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Theorem ruclem11 15646
 Description: Lemma for ruc 15649. 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 15641 . . . 4 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
6 1stcof 7728 . . . 4 (𝐺:ℕ0⟶(ℝ × ℝ) → (1st𝐺):ℕ0⟶ℝ)
75, 6syl 17 . . 3 (𝜑 → (1st𝐺):ℕ0⟶ℝ)
87frnd 6509 . 2 (𝜑 → ran (1st𝐺) ⊆ ℝ)
97fdmd 6512 . . . 4 (𝜑 → dom (1st𝐺) = ℕ0)
10 0nn0 11954 . . . . 5 0 ∈ ℕ0
11 ne0i 4235 . . . . 5 (0 ∈ ℕ0 → ℕ0 ≠ ∅)
1210, 11mp1i 13 . . . 4 (𝜑 → ℕ0 ≠ ∅)
139, 12eqnetrd 3018 . . 3 (𝜑 → dom (1st𝐺) ≠ ∅)
14 dm0rn0 5770 . . . 4 (dom (1st𝐺) = ∅ ↔ ran (1st𝐺) = ∅)
1514necon3bii 3003 . . 3 (dom (1st𝐺) ≠ ∅ ↔ ran (1st𝐺) ≠ ∅)
1613, 15sylib 221 . 2 (𝜑 → ran (1st𝐺) ≠ ∅)
17 fvco3 6755 . . . . . 6 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) = (1st ‘(𝐺𝑛)))
185, 17sylan 583 . . . . 5 ((𝜑𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) = (1st ‘(𝐺𝑛)))
191adantr 484 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝐹:ℕ⟶ℝ)
202adantr 484 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
21 simpr 488 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
2210a1i 11 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 0 ∈ ℕ0)
2319, 20, 3, 4, 21, 22ruclem10 15645 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺‘0)))
241, 2, 3, 4ruclem4 15640 . . . . . . . . . 10 (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
2524fveq2d 6666 . . . . . . . . 9 (𝜑 → (2nd ‘(𝐺‘0)) = (2nd ‘⟨0, 1⟩))
26 c0ex 10678 . . . . . . . . . 10 0 ∈ V
27 1ex 10680 . . . . . . . . . 10 1 ∈ V
2826, 27op2nd 7707 . . . . . . . . 9 (2nd ‘⟨0, 1⟩) = 1
2925, 28eqtrdi 2809 . . . . . . . 8 (𝜑 → (2nd ‘(𝐺‘0)) = 1)
3029adantr 484 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → (2nd ‘(𝐺‘0)) = 1)
3123, 30breqtrd 5061 . . . . . 6 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) < 1)
325ffvelrnda 6847 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ (ℝ × ℝ))
33 xp1st 7730 . . . . . . . 8 ((𝐺𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
3432, 33syl 17 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) ∈ ℝ)
35 1re 10684 . . . . . . 7 1 ∈ ℝ
36 ltle 10772 . . . . . . 7 (((1st ‘(𝐺𝑛)) ∈ ℝ ∧ 1 ∈ ℝ) → ((1st ‘(𝐺𝑛)) < 1 → (1st ‘(𝐺𝑛)) ≤ 1))
3734, 35, 36sylancl 589 . . . . . 6 ((𝜑𝑛 ∈ ℕ0) → ((1st ‘(𝐺𝑛)) < 1 → (1st ‘(𝐺𝑛)) ≤ 1))
3831, 37mpd 15 . . . . 5 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) ≤ 1)
3918, 38eqbrtrd 5057 . . . 4 ((𝜑𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) ≤ 1)
4039ralrimiva 3113 . . 3 (𝜑 → ∀𝑛 ∈ ℕ0 ((1st𝐺)‘𝑛) ≤ 1)
417ffnd 6503 . . . 4 (𝜑 → (1st𝐺) Fn ℕ0)
42 breq1 5038 . . . . 5 (𝑧 = ((1st𝐺)‘𝑛) → (𝑧 ≤ 1 ↔ ((1st𝐺)‘𝑛) ≤ 1))
4342ralrn 6850 . . . 4 ((1st𝐺) Fn ℕ0 → (∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1 ↔ ∀𝑛 ∈ ℕ0 ((1st𝐺)‘𝑛) ≤ 1))
4441, 43syl 17 . . 3 (𝜑 → (∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1 ↔ ∀𝑛 ∈ ℕ0 ((1st𝐺)‘𝑛) ≤ 1))
4540, 44mpbird 260 . 2 (𝜑 → ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1)
468, 16, 453jca 1125 1 (𝜑 → (ran (1st𝐺) ⊆ ℝ ∧ ran (1st𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ⦋csb 3807   ∪ cun 3858   ⊆ wss 3860  ∅c0 4227  ifcif 4423  {csn 4525  ⟨cop 4531   class class class wbr 5035   × cxp 5525  dom cdm 5527  ran crn 5528   ∘ ccom 5531   Fn wfn 6334  ⟶wf 6335  ‘cfv 6339  (class class class)co 7155   ∈ cmpo 7157  1st c1st 7696  2nd c2nd 7697  ℝcr 10579  0cc0 10580  1c1 10581   + caddc 10583   < clt 10718   ≤ cle 10719   / cdiv 11340  ℕcn 11679  2c2 11734  ℕ0cn0 11939  seqcseq 13423 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 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-cnex 10636  ax-resscn 10637  ax-1cn 10638  ax-icn 10639  ax-addcl 10640  ax-addrcl 10641  ax-mulcl 10642  ax-mulrcl 10643  ax-mulcom 10644  ax-addass 10645  ax-mulass 10646  ax-distr 10647  ax-i2m1 10648  ax-1ne0 10649  ax-1rid 10650  ax-rnegex 10651  ax-rrecex 10652  ax-cnre 10653  ax-pre-lttri 10654  ax-pre-lttrn 10655  ax-pre-ltadd 10656  ax-pre-mulgt0 10657 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 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-1st 7698  df-2nd 7699  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-er 8304  df-en 8533  df-dom 8534  df-sdom 8535  df-pnf 10720  df-mnf 10721  df-xr 10722  df-ltxr 10723  df-le 10724  df-sub 10915  df-neg 10916  df-div 11341  df-nn 11680  df-2 11742  df-n0 11940  df-z 12026  df-uz 12288  df-fz 12945  df-seq 13424 This theorem is referenced by:  ruclem12  15647
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