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Theorem ruclem8 16126
Description: Lemma for ruc 16132. The intervals of the 𝐺 sequence are all nonempty. (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
ruclem8 ((πœ‘ ∧ 𝑁 ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘)) < (2nd β€˜(πΊβ€˜π‘)))
Distinct variable groups:   π‘₯,π‘š,𝑦,𝐹   π‘š,𝐺,π‘₯,𝑦   π‘š,𝑁,π‘₯,𝑦
Allowed substitution hints:   πœ‘(π‘₯,𝑦,π‘š)   𝐢(π‘₯,𝑦,π‘š)   𝐷(π‘₯,𝑦,π‘š)

Proof of Theorem ruclem8
Dummy variables 𝑛 π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2fveq3 6852 . . . . 5 (π‘˜ = 0 β†’ (1st β€˜(πΊβ€˜π‘˜)) = (1st β€˜(πΊβ€˜0)))
2 2fveq3 6852 . . . . 5 (π‘˜ = 0 β†’ (2nd β€˜(πΊβ€˜π‘˜)) = (2nd β€˜(πΊβ€˜0)))
31, 2breq12d 5123 . . . 4 (π‘˜ = 0 β†’ ((1st β€˜(πΊβ€˜π‘˜)) < (2nd β€˜(πΊβ€˜π‘˜)) ↔ (1st β€˜(πΊβ€˜0)) < (2nd β€˜(πΊβ€˜0))))
43imbi2d 341 . . 3 (π‘˜ = 0 β†’ ((πœ‘ β†’ (1st β€˜(πΊβ€˜π‘˜)) < (2nd β€˜(πΊβ€˜π‘˜))) ↔ (πœ‘ β†’ (1st β€˜(πΊβ€˜0)) < (2nd β€˜(πΊβ€˜0)))))
5 2fveq3 6852 . . . . 5 (π‘˜ = 𝑛 β†’ (1st β€˜(πΊβ€˜π‘˜)) = (1st β€˜(πΊβ€˜π‘›)))
6 2fveq3 6852 . . . . 5 (π‘˜ = 𝑛 β†’ (2nd β€˜(πΊβ€˜π‘˜)) = (2nd β€˜(πΊβ€˜π‘›)))
75, 6breq12d 5123 . . . 4 (π‘˜ = 𝑛 β†’ ((1st β€˜(πΊβ€˜π‘˜)) < (2nd β€˜(πΊβ€˜π‘˜)) ↔ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›))))
87imbi2d 341 . . 3 (π‘˜ = 𝑛 β†’ ((πœ‘ β†’ (1st β€˜(πΊβ€˜π‘˜)) < (2nd β€˜(πΊβ€˜π‘˜))) ↔ (πœ‘ β†’ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))))
9 2fveq3 6852 . . . . 5 (π‘˜ = (𝑛 + 1) β†’ (1st β€˜(πΊβ€˜π‘˜)) = (1st β€˜(πΊβ€˜(𝑛 + 1))))
10 2fveq3 6852 . . . . 5 (π‘˜ = (𝑛 + 1) β†’ (2nd β€˜(πΊβ€˜π‘˜)) = (2nd β€˜(πΊβ€˜(𝑛 + 1))))
119, 10breq12d 5123 . . . 4 (π‘˜ = (𝑛 + 1) β†’ ((1st β€˜(πΊβ€˜π‘˜)) < (2nd β€˜(πΊβ€˜π‘˜)) ↔ (1st β€˜(πΊβ€˜(𝑛 + 1))) < (2nd β€˜(πΊβ€˜(𝑛 + 1)))))
1211imbi2d 341 . . 3 (π‘˜ = (𝑛 + 1) β†’ ((πœ‘ β†’ (1st β€˜(πΊβ€˜π‘˜)) < (2nd β€˜(πΊβ€˜π‘˜))) ↔ (πœ‘ β†’ (1st β€˜(πΊβ€˜(𝑛 + 1))) < (2nd β€˜(πΊβ€˜(𝑛 + 1))))))
13 2fveq3 6852 . . . . 5 (π‘˜ = 𝑁 β†’ (1st β€˜(πΊβ€˜π‘˜)) = (1st β€˜(πΊβ€˜π‘)))
14 2fveq3 6852 . . . . 5 (π‘˜ = 𝑁 β†’ (2nd β€˜(πΊβ€˜π‘˜)) = (2nd β€˜(πΊβ€˜π‘)))
1513, 14breq12d 5123 . . . 4 (π‘˜ = 𝑁 β†’ ((1st β€˜(πΊβ€˜π‘˜)) < (2nd β€˜(πΊβ€˜π‘˜)) ↔ (1st β€˜(πΊβ€˜π‘)) < (2nd β€˜(πΊβ€˜π‘))))
1615imbi2d 341 . . 3 (π‘˜ = 𝑁 β†’ ((πœ‘ β†’ (1st β€˜(πΊβ€˜π‘˜)) < (2nd β€˜(πΊβ€˜π‘˜))) ↔ (πœ‘ β†’ (1st β€˜(πΊβ€˜π‘)) < (2nd β€˜(πΊβ€˜π‘)))))
17 0lt1 11684 . . . . 5 0 < 1
1817a1i 11 . . . 4 (πœ‘ β†’ 0 < 1)
19 ruc.1 . . . . . . 7 (πœ‘ β†’ 𝐹:β„•βŸΆβ„)
20 ruc.2 . . . . . . 7 (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‘₯)⟩)))
21 ruc.4 . . . . . . 7 𝐢 = ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹)
22 ruc.5 . . . . . . 7 𝐺 = seq0(𝐷, 𝐢)
2319, 20, 21, 22ruclem4 16123 . . . . . 6 (πœ‘ β†’ (πΊβ€˜0) = ⟨0, 1⟩)
2423fveq2d 6851 . . . . 5 (πœ‘ β†’ (1st β€˜(πΊβ€˜0)) = (1st β€˜βŸ¨0, 1⟩))
25 c0ex 11156 . . . . . 6 0 ∈ V
26 1ex 11158 . . . . . 6 1 ∈ V
2725, 26op1st 7934 . . . . 5 (1st β€˜βŸ¨0, 1⟩) = 0
2824, 27eqtrdi 2793 . . . 4 (πœ‘ β†’ (1st β€˜(πΊβ€˜0)) = 0)
2923fveq2d 6851 . . . . 5 (πœ‘ β†’ (2nd β€˜(πΊβ€˜0)) = (2nd β€˜βŸ¨0, 1⟩))
3025, 26op2nd 7935 . . . . 5 (2nd β€˜βŸ¨0, 1⟩) = 1
3129, 30eqtrdi 2793 . . . 4 (πœ‘ β†’ (2nd β€˜(πΊβ€˜0)) = 1)
3218, 28, 313brtr4d 5142 . . 3 (πœ‘ β†’ (1st β€˜(πΊβ€˜0)) < (2nd β€˜(πΊβ€˜0)))
3319adantr 482 . . . . . . . . 9 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))) β†’ 𝐹:β„•βŸΆβ„)
3420adantr 482 . . . . . . . . 9 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))) 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‘₯)⟩)))
3519, 20, 21, 22ruclem6 16124 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐺:β„•0⟢(ℝ Γ— ℝ))
3635ffvelcdmda 7040 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ))
3736adantrr 716 . . . . . . . . . 10 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))) β†’ (πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ))
38 xp1st 7958 . . . . . . . . . 10 ((πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ) β†’ (1st β€˜(πΊβ€˜π‘›)) ∈ ℝ)
3937, 38syl 17 . . . . . . . . 9 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))) β†’ (1st β€˜(πΊβ€˜π‘›)) ∈ ℝ)
40 xp2nd 7959 . . . . . . . . . 10 ((πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜(πΊβ€˜π‘›)) ∈ ℝ)
4137, 40syl 17 . . . . . . . . 9 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))) β†’ (2nd β€˜(πΊβ€˜π‘›)) ∈ ℝ)
42 nn0p1nn 12459 . . . . . . . . . . 11 (𝑛 ∈ β„•0 β†’ (𝑛 + 1) ∈ β„•)
43 ffvelcdm 7037 . . . . . . . . . . 11 ((𝐹:β„•βŸΆβ„ ∧ (𝑛 + 1) ∈ β„•) β†’ (πΉβ€˜(𝑛 + 1)) ∈ ℝ)
4419, 42, 43syl2an 597 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (πΉβ€˜(𝑛 + 1)) ∈ ℝ)
4544adantrr 716 . . . . . . . . 9 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))) β†’ (πΉβ€˜(𝑛 + 1)) ∈ ℝ)
46 eqid 2737 . . . . . . . . 9 (1st β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1)))) = (1st β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1))))
47 eqid 2737 . . . . . . . . 9 (2nd β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1)))) = (2nd β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1))))
48 simprr 772 . . . . . . . . 9 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))) β†’ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))
4933, 34, 39, 41, 45, 46, 47, 48ruclem2 16121 . . . . . . . 8 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))) β†’ ((1st β€˜(πΊβ€˜π‘›)) ≀ (1st β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1)))) ∧ (1st β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1)))) < (2nd β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1)))) ∧ (2nd β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1)))) ≀ (2nd β€˜(πΊβ€˜π‘›))))
5049simp2d 1144 . . . . . . 7 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))) β†’ (1st β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1)))) < (2nd β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1)))))
5119, 20, 21, 22ruclem7 16125 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (πΊβ€˜(𝑛 + 1)) = ((πΊβ€˜π‘›)𝐷(πΉβ€˜(𝑛 + 1))))
5251adantrr 716 . . . . . . . . 9 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))) β†’ (πΊβ€˜(𝑛 + 1)) = ((πΊβ€˜π‘›)𝐷(πΉβ€˜(𝑛 + 1))))
53 1st2nd2 7965 . . . . . . . . . . 11 ((πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ) β†’ (πΊβ€˜π‘›) = ⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩)
5437, 53syl 17 . . . . . . . . . 10 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))) β†’ (πΊβ€˜π‘›) = ⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩)
5554oveq1d 7377 . . . . . . . . 9 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))) β†’ ((πΊβ€˜π‘›)𝐷(πΉβ€˜(𝑛 + 1))) = (⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1))))
5652, 55eqtrd 2777 . . . . . . . 8 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))) β†’ (πΊβ€˜(𝑛 + 1)) = (⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1))))
5756fveq2d 6851 . . . . . . 7 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))) β†’ (1st β€˜(πΊβ€˜(𝑛 + 1))) = (1st β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1)))))
5856fveq2d 6851 . . . . . . 7 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))) β†’ (2nd β€˜(πΊβ€˜(𝑛 + 1))) = (2nd β€˜(⟨(1st β€˜(πΊβ€˜π‘›)), (2nd β€˜(πΊβ€˜π‘›))⟩𝐷(πΉβ€˜(𝑛 + 1)))))
5950, 57, 583brtr4d 5142 . . . . . 6 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))) β†’ (1st β€˜(πΊβ€˜(𝑛 + 1))) < (2nd β€˜(πΊβ€˜(𝑛 + 1))))
6059expr 458 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ ((1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)) β†’ (1st β€˜(πΊβ€˜(𝑛 + 1))) < (2nd β€˜(πΊβ€˜(𝑛 + 1)))))
6160expcom 415 . . . 4 (𝑛 ∈ β„•0 β†’ (πœ‘ β†’ ((1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)) β†’ (1st β€˜(πΊβ€˜(𝑛 + 1))) < (2nd β€˜(πΊβ€˜(𝑛 + 1))))))
6261a2d 29 . . 3 (𝑛 ∈ β„•0 β†’ ((πœ‘ β†’ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›))) β†’ (πœ‘ β†’ (1st β€˜(πΊβ€˜(𝑛 + 1))) < (2nd β€˜(πΊβ€˜(𝑛 + 1))))))
634, 8, 12, 16, 32, 62nn0ind 12605 . 2 (𝑁 ∈ β„•0 β†’ (πœ‘ β†’ (1st β€˜(πΊβ€˜π‘)) < (2nd β€˜(πΊβ€˜π‘))))
6463impcom 409 1 ((πœ‘ ∧ 𝑁 ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘)) < (2nd β€˜(πΊβ€˜π‘)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  β¦‹csb 3860   βˆͺ cun 3913  ifcif 4491  {csn 4591  βŸ¨cop 4597   class class class wbr 5110   Γ— cxp 5636  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:  ruclem9  16127  ruclem10  16128  ruclem12  16130
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