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Theorem caucvgprprlemell 6811
Description: Lemma for caucvgprpr 6838. 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 5544 . . . . . . . 8 (𝑙 = 𝑋 → (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) = (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )))
21breq2d 3801 . . . . . . 7 (𝑙 = 𝑋 → (𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) ↔ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))))
32abbidv 2169 . . . . . 6 (𝑙 = 𝑋 → {𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))} = {𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))})
41breq1d 3799 . . . . . . 7 (𝑙 = 𝑋 → ((𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞 ↔ (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞))
54abbidv 2169 . . . . . 6 (𝑙 = 𝑋 → {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞})
63, 5opeq12d 3582 . . . . 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 3799 . . . 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 2342 . . 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 5206 . . . 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 6489 . . . . . 6 Q ∈ V
1211rabex 3926 . . . . 5 {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)} ∈ V
1311rabex 3926 . . . . 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 5798 . . . 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 2074 . . 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 2720 . 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 3574 . . . . . . . . . . . 12 (𝑟 = 𝑎 → ⟨𝑟, 1𝑜⟩ = ⟨𝑎, 1𝑜⟩)
1817eceq1d 6170 . . . . . . . . . . 11 (𝑟 = 𝑎 → [⟨𝑟, 1𝑜⟩] ~Q = [⟨𝑎, 1𝑜⟩] ~Q )
1918fveq2d 5207 . . . . . . . . . 10 (𝑟 = 𝑎 → (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))
2019oveq2d 5553 . . . . . . . . 9 (𝑟 = 𝑎 → (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) = (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
2120breq2d 3801 . . . . . . . 8 (𝑟 = 𝑎 → (𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) ↔ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))))
2221abbidv 2169 . . . . . . 7 (𝑟 = 𝑎 → {𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))} = {𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))})
2320breq1d 3799 . . . . . . . 8 (𝑟 = 𝑎 → ((𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞 ↔ (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞))
2423abbidv 2169 . . . . . . 7 (𝑟 = 𝑎 → {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞})
2522, 24opeq12d 3582 . . . . . 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 5203 . . . . . 6 (𝑟 = 𝑎 → (𝐹𝑟) = (𝐹𝑎))
2725, 26breq12d 3802 . . . . 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 2549 . . . 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 3574 . . . . . . . . . . . 12 (𝑎 = 𝑏 → ⟨𝑎, 1𝑜⟩ = ⟨𝑏, 1𝑜⟩)
3029eceq1d 6170 . . . . . . . . . . 11 (𝑎 = 𝑏 → [⟨𝑎, 1𝑜⟩] ~Q = [⟨𝑏, 1𝑜⟩] ~Q )
3130fveq2d 5207 . . . . . . . . . 10 (𝑎 = 𝑏 → (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))
3231oveq2d 5553 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) = (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )))
3332breq2d 3801 . . . . . . . 8 (𝑎 = 𝑏 → (𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ↔ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))))
3433abbidv 2169 . . . . . . 7 (𝑎 = 𝑏 → {𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))} = {𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))})
3532breq1d 3799 . . . . . . . 8 (𝑎 = 𝑏 → ((𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞 ↔ (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞))
3635abbidv 2169 . . . . . . 7 (𝑎 = 𝑏 → {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞})
3734, 36opeq12d 3582 . . . . . 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 5203 . . . . . 6 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
3937, 38breq12d 3802 . . . . 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 2549 . . . 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 177 . . 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 438 . 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 177 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 101  wb 102   = wceq 1257  wcel 1407  {cab 2040  wrex 2322  {crab 2325  cop 3403   class class class wbr 3789  cfv 4927  (class class class)co 5537  1st c1st 5790  1𝑜c1o 6022  [cec 6132  Ncnpi 6398   ~Q ceq 6405  Qcnq 6406   +Q cplq 6408  *Qcrq 6410   <Q cltq 6411   +P cpp 6419  <P cltp 6421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-coll 3897  ax-sep 3900  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-iinf 4336
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-reu 2328  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-int 3641  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-id 4055  df-iom 4339  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-ov 5540  df-1st 5792  df-ec 6136  df-qs 6140  df-ni 6430  df-nqqs 6474
This theorem is referenced by:  caucvgprprlemopl  6823  caucvgprprlemlol  6824  caucvgprprlemdisj  6828  caucvgprprlemloc  6829
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