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Theorem ruclem11 16277
Description: Lemma for ruc 16280. 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 16272 . . . 4 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
6 1stcof 8045 . . . 4 (𝐺:ℕ0⟶(ℝ × ℝ) → (1st𝐺):ℕ0⟶ℝ)
75, 6syl 17 . . 3 (𝜑 → (1st𝐺):ℕ0⟶ℝ)
87frnd 6743 . 2 (𝜑 → ran (1st𝐺) ⊆ ℝ)
97fdmd 6745 . . . 4 (𝜑 → dom (1st𝐺) = ℕ0)
10 0nn0 12543 . . . . 5 0 ∈ ℕ0
11 ne0i 4340 . . . . 5 (0 ∈ ℕ0 → ℕ0 ≠ ∅)
1210, 11mp1i 13 . . . 4 (𝜑 → ℕ0 ≠ ∅)
139, 12eqnetrd 3007 . . 3 (𝜑 → dom (1st𝐺) ≠ ∅)
14 dm0rn0 5934 . . . 4 (dom (1st𝐺) = ∅ ↔ ran (1st𝐺) = ∅)
1514necon3bii 2992 . . 3 (dom (1st𝐺) ≠ ∅ ↔ ran (1st𝐺) ≠ ∅)
1613, 15sylib 218 . 2 (𝜑 → ran (1st𝐺) ≠ ∅)
17 fvco3 7007 . . . . . 6 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) = (1st ‘(𝐺𝑛)))
185, 17sylan 580 . . . . 5 ((𝜑𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) = (1st ‘(𝐺𝑛)))
191adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝐹:ℕ⟶ℝ)
202adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
21 simpr 484 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
2210a1i 11 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 0 ∈ ℕ0)
2319, 20, 3, 4, 21, 22ruclem10 16276 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺‘0)))
241, 2, 3, 4ruclem4 16271 . . . . . . . . . 10 (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
2524fveq2d 6909 . . . . . . . . 9 (𝜑 → (2nd ‘(𝐺‘0)) = (2nd ‘⟨0, 1⟩))
26 c0ex 11256 . . . . . . . . . 10 0 ∈ V
27 1ex 11258 . . . . . . . . . 10 1 ∈ V
2826, 27op2nd 8024 . . . . . . . . 9 (2nd ‘⟨0, 1⟩) = 1
2925, 28eqtrdi 2792 . . . . . . . 8 (𝜑 → (2nd ‘(𝐺‘0)) = 1)
3029adantr 480 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → (2nd ‘(𝐺‘0)) = 1)
3123, 30breqtrd 5168 . . . . . 6 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) < 1)
325ffvelcdmda 7103 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ (ℝ × ℝ))
33 xp1st 8047 . . . . . . . 8 ((𝐺𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
3432, 33syl 17 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) ∈ ℝ)
35 1re 11262 . . . . . . 7 1 ∈ ℝ
36 ltle 11350 . . . . . . 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 5164 . . . 4 ((𝜑𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) ≤ 1)
4039ralrimiva 3145 . . 3 (𝜑 → ∀𝑛 ∈ ℕ0 ((1st𝐺)‘𝑛) ≤ 1)
417ffnd 6736 . . . 4 (𝜑 → (1st𝐺) Fn ℕ0)
42 breq1 5145 . . . . 5 (𝑧 = ((1st𝐺)‘𝑛) → (𝑧 ≤ 1 ↔ ((1st𝐺)‘𝑛) ≤ 1))
4342ralrn 7107 . . . 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 257 . 2 (𝜑 → ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1)
468, 16, 453jca 1128 1 (𝜑 → (ran (1st𝐺) ⊆ ℝ ∧ ran (1st𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wne 2939  wral 3060  csb 3898  cun 3948  wss 3950  c0 4332  ifcif 4524  {csn 4625  cop 4631   class class class wbr 5142   × cxp 5682  dom cdm 5684  ran crn 5685  ccom 5688   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432  cmpo 7434  1st c1st 8013  2nd c2nd 8014  cr 11155  0cc0 11156  1c1 11157   + caddc 11159   < clt 11296  cle 11297   / cdiv 11921  cn 12267  2c2 12322  0cn0 12528  seqcseq 14043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-n0 12529  df-z 12616  df-uz 12880  df-fz 13549  df-seq 14044
This theorem is referenced by:  ruclem12  16278
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