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Theorem caucvgprprlemell 7007
Description: Lemma for caucvgprpr 7034. Membership in the lower cut of the putative limit. (Contributed by Jim Kingdon, 21-Jan-2021.)
Hypothesis
Ref Expression
caucvgprprlemell.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩
Assertion
Ref Expression
caucvgprprlemell (𝑋 ∈ (1st𝐿) ↔ (𝑋Q ∧ ∃𝑏N ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)))
Distinct variable groups:   𝐹,𝑏   𝐹,𝑙,𝑟   𝑢,𝐹,𝑟   𝑋,𝑏,𝑝   𝑋,𝑙,𝑟,𝑝   𝑢,𝑋,𝑝   𝑋,𝑞,𝑏   𝑞,𝑙,𝑟   𝑢,𝑞
Allowed substitution hints:   𝐹(𝑞,𝑝)   𝐿(𝑢,𝑟,𝑞,𝑝,𝑏,𝑙)

Proof of Theorem caucvgprprlemell
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 oveq1 5571 . . . . . . . 8 (𝑙 = 𝑋 → (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) = (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )))
21breq2d 3817 . . . . . . 7 (𝑙 = 𝑋 → (𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) ↔ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))))
32abbidv 2200 . . . . . 6 (𝑙 = 𝑋 → {𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))} = {𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))})
41breq1d 3815 . . . . . . 7 (𝑙 = 𝑋 → ((𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞 ↔ (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞))
54abbidv 2200 . . . . . 6 (𝑙 = 𝑋 → {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞})
63, 5opeq12d 3598 . . . . 5 (𝑙 = 𝑋 → ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩)
76breq1d 3815 . . . 4 (𝑙 = 𝑋 → (⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟) ↔ ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)))
87rexbidv 2374 . . 3 (𝑙 = 𝑋 → (∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟) ↔ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)))
9 caucvgprprlemell.lim . . . . 5 𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩
109fveq2i 5233 . . . 4 (1st𝐿) = (1st ‘⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩)
11 nqex 6685 . . . . . 6 Q ∈ V
1211rabex 3942 . . . . 5 {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)} ∈ V
1311rabex 3942 . . . . 5 {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩} ∈ V
1412, 13op1st 5825 . . . 4 (1st ‘⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩) = {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}
1510, 14eqtri 2103 . . 3 (1st𝐿) = {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}
168, 15elrab2 2760 . 2 (𝑋 ∈ (1st𝐿) ↔ (𝑋Q ∧ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)))
17 opeq1 3590 . . . . . . . . . . . 12 (𝑟 = 𝑎 → ⟨𝑟, 1𝑜⟩ = ⟨𝑎, 1𝑜⟩)
1817eceq1d 6230 . . . . . . . . . . 11 (𝑟 = 𝑎 → [⟨𝑟, 1𝑜⟩] ~Q = [⟨𝑎, 1𝑜⟩] ~Q )
1918fveq2d 5234 . . . . . . . . . 10 (𝑟 = 𝑎 → (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))
2019oveq2d 5580 . . . . . . . . 9 (𝑟 = 𝑎 → (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) = (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
2120breq2d 3817 . . . . . . . 8 (𝑟 = 𝑎 → (𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) ↔ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))))
2221abbidv 2200 . . . . . . 7 (𝑟 = 𝑎 → {𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))} = {𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))})
2320breq1d 3815 . . . . . . . 8 (𝑟 = 𝑎 → ((𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞 ↔ (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞))
2423abbidv 2200 . . . . . . 7 (𝑟 = 𝑎 → {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞})
2522, 24opeq12d 3598 . . . . . 6 (𝑟 = 𝑎 → ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩)
26 fveq2 5230 . . . . . 6 (𝑟 = 𝑎 → (𝐹𝑟) = (𝐹𝑎))
2725, 26breq12d 3818 . . . . 5 (𝑟 = 𝑎 → (⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟) ↔ ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑎)))
2827cbvrexv 2583 . . . 4 (∃𝑟N ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟) ↔ ∃𝑎N ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑎))
29 opeq1 3590 . . . . . . . . . . . 12 (𝑎 = 𝑏 → ⟨𝑎, 1𝑜⟩ = ⟨𝑏, 1𝑜⟩)
3029eceq1d 6230 . . . . . . . . . . 11 (𝑎 = 𝑏 → [⟨𝑎, 1𝑜⟩] ~Q = [⟨𝑏, 1𝑜⟩] ~Q )
3130fveq2d 5234 . . . . . . . . . 10 (𝑎 = 𝑏 → (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))
3231oveq2d 5580 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) = (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )))
3332breq2d 3817 . . . . . . . 8 (𝑎 = 𝑏 → (𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ↔ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))))
3433abbidv 2200 . . . . . . 7 (𝑎 = 𝑏 → {𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))} = {𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))})
3532breq1d 3815 . . . . . . . 8 (𝑎 = 𝑏 → ((𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞 ↔ (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞))
3635abbidv 2200 . . . . . . 7 (𝑎 = 𝑏 → {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞})
3734, 36opeq12d 3598 . . . . . 6 (𝑎 = 𝑏 → ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩)
38 fveq2 5230 . . . . . 6 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
3937, 38breq12d 3818 . . . . 5 (𝑎 = 𝑏 → (⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑎) ↔ ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)))
4039cbvrexv 2583 . . . 4 (∃𝑎N ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑎) ↔ ∃𝑏N ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))
4128, 40bitri 182 . . 3 (∃𝑟N ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟) ↔ ∃𝑏N ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏))
4241anbi2i 445 . 2 ((𝑋Q ∧ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)) ↔ (𝑋Q ∧ ∃𝑏N ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)))
4316, 42bitri 182 1 (𝑋 ∈ (1st𝐿) ↔ (𝑋Q ∧ ∃𝑏N ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1285  wcel 1434  {cab 2069  wrex 2354  {crab 2357  cop 3419   class class class wbr 3805  cfv 4952  (class class class)co 5564  1st c1st 5817  1𝑜c1o 6079  [cec 6192  Ncnpi 6594   ~Q ceq 6601  Qcnq 6602   +Q cplq 6604  *Qcrq 6606   <Q cltq 6607   +P cpp 6615  <P cltp 6617
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5567  df-1st 5819  df-ec 6196  df-qs 6200  df-ni 6626  df-nqqs 6670
This theorem is referenced by:  caucvgprprlemopl  7019  caucvgprprlemlol  7020  caucvgprprlemdisj  7024  caucvgprprlemloc  7025
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