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Theorem ruclem11 16129
Description: Lemma for ruc 16132. 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 16124 . . . 4 (πœ‘ β†’ 𝐺:β„•0⟢(ℝ Γ— ℝ))
6 1stcof 7956 . . . 4 (𝐺:β„•0⟢(ℝ Γ— ℝ) β†’ (1st ∘ 𝐺):β„•0βŸΆβ„)
75, 6syl 17 . . 3 (πœ‘ β†’ (1st ∘ 𝐺):β„•0βŸΆβ„)
87frnd 6681 . 2 (πœ‘ β†’ ran (1st ∘ 𝐺) βŠ† ℝ)
97fdmd 6684 . . . 4 (πœ‘ β†’ dom (1st ∘ 𝐺) = β„•0)
10 0nn0 12435 . . . . 5 0 ∈ β„•0
11 ne0i 4299 . . . . 5 (0 ∈ β„•0 β†’ β„•0 β‰  βˆ…)
1210, 11mp1i 13 . . . 4 (πœ‘ β†’ β„•0 β‰  βˆ…)
139, 12eqnetrd 3012 . . 3 (πœ‘ β†’ dom (1st ∘ 𝐺) β‰  βˆ…)
14 dm0rn0 5885 . . . 4 (dom (1st ∘ 𝐺) = βˆ… ↔ ran (1st ∘ 𝐺) = βˆ…)
1514necon3bii 2997 . . 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 6945 . . . . . 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 16128 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜0)))
241, 2, 3, 4ruclem4 16123 . . . . . . . . . 10 (πœ‘ β†’ (πΊβ€˜0) = ⟨0, 1⟩)
2524fveq2d 6851 . . . . . . . . 9 (πœ‘ β†’ (2nd β€˜(πΊβ€˜0)) = (2nd β€˜βŸ¨0, 1⟩))
26 c0ex 11156 . . . . . . . . . 10 0 ∈ V
27 1ex 11158 . . . . . . . . . 10 1 ∈ V
2826, 27op2nd 7935 . . . . . . . . 9 (2nd β€˜βŸ¨0, 1⟩) = 1
2925, 28eqtrdi 2793 . . . . . . . 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 5136 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘›)) < 1)
325ffvelcdmda 7040 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ))
33 xp1st 7958 . . . . . . . 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 11162 . . . . . . 7 1 ∈ ℝ
36 ltle 11250 . . . . . . 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 5132 . . . 4 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ ((1st ∘ 𝐺)β€˜π‘›) ≀ 1)
4039ralrimiva 3144 . . 3 (πœ‘ β†’ βˆ€π‘› ∈ β„•0 ((1st ∘ 𝐺)β€˜π‘›) ≀ 1)
417ffnd 6674 . . . 4 (πœ‘ β†’ (1st ∘ 𝐺) Fn β„•0)
42 breq1 5113 . . . . 5 (𝑧 = ((1st ∘ 𝐺)β€˜π‘›) β†’ (𝑧 ≀ 1 ↔ ((1st ∘ 𝐺)β€˜π‘›) ≀ 1))
4342ralrn 7043 . . . 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 2944  βˆ€wral 3065  β¦‹csb 3860   βˆͺ cun 3913   βŠ† wss 3915  βˆ…c0 4287  ifcif 4491  {csn 4591  βŸ¨cop 4597   class class class wbr 5110   Γ— cxp 5636  dom cdm 5638  ran crn 5639   ∘ ccom 5642   Fn wfn 6496  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362   ∈ cmpo 7364  1st c1st 7924  2nd c2nd 7925  β„cr 11057  0cc0 11058  1c1 11059   + caddc 11061   < clt 11196   ≀ cle 11197   / cdiv 11819  β„•cn 12160  2c2 12215  β„•0cn0 12420  seqcseq 13913
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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-n0 12421  df-z 12507  df-uz 12771  df-fz 13432  df-seq 13914
This theorem is referenced by:  ruclem12  16130
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