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Theorem ruclem11 14808
 Description: Lemma for ruc 14811. Closure lemmas for supremum. (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
ruclem11 (𝜑 → (ran (1st𝐺) ⊆ ℝ ∧ ran (1st𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1))
Distinct variable groups:   𝑥,𝑚,𝑦   𝑧,𝐶   𝑧,𝑚,𝐹,𝑥,𝑦   𝑚,𝐺,𝑥,𝑦,𝑧   𝜑,𝑧   𝑧,𝐷
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑚)   𝐶(𝑥,𝑦,𝑚)   𝐷(𝑥,𝑦,𝑚)

Proof of Theorem ruclem11
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 ruc.1 . . . . 5 (𝜑𝐹:ℕ⟶ℝ)
2 ruc.2 . . . . 5 (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
3 ruc.4 . . . . 5 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
4 ruc.5 . . . . 5 𝐺 = seq0(𝐷, 𝐶)
51, 2, 3, 4ruclem6 14803 . . . 4 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
6 1stcof 7087 . . . 4 (𝐺:ℕ0⟶(ℝ × ℝ) → (1st𝐺):ℕ0⟶ℝ)
75, 6syl 17 . . 3 (𝜑 → (1st𝐺):ℕ0⟶ℝ)
8 frn 5966 . . 3 ((1st𝐺):ℕ0⟶ℝ → ran (1st𝐺) ⊆ ℝ)
97, 8syl 17 . 2 (𝜑 → ran (1st𝐺) ⊆ ℝ)
10 fdm 5964 . . . . 5 ((1st𝐺):ℕ0⟶ℝ → dom (1st𝐺) = ℕ0)
117, 10syl 17 . . . 4 (𝜑 → dom (1st𝐺) = ℕ0)
12 0nn0 11184 . . . . 5 0 ∈ ℕ0
13 ne0i 3880 . . . . 5 (0 ∈ ℕ0 → ℕ0 ≠ ∅)
1412, 13mp1i 13 . . . 4 (𝜑 → ℕ0 ≠ ∅)
1511, 14eqnetrd 2849 . . 3 (𝜑 → dom (1st𝐺) ≠ ∅)
16 dm0rn0 5263 . . . 4 (dom (1st𝐺) = ∅ ↔ ran (1st𝐺) = ∅)
1716necon3bii 2834 . . 3 (dom (1st𝐺) ≠ ∅ ↔ ran (1st𝐺) ≠ ∅)
1815, 17sylib 207 . 2 (𝜑 → ran (1st𝐺) ≠ ∅)
19 fvco3 6185 . . . . . 6 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) = (1st ‘(𝐺𝑛)))
205, 19sylan 487 . . . . 5 ((𝜑𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) = (1st ‘(𝐺𝑛)))
211adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝐹:ℕ⟶ℝ)
222adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
23 simpr 476 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
2412a1i 11 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 0 ∈ ℕ0)
2521, 22, 3, 4, 23, 24ruclem10 14807 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺‘0)))
261, 2, 3, 4ruclem4 14802 . . . . . . . . . 10 (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
2726fveq2d 6107 . . . . . . . . 9 (𝜑 → (2nd ‘(𝐺‘0)) = (2nd ‘⟨0, 1⟩))
28 c0ex 9913 . . . . . . . . . 10 0 ∈ V
29 1ex 9914 . . . . . . . . . 10 1 ∈ V
3028, 29op2nd 7068 . . . . . . . . 9 (2nd ‘⟨0, 1⟩) = 1
3127, 30syl6eq 2660 . . . . . . . 8 (𝜑 → (2nd ‘(𝐺‘0)) = 1)
3231adantr 480 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → (2nd ‘(𝐺‘0)) = 1)
3325, 32breqtrd 4609 . . . . . 6 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) < 1)
345ffvelrnda 6267 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ (ℝ × ℝ))
35 xp1st 7089 . . . . . . . 8 ((𝐺𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
3634, 35syl 17 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) ∈ ℝ)
37 1re 9918 . . . . . . 7 1 ∈ ℝ
38 ltle 10005 . . . . . . 7 (((1st ‘(𝐺𝑛)) ∈ ℝ ∧ 1 ∈ ℝ) → ((1st ‘(𝐺𝑛)) < 1 → (1st ‘(𝐺𝑛)) ≤ 1))
3936, 37, 38sylancl 693 . . . . . 6 ((𝜑𝑛 ∈ ℕ0) → ((1st ‘(𝐺𝑛)) < 1 → (1st ‘(𝐺𝑛)) ≤ 1))
4033, 39mpd 15 . . . . 5 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) ≤ 1)
4120, 40eqbrtrd 4605 . . . 4 ((𝜑𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) ≤ 1)
4241ralrimiva 2949 . . 3 (𝜑 → ∀𝑛 ∈ ℕ0 ((1st𝐺)‘𝑛) ≤ 1)
43 ffn 5958 . . . . 5 ((1st𝐺):ℕ0⟶ℝ → (1st𝐺) Fn ℕ0)
447, 43syl 17 . . . 4 (𝜑 → (1st𝐺) Fn ℕ0)
45 breq1 4586 . . . . 5 (𝑧 = ((1st𝐺)‘𝑛) → (𝑧 ≤ 1 ↔ ((1st𝐺)‘𝑛) ≤ 1))
4645ralrn 6270 . . . 4 ((1st𝐺) Fn ℕ0 → (∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1 ↔ ∀𝑛 ∈ ℕ0 ((1st𝐺)‘𝑛) ≤ 1))
4744, 46syl 17 . . 3 (𝜑 → (∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1 ↔ ∀𝑛 ∈ ℕ0 ((1st𝐺)‘𝑛) ≤ 1))
4842, 47mpbird 246 . 2 (𝜑 → ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1)
499, 18, 483jca 1235 1 (𝜑 → (ran (1st𝐺) ⊆ ℝ ∧ ran (1st𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ⦋csb 3499   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  ifcif 4036  {csn 4125  ⟨cop 4131   class class class wbr 4583   × cxp 5036  dom cdm 5038  ran crn 5039   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954   / cdiv 10563  ℕcn 10897  2c2 10947  ℕ0cn0 11169  seqcseq 12663 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664 This theorem is referenced by:  ruclem12  14809
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