbr 4577   × cxp 5026  wf 5786  cfv 5790  (class class class)co 6527  cmpt2 6529  1st c1st 7034  2nd c2nd 7035  cr 9791  0cc0 9792  1c1 9793   + caddc 9795   < clt 9930   / cdiv 10533  cn 10867  2c2 10917  0cn0 11139  cz 11210  cuz 11519  seqcseq 12618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-seq 12619
This theorem is referenced by:  ruclem8  14751  ruclem9  14752  ruclem10  14753  ruclem11  14754  ruclem12  14755
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