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Theorem ruclem11 16188
Description: Lemma for ruc 16191. Closure lemmas for supremum. (Contributed by Mario Carneiro, 28-May-2014.)
Hypotheses
Ref Expression
ruc.1 (πœ‘ β†’ 𝐹:β„•βŸΆβ„)
ruc.2 (πœ‘ β†’ 𝐷 = (π‘₯ ∈ (ℝ Γ— ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st β€˜π‘₯) + (2nd β€˜π‘₯)) / 2) / π‘šβ¦Œif(π‘š < 𝑦, ⟨(1st β€˜π‘₯), π‘šβŸ©, ⟨((π‘š + (2nd β€˜π‘₯)) / 2), (2nd β€˜π‘₯)⟩)))
ruc.4 𝐢 = ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹)
ruc.5 𝐺 = seq0(𝐷, 𝐢)
Assertion
Ref Expression
ruclem11 (πœ‘ β†’ (ran (1st ∘ 𝐺) βŠ† ℝ ∧ ran (1st ∘ 𝐺) β‰  βˆ… ∧ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 1))
Distinct variable groups:   π‘₯,π‘š,𝑦   𝑧,𝐢   𝑧,π‘š,𝐹,π‘₯,𝑦   π‘š,𝐺,π‘₯,𝑦,𝑧   πœ‘,𝑧   𝑧,𝐷
Allowed substitution hints:   πœ‘(π‘₯,𝑦,π‘š)   𝐢(π‘₯,𝑦,π‘š)   𝐷(π‘₯,𝑦,π‘š)

Proof of Theorem ruclem11
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 ruc.1 . . . . 5 (πœ‘ β†’ 𝐹:β„•βŸΆβ„)
2 ruc.2 . . . . 5 (πœ‘ β†’ 𝐷 = (π‘₯ ∈ (ℝ Γ— ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st β€˜π‘₯) + (2nd β€˜π‘₯)) / 2) / π‘šβ¦Œif(π‘š < 𝑦, ⟨(1st β€˜π‘₯), π‘šβŸ©, ⟨((π‘š + (2nd β€˜π‘₯)) / 2), (2nd β€˜π‘₯)⟩)))
3 ruc.4 . . . . 5 𝐢 = ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹)
4 ruc.5 . . . . 5 𝐺 = seq0(𝐷, 𝐢)
51, 2, 3, 4ruclem6 16183 . . . 4 (πœ‘ β†’ 𝐺:β„•0⟢(ℝ Γ— ℝ))
6 1stcof 8008 . . . 4 (𝐺:β„•0⟢(ℝ Γ— ℝ) β†’ (1st ∘ 𝐺):β„•0βŸΆβ„)
75, 6syl 17 . . 3 (πœ‘ β†’ (1st ∘ 𝐺):β„•0βŸΆβ„)
87frnd 6725 . 2 (πœ‘ β†’ ran (1st ∘ 𝐺) βŠ† ℝ)
97fdmd 6728 . . . 4 (πœ‘ β†’ dom (1st ∘ 𝐺) = β„•0)
10 0nn0 12492 . . . . 5 0 ∈ β„•0
11 ne0i 4334 . . . . 5 (0 ∈ β„•0 β†’ β„•0 β‰  βˆ…)
1210, 11mp1i 13 . . . 4 (πœ‘ β†’ β„•0 β‰  βˆ…)
139, 12eqnetrd 3007 . . 3 (πœ‘ β†’ dom (1st ∘ 𝐺) β‰  βˆ…)
14 dm0rn0 5924 . . . 4 (dom (1st ∘ 𝐺) = βˆ… ↔ ran (1st ∘ 𝐺) = βˆ…)
1514necon3bii 2992 . . 3 (dom (1st ∘ 𝐺) β‰  βˆ… ↔ ran (1st ∘ 𝐺) β‰  βˆ…)
1613, 15sylib 217 . 2 (πœ‘ β†’ ran (1st ∘ 𝐺) β‰  βˆ…)
17 fvco3 6990 . . . . . 6 ((𝐺:β„•0⟢(ℝ Γ— ℝ) ∧ 𝑛 ∈ β„•0) β†’ ((1st ∘ 𝐺)β€˜π‘›) = (1st β€˜(πΊβ€˜π‘›)))
185, 17sylan 579 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ ((1st ∘ 𝐺)β€˜π‘›) = (1st β€˜(πΊβ€˜π‘›)))
191adantr 480 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ 𝐹:β„•βŸΆβ„)
202adantr 480 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ 𝐷 = (π‘₯ ∈ (ℝ Γ— ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st β€˜π‘₯) + (2nd β€˜π‘₯)) / 2) / π‘šβ¦Œif(π‘š < 𝑦, ⟨(1st β€˜π‘₯), π‘šβŸ©, ⟨((π‘š + (2nd β€˜π‘₯)) / 2), (2nd β€˜π‘₯)⟩)))
21 simpr 484 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ 𝑛 ∈ β„•0)
2210a1i 11 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ 0 ∈ β„•0)
2319, 20, 3, 4, 21, 22ruclem10 16187 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜0)))
241, 2, 3, 4ruclem4 16182 . . . . . . . . . 10 (πœ‘ β†’ (πΊβ€˜0) = ⟨0, 1⟩)
2524fveq2d 6895 . . . . . . . . 9 (πœ‘ β†’ (2nd β€˜(πΊβ€˜0)) = (2nd β€˜βŸ¨0, 1⟩))
26 c0ex 11213 . . . . . . . . . 10 0 ∈ V
27 1ex 11215 . . . . . . . . . 10 1 ∈ V
2826, 27op2nd 7987 . . . . . . . . 9 (2nd β€˜βŸ¨0, 1⟩) = 1
2925, 28eqtrdi 2787 . . . . . . . 8 (πœ‘ β†’ (2nd β€˜(πΊβ€˜0)) = 1)
3029adantr 480 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (2nd β€˜(πΊβ€˜0)) = 1)
3123, 30breqtrd 5174 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘›)) < 1)
325ffvelcdmda 7086 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ))
33 xp1st 8010 . . . . . . . 8 ((πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ) β†’ (1st β€˜(πΊβ€˜π‘›)) ∈ ℝ)
3432, 33syl 17 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘›)) ∈ ℝ)
35 1re 11219 . . . . . . 7 1 ∈ ℝ
36 ltle 11307 . . . . . . 7 (((1st β€˜(πΊβ€˜π‘›)) ∈ ℝ ∧ 1 ∈ ℝ) β†’ ((1st β€˜(πΊβ€˜π‘›)) < 1 β†’ (1st β€˜(πΊβ€˜π‘›)) ≀ 1))
3734, 35, 36sylancl 585 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ ((1st β€˜(πΊβ€˜π‘›)) < 1 β†’ (1st β€˜(πΊβ€˜π‘›)) ≀ 1))
3831, 37mpd 15 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘›)) ≀ 1)
3918, 38eqbrtrd 5170 . . . 4 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ ((1st ∘ 𝐺)β€˜π‘›) ≀ 1)
4039ralrimiva 3145 . . 3 (πœ‘ β†’ βˆ€π‘› ∈ β„•0 ((1st ∘ 𝐺)β€˜π‘›) ≀ 1)
417ffnd 6718 . . . 4 (πœ‘ β†’ (1st ∘ 𝐺) Fn β„•0)
42 breq1 5151 . . . . 5 (𝑧 = ((1st ∘ 𝐺)β€˜π‘›) β†’ (𝑧 ≀ 1 ↔ ((1st ∘ 𝐺)β€˜π‘›) ≀ 1))
4342ralrn 7089 . . . 4 ((1st ∘ 𝐺) Fn β„•0 β†’ (βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 1 ↔ βˆ€π‘› ∈ β„•0 ((1st ∘ 𝐺)β€˜π‘›) ≀ 1))
4441, 43syl 17 . . 3 (πœ‘ β†’ (βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 1 ↔ βˆ€π‘› ∈ β„•0 ((1st ∘ 𝐺)β€˜π‘›) ≀ 1))
4540, 44mpbird 257 . 2 (πœ‘ β†’ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 1)
468, 16, 453jca 1127 1 (πœ‘ β†’ (ran (1st ∘ 𝐺) βŠ† ℝ ∧ ran (1st ∘ 𝐺) β‰  βˆ… ∧ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 1))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  βˆ€wral 3060  β¦‹csb 3893   βˆͺ cun 3946   βŠ† wss 3948  βˆ…c0 4322  ifcif 4528  {csn 4628  βŸ¨cop 4634   class class class wbr 5148   Γ— cxp 5674  dom cdm 5676  ran crn 5677   ∘ ccom 5680   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7412   ∈ cmpo 7414  1st c1st 7976  2nd c2nd 7977  β„cr 11112  0cc0 11113  1c1 11114   + caddc 11116   < clt 11253   ≀ cle 11254   / cdiv 11876  β„•cn 12217  2c2 12272  β„•0cn0 12477  seqcseq 13971
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 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190
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 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-er 8706  df-en 8943  df-dom 8944  df-sdom 8945  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-nn 12218  df-2 12280  df-n0 12478  df-z 12564  df-uz 12828  df-fz 13490  df-seq 13972
This theorem is referenced by:  ruclem12  16189
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