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Theorem ruclem11 16187
Description: Lemma for ruc 16190. 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 16182 . . . 4 (πœ‘ β†’ 𝐺:β„•0⟢(ℝ Γ— ℝ))
6 1stcof 8007 . . . 4 (𝐺:β„•0⟢(ℝ Γ— ℝ) β†’ (1st ∘ 𝐺):β„•0βŸΆβ„)
75, 6syl 17 . . 3 (πœ‘ β†’ (1st ∘ 𝐺):β„•0βŸΆβ„)
87frnd 6724 . 2 (πœ‘ β†’ ran (1st ∘ 𝐺) βŠ† ℝ)
97fdmd 6727 . . . 4 (πœ‘ β†’ dom (1st ∘ 𝐺) = β„•0)
10 0nn0 12491 . . . . 5 0 ∈ β„•0
11 ne0i 4333 . . . . 5 (0 ∈ β„•0 β†’ β„•0 β‰  βˆ…)
1210, 11mp1i 13 . . . 4 (πœ‘ β†’ β„•0 β‰  βˆ…)
139, 12eqnetrd 3006 . . 3 (πœ‘ β†’ dom (1st ∘ 𝐺) β‰  βˆ…)
14 dm0rn0 5923 . . . 4 (dom (1st ∘ 𝐺) = βˆ… ↔ ran (1st ∘ 𝐺) = βˆ…)
1514necon3bii 2991 . . 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 6989 . . . . . 6 ((𝐺:β„•0⟢(ℝ Γ— ℝ) ∧ 𝑛 ∈ β„•0) β†’ ((1st ∘ 𝐺)β€˜π‘›) = (1st β€˜(πΊβ€˜π‘›)))
185, 17sylan 578 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ ((1st ∘ 𝐺)β€˜π‘›) = (1st β€˜(πΊβ€˜π‘›)))
191adantr 479 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ 𝐹:β„•βŸΆβ„)
202adantr 479 . . . . . . . 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 483 . . . . . . . 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 16186 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜0)))
241, 2, 3, 4ruclem4 16181 . . . . . . . . . 10 (πœ‘ β†’ (πΊβ€˜0) = ⟨0, 1⟩)
2524fveq2d 6894 . . . . . . . . 9 (πœ‘ β†’ (2nd β€˜(πΊβ€˜0)) = (2nd β€˜βŸ¨0, 1⟩))
26 c0ex 11212 . . . . . . . . . 10 0 ∈ V
27 1ex 11214 . . . . . . . . . 10 1 ∈ V
2826, 27op2nd 7986 . . . . . . . . 9 (2nd β€˜βŸ¨0, 1⟩) = 1
2925, 28eqtrdi 2786 . . . . . . . 8 (πœ‘ β†’ (2nd β€˜(πΊβ€˜0)) = 1)
3029adantr 479 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (2nd β€˜(πΊβ€˜0)) = 1)
3123, 30breqtrd 5173 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘›)) < 1)
325ffvelcdmda 7085 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ))
33 xp1st 8009 . . . . . . . 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 11218 . . . . . . 7 1 ∈ ℝ
36 ltle 11306 . . . . . . 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 584 . . . . . 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 5169 . . . 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 6717 . . . 4 (πœ‘ β†’ (1st ∘ 𝐺) Fn β„•0)
42 breq1 5150 . . . . 5 (𝑧 = ((1st ∘ 𝐺)β€˜π‘›) β†’ (𝑧 ≀ 1 ↔ ((1st ∘ 𝐺)β€˜π‘›) ≀ 1))
4342ralrn 7088 . . . 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 256 . 2 (πœ‘ β†’ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 1)
468, 16, 453jca 1126 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  β¦‹csb 3892   βˆͺ cun 3945   βŠ† wss 3947  βˆ…c0 4321  ifcif 4527  {csn 4627  βŸ¨cop 4633   class class class wbr 5147   Γ— cxp 5673  dom cdm 5675  ran crn 5676   ∘ ccom 5679   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ∈ cmpo 7413  1st c1st 7975  2nd c2nd 7976  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115   < clt 11252   ≀ cle 11253   / cdiv 11875  β„•cn 12216  2c2 12271  β„•0cn0 12476  seqcseq 13970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-seq 13971
This theorem is referenced by:  ruclem12  16188
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