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Theorem ruclem8 16184
Description: Lemma for ruc 16190. 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 6889 . . . . 5 (π‘˜ = 0 β†’ (1st β€˜(πΊβ€˜π‘˜)) = (1st β€˜(πΊβ€˜0)))
2 2fveq3 6889 . . . . 5 (π‘˜ = 0 β†’ (2nd β€˜(πΊβ€˜π‘˜)) = (2nd β€˜(πΊβ€˜0)))
31, 2breq12d 5154 . . . 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 340 . . 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 6889 . . . . 5 (π‘˜ = 𝑛 β†’ (1st β€˜(πΊβ€˜π‘˜)) = (1st β€˜(πΊβ€˜π‘›)))
6 2fveq3 6889 . . . . 5 (π‘˜ = 𝑛 β†’ (2nd β€˜(πΊβ€˜π‘˜)) = (2nd β€˜(πΊβ€˜π‘›)))
75, 6breq12d 5154 . . . 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 340 . . 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 6889 . . . . 5 (π‘˜ = (𝑛 + 1) β†’ (1st β€˜(πΊβ€˜π‘˜)) = (1st β€˜(πΊβ€˜(𝑛 + 1))))
10 2fveq3 6889 . . . . 5 (π‘˜ = (𝑛 + 1) β†’ (2nd β€˜(πΊβ€˜π‘˜)) = (2nd β€˜(πΊβ€˜(𝑛 + 1))))
119, 10breq12d 5154 . . . 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 340 . . 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 6889 . . . . 5 (π‘˜ = 𝑁 β†’ (1st β€˜(πΊβ€˜π‘˜)) = (1st β€˜(πΊβ€˜π‘)))
14 2fveq3 6889 . . . . 5 (π‘˜ = 𝑁 β†’ (2nd β€˜(πΊβ€˜π‘˜)) = (2nd β€˜(πΊβ€˜π‘)))
1513, 14breq12d 5154 . . . 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 340 . . 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 11737 . . . . 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 16181 . . . . . 6 (πœ‘ β†’ (πΊβ€˜0) = ⟨0, 1⟩)
2423fveq2d 6888 . . . . 5 (πœ‘ β†’ (1st β€˜(πΊβ€˜0)) = (1st β€˜βŸ¨0, 1⟩))
25 c0ex 11209 . . . . . 6 0 ∈ V
26 1ex 11211 . . . . . 6 1 ∈ V
2725, 26op1st 7979 . . . . 5 (1st β€˜βŸ¨0, 1⟩) = 0
2824, 27eqtrdi 2782 . . . 4 (πœ‘ β†’ (1st β€˜(πΊβ€˜0)) = 0)
2923fveq2d 6888 . . . . 5 (πœ‘ β†’ (2nd β€˜(πΊβ€˜0)) = (2nd β€˜βŸ¨0, 1⟩))
3025, 26op2nd 7980 . . . . 5 (2nd β€˜βŸ¨0, 1⟩) = 1
3129, 30eqtrdi 2782 . . . 4 (πœ‘ β†’ (2nd β€˜(πΊβ€˜0)) = 1)
3218, 28, 313brtr4d 5173 . . 3 (πœ‘ β†’ (1st β€˜(πΊβ€˜0)) < (2nd β€˜(πΊβ€˜0)))
3319adantr 480 . . . . . . . . 9 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))) β†’ 𝐹:β„•βŸΆβ„)
3420adantr 480 . . . . . . . . 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 16182 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐺:β„•0⟢(ℝ Γ— ℝ))
3635ffvelcdmda 7079 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ))
3736adantrr 714 . . . . . . . . . 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 8003 . . . . . . . . . 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 8004 . . . . . . . . . 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 12512 . . . . . . . . . . 11 (𝑛 ∈ β„•0 β†’ (𝑛 + 1) ∈ β„•)
43 ffvelcdm 7076 . . . . . . . . . . 11 ((𝐹:β„•βŸΆβ„ ∧ (𝑛 + 1) ∈ β„•) β†’ (πΉβ€˜(𝑛 + 1)) ∈ ℝ)
4419, 42, 43syl2an 595 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (πΉβ€˜(𝑛 + 1)) ∈ ℝ)
4544adantrr 714 . . . . . . . . 9 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))) β†’ (πΉβ€˜(𝑛 + 1)) ∈ ℝ)
46 eqid 2726 . . . . . . . . 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 2726 . . . . . . . . 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 770 . . . . . . . . 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 16179 . . . . . . . 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 1140 . . . . . . 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 16183 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (πΊβ€˜(𝑛 + 1)) = ((πΊβ€˜π‘›)𝐷(πΉβ€˜(𝑛 + 1))))
5251adantrr 714 . . . . . . . . 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 8010 . . . . . . . . . . 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 7419 . . . . . . . . 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 2766 . . . . . . . 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 6888 . . . . . . 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 6888 . . . . . . 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 5173 . . . . . 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 456 . . . . 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 413 . . . 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 12658 . 2 (𝑁 ∈ β„•0 β†’ (πœ‘ β†’ (1st β€˜(πΊβ€˜π‘)) < (2nd β€˜(πΊβ€˜π‘))))
6463impcom 407 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 395   = wceq 1533   ∈ wcel 2098  β¦‹csb 3888   βˆͺ cun 3941  ifcif 4523  {csn 4623  βŸ¨cop 4629   class class class wbr 5141   Γ— cxp 5667  βŸΆwf 6532  β€˜cfv 6536  (class class class)co 7404   ∈ cmpo 7406  1st c1st 7969  2nd c2nd 7970  β„cr 11108  0cc0 11109  1c1 11110   + caddc 11112   < clt 11249   ≀ cle 11250   / cdiv 11872  β„•cn 12213  2c2 12268  β„•0cn0 12473  seqcseq 13969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-div 11873  df-nn 12214  df-2 12276  df-n0 12474  df-z 12560  df-uz 12824  df-fz 13488  df-seq 13970
This theorem is referenced by:  ruclem9  16185  ruclem10  16186  ruclem12  16188
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