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Theorem ruclem11 16190
Description: Lemma for ruc 16193. 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 16185 . . . 4 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
6 1stcof 8009 . . . 4 (𝐺:ℕ0⟶(ℝ × ℝ) → (1st𝐺):ℕ0⟶ℝ)
75, 6syl 17 . . 3 (𝜑 → (1st𝐺):ℕ0⟶ℝ)
87frnd 6725 . 2 (𝜑 → ran (1st𝐺) ⊆ ℝ)
97fdmd 6728 . . . 4 (𝜑 → dom (1st𝐺) = ℕ0)
10 0nn0 12494 . . . . 5 0 ∈ ℕ0
11 ne0i 4334 . . . . 5 (0 ∈ ℕ0 → ℕ0 ≠ ∅)
1210, 11mp1i 13 . . . 4 (𝜑 → ℕ0 ≠ ∅)
139, 12eqnetrd 3007 . . 3 (𝜑 → dom (1st𝐺) ≠ ∅)
14 dm0rn0 5924 . . . 4 (dom (1st𝐺) = ∅ ↔ ran (1st𝐺) = ∅)
1514necon3bii 2992 . . 3 (dom (1st𝐺) ≠ ∅ ↔ ran (1st𝐺) ≠ ∅)
1613, 15sylib 217 . 2 (𝜑 → ran (1st𝐺) ≠ ∅)
17 fvco3 6990 . . . . . 6 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) = (1st ‘(𝐺𝑛)))
185, 17sylan 579 . . . . 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 16189 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺‘0)))
241, 2, 3, 4ruclem4 16184 . . . . . . . . . 10 (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
2524fveq2d 6895 . . . . . . . . 9 (𝜑 → (2nd ‘(𝐺‘0)) = (2nd ‘⟨0, 1⟩))
26 c0ex 11215 . . . . . . . . . 10 0 ∈ V
27 1ex 11217 . . . . . . . . . 10 1 ∈ V
2826, 27op2nd 7988 . . . . . . . . 9 (2nd ‘⟨0, 1⟩) = 1
2925, 28eqtrdi 2787 . . . . . . . 8 (𝜑 → (2nd ‘(𝐺‘0)) = 1)
3029adantr 480 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → (2nd ‘(𝐺‘0)) = 1)
3123, 30breqtrd 5174 . . . . . 6 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) < 1)
325ffvelcdmda 7086 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ (ℝ × ℝ))
33 xp1st 8011 . . . . . . . 8 ((𝐺𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
3432, 33syl 17 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) ∈ ℝ)
35 1re 11221 . . . . . . 7 1 ∈ ℝ
36 ltle 11309 . . . . . . 7 (((1st ‘(𝐺𝑛)) ∈ ℝ ∧ 1 ∈ ℝ) → ((1st ‘(𝐺𝑛)) < 1 → (1st ‘(𝐺𝑛)) ≤ 1))
3734, 35, 36sylancl 585 . . . . . 6 ((𝜑𝑛 ∈ ℕ0) → ((1st ‘(𝐺𝑛)) < 1 → (1st ‘(𝐺𝑛)) ≤ 1))
3831, 37mpd 15 . . . . 5 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) ≤ 1)
3918, 38eqbrtrd 5170 . . . 4 ((𝜑𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) ≤ 1)
4039ralrimiva 3145 . . 3 (𝜑 → ∀𝑛 ∈ ℕ0 ((1st𝐺)‘𝑛) ≤ 1)
417ffnd 6718 . . . 4 (𝜑 → (1st𝐺) Fn ℕ0)
42 breq1 5151 . . . . 5 (𝑧 = ((1st𝐺)‘𝑛) → (𝑧 ≤ 1 ↔ ((1st𝐺)‘𝑛) ≤ 1))
4342ralrn 7089 . . . 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 1127 1 (𝜑 → (ran (1st𝐺) ⊆ ℝ ∧ ran (1st𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wne 2939  wral 3060  csb 3893  cun 3946  wss 3948  c0 4322  ifcif 4528  {csn 4628  cop 4634   class class class wbr 5148   × cxp 5674  dom cdm 5676  ran crn 5677  ccom 5680   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7412  cmpo 7414  1st c1st 7977  2nd c2nd 7978  cr 11115  0cc0 11116  1c1 11117   + caddc 11119   < clt 11255  cle 11256   / cdiv 11878  cn 12219  2c2 12274  0cn0 12479  seqcseq 13973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-n0 12480  df-z 12566  df-uz 12830  df-fz 13492  df-seq 13974
This theorem is referenced by:  ruclem12  16191
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