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Theorem ruclem11 15431
Description: Lemma for ruc 15434. 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 15426 . . . 4 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
6 1stcof 7580 . . . 4 (𝐺:ℕ0⟶(ℝ × ℝ) → (1st𝐺):ℕ0⟶ℝ)
75, 6syl 17 . . 3 (𝜑 → (1st𝐺):ℕ0⟶ℝ)
87frnd 6394 . 2 (𝜑 → ran (1st𝐺) ⊆ ℝ)
97fdmd 6396 . . . 4 (𝜑 → dom (1st𝐺) = ℕ0)
10 0nn0 11765 . . . . 5 0 ∈ ℕ0
11 ne0i 4224 . . . . 5 (0 ∈ ℕ0 → ℕ0 ≠ ∅)
1210, 11mp1i 13 . . . 4 (𝜑 → ℕ0 ≠ ∅)
139, 12eqnetrd 3051 . . 3 (𝜑 → dom (1st𝐺) ≠ ∅)
14 dm0rn0 5684 . . . 4 (dom (1st𝐺) = ∅ ↔ ran (1st𝐺) = ∅)
1514necon3bii 3036 . . 3 (dom (1st𝐺) ≠ ∅ ↔ ran (1st𝐺) ≠ ∅)
1613, 15sylib 219 . 2 (𝜑 → ran (1st𝐺) ≠ ∅)
17 fvco3 6632 . . . . . 6 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) = (1st ‘(𝐺𝑛)))
185, 17sylan 580 . . . . 5 ((𝜑𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) = (1st ‘(𝐺𝑛)))
191adantr 481 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝐹:ℕ⟶ℝ)
202adantr 481 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
21 simpr 485 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
2210a1i 11 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 0 ∈ ℕ0)
2319, 20, 3, 4, 21, 22ruclem10 15430 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺‘0)))
241, 2, 3, 4ruclem4 15425 . . . . . . . . . 10 (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
2524fveq2d 6547 . . . . . . . . 9 (𝜑 → (2nd ‘(𝐺‘0)) = (2nd ‘⟨0, 1⟩))
26 c0ex 10486 . . . . . . . . . 10 0 ∈ V
27 1ex 10488 . . . . . . . . . 10 1 ∈ V
2826, 27op2nd 7559 . . . . . . . . 9 (2nd ‘⟨0, 1⟩) = 1
2925, 28syl6eq 2847 . . . . . . . 8 (𝜑 → (2nd ‘(𝐺‘0)) = 1)
3029adantr 481 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → (2nd ‘(𝐺‘0)) = 1)
3123, 30breqtrd 4992 . . . . . 6 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) < 1)
325ffvelrnda 6721 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ (ℝ × ℝ))
33 xp1st 7582 . . . . . . . 8 ((𝐺𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
3432, 33syl 17 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) ∈ ℝ)
35 1re 10492 . . . . . . 7 1 ∈ ℝ
36 ltle 10581 . . . . . . 7 (((1st ‘(𝐺𝑛)) ∈ ℝ ∧ 1 ∈ ℝ) → ((1st ‘(𝐺𝑛)) < 1 → (1st ‘(𝐺𝑛)) ≤ 1))
3734, 35, 36sylancl 586 . . . . . 6 ((𝜑𝑛 ∈ ℕ0) → ((1st ‘(𝐺𝑛)) < 1 → (1st ‘(𝐺𝑛)) ≤ 1))
3831, 37mpd 15 . . . . 5 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) ≤ 1)
3918, 38eqbrtrd 4988 . . . 4 ((𝜑𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) ≤ 1)
4039ralrimiva 3149 . . 3 (𝜑 → ∀𝑛 ∈ ℕ0 ((1st𝐺)‘𝑛) ≤ 1)
417ffnd 6388 . . . 4 (𝜑 → (1st𝐺) Fn ℕ0)
42 breq1 4969 . . . . 5 (𝑧 = ((1st𝐺)‘𝑛) → (𝑧 ≤ 1 ↔ ((1st𝐺)‘𝑛) ≤ 1))
4342ralrn 6724 . . . 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 258 . 2 (𝜑 → ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1)
468, 16, 453jca 1121 1 (𝜑 → (ran (1st𝐺) ⊆ ℝ ∧ ran (1st𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wne 2984  wral 3105  csb 3815  cun 3861  wss 3863  c0 4215  ifcif 4385  {csn 4476  cop 4482   class class class wbr 4966   × cxp 5446  dom cdm 5448  ran crn 5449  ccom 5452   Fn wfn 6225  wf 6226  cfv 6230  (class class class)co 7021  cmpo 7023  1st c1st 7548  2nd c2nd 7549  cr 10387  0cc0 10388  1c1 10389   + caddc 10391   < clt 10526  cle 10527   / cdiv 11150  cn 11491  2c2 11545  0cn0 11750  seqcseq 13224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324  ax-cnex 10444  ax-resscn 10445  ax-1cn 10446  ax-icn 10447  ax-addcl 10448  ax-addrcl 10449  ax-mulcl 10450  ax-mulrcl 10451  ax-mulcom 10452  ax-addass 10453  ax-mulass 10454  ax-distr 10455  ax-i2m1 10456  ax-1ne0 10457  ax-1rid 10458  ax-rnegex 10459  ax-rrecex 10460  ax-cnre 10461  ax-pre-lttri 10462  ax-pre-lttrn 10463  ax-pre-ltadd 10464  ax-pre-mulgt0 10465
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-pred 6028  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-riota 6982  df-ov 7024  df-oprab 7025  df-mpo 7026  df-om 7442  df-1st 7550  df-2nd 7551  df-wrecs 7803  df-recs 7865  df-rdg 7903  df-er 8144  df-en 8363  df-dom 8364  df-sdom 8365  df-pnf 10528  df-mnf 10529  df-xr 10530  df-ltxr 10531  df-le 10532  df-sub 10724  df-neg 10725  df-div 11151  df-nn 11492  df-2 11553  df-n0 11751  df-z 11835  df-uz 12099  df-fz 12748  df-seq 13225
This theorem is referenced by:  ruclem12  15432
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