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Theorem ruclem8 16221
Description: Lemma for ruc 16227. 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 6907 . . . . 5 (π‘˜ = 0 β†’ (1st β€˜(πΊβ€˜π‘˜)) = (1st β€˜(πΊβ€˜0)))
2 2fveq3 6907 . . . . 5 (π‘˜ = 0 β†’ (2nd β€˜(πΊβ€˜π‘˜)) = (2nd β€˜(πΊβ€˜0)))
31, 2breq12d 5165 . . . 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 339 . . 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 6907 . . . . 5 (π‘˜ = 𝑛 β†’ (1st β€˜(πΊβ€˜π‘˜)) = (1st β€˜(πΊβ€˜π‘›)))
6 2fveq3 6907 . . . . 5 (π‘˜ = 𝑛 β†’ (2nd β€˜(πΊβ€˜π‘˜)) = (2nd β€˜(πΊβ€˜π‘›)))
75, 6breq12d 5165 . . . 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 339 . . 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 6907 . . . . 5 (π‘˜ = (𝑛 + 1) β†’ (1st β€˜(πΊβ€˜π‘˜)) = (1st β€˜(πΊβ€˜(𝑛 + 1))))
10 2fveq3 6907 . . . . 5 (π‘˜ = (𝑛 + 1) β†’ (2nd β€˜(πΊβ€˜π‘˜)) = (2nd β€˜(πΊβ€˜(𝑛 + 1))))
119, 10breq12d 5165 . . . 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 339 . . 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 6907 . . . . 5 (π‘˜ = 𝑁 β†’ (1st β€˜(πΊβ€˜π‘˜)) = (1st β€˜(πΊβ€˜π‘)))
14 2fveq3 6907 . . . . 5 (π‘˜ = 𝑁 β†’ (2nd β€˜(πΊβ€˜π‘˜)) = (2nd β€˜(πΊβ€˜π‘)))
1513, 14breq12d 5165 . . . 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 339 . . 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 11774 . . . . 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 16218 . . . . . 6 (πœ‘ β†’ (πΊβ€˜0) = ⟨0, 1⟩)
2423fveq2d 6906 . . . . 5 (πœ‘ β†’ (1st β€˜(πΊβ€˜0)) = (1st β€˜βŸ¨0, 1⟩))
25 c0ex 11246 . . . . . 6 0 ∈ V
26 1ex 11248 . . . . . 6 1 ∈ V
2725, 26op1st 8007 . . . . 5 (1st β€˜βŸ¨0, 1⟩) = 0
2824, 27eqtrdi 2784 . . . 4 (πœ‘ β†’ (1st β€˜(πΊβ€˜0)) = 0)
2923fveq2d 6906 . . . . 5 (πœ‘ β†’ (2nd β€˜(πΊβ€˜0)) = (2nd β€˜βŸ¨0, 1⟩))
3025, 26op2nd 8008 . . . . 5 (2nd β€˜βŸ¨0, 1⟩) = 1
3129, 30eqtrdi 2784 . . . 4 (πœ‘ β†’ (2nd β€˜(πΊβ€˜0)) = 1)
3218, 28, 313brtr4d 5184 . . 3 (πœ‘ β†’ (1st β€˜(πΊβ€˜0)) < (2nd β€˜(πΊβ€˜0)))
3319adantr 479 . . . . . . . . 9 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))) β†’ 𝐹:β„•βŸΆβ„)
3420adantr 479 . . . . . . . . 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 16219 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐺:β„•0⟢(ℝ Γ— ℝ))
3635ffvelcdmda 7099 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ))
3736adantrr 715 . . . . . . . . . 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 8031 . . . . . . . . . 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 8032 . . . . . . . . . 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 12549 . . . . . . . . . . 11 (𝑛 ∈ β„•0 β†’ (𝑛 + 1) ∈ β„•)
43 ffvelcdm 7096 . . . . . . . . . . 11 ((𝐹:β„•βŸΆβ„ ∧ (𝑛 + 1) ∈ β„•) β†’ (πΉβ€˜(𝑛 + 1)) ∈ ℝ)
4419, 42, 43syl2an 594 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (πΉβ€˜(𝑛 + 1)) ∈ ℝ)
4544adantrr 715 . . . . . . . . 9 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (1st β€˜(πΊβ€˜π‘›)) < (2nd β€˜(πΊβ€˜π‘›)))) β†’ (πΉβ€˜(𝑛 + 1)) ∈ ℝ)
46 eqid 2728 . . . . . . . . 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 2728 . . . . . . . . 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 771 . . . . . . . . 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 16216 . . . . . . . 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 16220 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ (πΊβ€˜(𝑛 + 1)) = ((πΊβ€˜π‘›)𝐷(πΉβ€˜(𝑛 + 1))))
5251adantrr 715 . . . . . . . . 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 8038 . . . . . . . . . . 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 7441 . . . . . . . . 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 2768 . . . . . . . 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 6906 . . . . . . 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 6906 . . . . . . 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 5184 . . . . . 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 455 . . . . 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 412 . . . 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 12695 . 2 (𝑁 ∈ β„•0 β†’ (πœ‘ β†’ (1st β€˜(πΊβ€˜π‘)) < (2nd β€˜(πΊβ€˜π‘))))
6463impcom 406 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 394   = wceq 1533   ∈ wcel 2098  β¦‹csb 3894   βˆͺ cun 3947  ifcif 4532  {csn 4632  βŸ¨cop 4638   class class class wbr 5152   Γ— cxp 5680  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426   ∈ cmpo 7428  1st c1st 7997  2nd c2nd 7998  β„cr 11145  0cc0 11146  1c1 11147   + caddc 11149   < clt 11286   ≀ cle 11287   / cdiv 11909  β„•cn 12250  2c2 12305  β„•0cn0 12510  seqcseq 14006
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12251  df-2 12313  df-n0 12511  df-z 12597  df-uz 12861  df-fz 13525  df-seq 14007
This theorem is referenced by:  ruclem9  16222  ruclem10  16223  ruclem12  16225
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