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Theorem ruclem11 16183
Description: Lemma for ruc 16186. 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 16178 . . . 4 (πœ‘ β†’ 𝐺:β„•0⟢(ℝ Γ— ℝ))
6 1stcof 8005 . . . 4 (𝐺:β„•0⟢(ℝ Γ— ℝ) β†’ (1st ∘ 𝐺):β„•0βŸΆβ„)
75, 6syl 17 . . 3 (πœ‘ β†’ (1st ∘ 𝐺):β„•0βŸΆβ„)
87frnd 6726 . 2 (πœ‘ β†’ ran (1st ∘ 𝐺) βŠ† ℝ)
97fdmd 6729 . . . 4 (πœ‘ β†’ dom (1st ∘ 𝐺) = β„•0)
10 0nn0 12487 . . . . 5 0 ∈ β„•0
11 ne0i 4335 . . . . 5 (0 ∈ β„•0 β†’ β„•0 β‰  βˆ…)
1210, 11mp1i 13 . . . 4 (πœ‘ β†’ β„•0 β‰  βˆ…)
139, 12eqnetrd 3009 . . 3 (πœ‘ β†’ dom (1st ∘ 𝐺) β‰  βˆ…)
14 dm0rn0 5925 . . . 4 (dom (1st ∘ 𝐺) = βˆ… ↔ ran (1st ∘ 𝐺) = βˆ…)
1514necon3bii 2994 . . 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 6991 . . . . . 6 ((𝐺:β„•0⟢(ℝ Γ— ℝ) ∧ 𝑛 ∈ β„•0) β†’ ((1st ∘ 𝐺)β€˜π‘›) = (1st β€˜(πΊβ€˜π‘›)))
185, 17sylan 581 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ ((1st ∘ 𝐺)β€˜π‘›) = (1st β€˜(πΊβ€˜π‘›)))
191adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ 𝐹:β„•βŸΆβ„)
202adantr 482 . . . . . . . 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 486 . . . . . . . 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 16182 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜0)))
241, 2, 3, 4ruclem4 16177 . . . . . . . . . 10 (πœ‘ β†’ (πΊβ€˜0) = ⟨0, 1⟩)
2524fveq2d 6896 . . . . . . . . 9 (πœ‘ β†’ (2nd β€˜(πΊβ€˜0)) = (2nd β€˜βŸ¨0, 1⟩))
26 c0ex 11208 . . . . . . . . . 10 0 ∈ V
27 1ex 11210 . . . . . . . . . 10 1 ∈ V
2826, 27op2nd 7984 . . . . . . . . 9 (2nd β€˜βŸ¨0, 1⟩) = 1
2925, 28eqtrdi 2789 . . . . . . . 8 (πœ‘ β†’ (2nd β€˜(πΊβ€˜0)) = 1)
3029adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (2nd β€˜(πΊβ€˜0)) = 1)
3123, 30breqtrd 5175 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘›)) < 1)
325ffvelcdmda 7087 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ))
33 xp1st 8007 . . . . . . . 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 11214 . . . . . . 7 1 ∈ ℝ
36 ltle 11302 . . . . . . 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 587 . . . . . 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 5171 . . . 4 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ ((1st ∘ 𝐺)β€˜π‘›) ≀ 1)
4039ralrimiva 3147 . . 3 (πœ‘ β†’ βˆ€π‘› ∈ β„•0 ((1st ∘ 𝐺)β€˜π‘›) ≀ 1)
417ffnd 6719 . . . 4 (πœ‘ β†’ (1st ∘ 𝐺) Fn β„•0)
42 breq1 5152 . . . . 5 (𝑧 = ((1st ∘ 𝐺)β€˜π‘›) β†’ (𝑧 ≀ 1 ↔ ((1st ∘ 𝐺)β€˜π‘›) ≀ 1))
4342ralrn 7090 . . . 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 1129 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  β¦‹csb 3894   βˆͺ cun 3947   βŠ† wss 3949  βˆ…c0 4323  ifcif 4529  {csn 4629  βŸ¨cop 4635   class class class wbr 5149   Γ— cxp 5675  dom cdm 5677  ran crn 5678   ∘ ccom 5681   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  1st c1st 7973  2nd c2nd 7974  β„cr 11109  0cc0 11110  1c1 11111   + caddc 11113   < clt 11248   ≀ cle 11249   / cdiv 11871  β„•cn 12212  2c2 12267  β„•0cn0 12472  seqcseq 13966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-seq 13967
This theorem is referenced by:  ruclem12  16184
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