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Theorem mnringvald 42967
Description: Value of the monoid ring function. (Contributed by Rohan Ridenour, 14-May-2024.)
Hypotheses
Ref Expression
mnringvald.1 𝐹 = (𝑅 MndRing 𝑀)
mnringvald.2 Β· = (.rβ€˜π‘…)
mnringvald.3 0 = (0gβ€˜π‘…)
mnringvald.4 𝐴 = (Baseβ€˜π‘€)
mnringvald.5 + = (+gβ€˜π‘€)
mnringvald.6 𝑉 = (𝑅 freeLMod 𝐴)
mnringvald.7 𝐡 = (Baseβ€˜π‘‰)
mnringvald.8 (πœ‘ β†’ 𝑅 ∈ π‘ˆ)
mnringvald.9 (πœ‘ β†’ 𝑀 ∈ π‘Š)
Assertion
Ref Expression
mnringvald (πœ‘ β†’ 𝐹 = (𝑉 sSet ⟨(.rβ€˜ndx), (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (𝑉 Ξ£g (π‘Ž ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (π‘Ž + 𝑏), ((π‘₯β€˜π‘Ž) Β· (π‘¦β€˜π‘)), 0 )))))⟩))
Distinct variable groups:   𝑅,π‘Ž,𝑏,𝑖,π‘₯,𝑦   𝑀,π‘Ž,𝑏,𝑖,π‘₯,𝑦
Allowed substitution hints:   πœ‘(π‘₯,𝑦,𝑖,π‘Ž,𝑏)   𝐴(π‘₯,𝑦,𝑖,π‘Ž,𝑏)   𝐡(π‘₯,𝑦,𝑖,π‘Ž,𝑏)   + (π‘₯,𝑦,𝑖,π‘Ž,𝑏)   Β· (π‘₯,𝑦,𝑖,π‘Ž,𝑏)   π‘ˆ(π‘₯,𝑦,𝑖,π‘Ž,𝑏)   𝐹(π‘₯,𝑦,𝑖,π‘Ž,𝑏)   𝑉(π‘₯,𝑦,𝑖,π‘Ž,𝑏)   π‘Š(π‘₯,𝑦,𝑖,π‘Ž,𝑏)   0 (π‘₯,𝑦,𝑖,π‘Ž,𝑏)

Proof of Theorem mnringvald
Dummy variables π‘š π‘Ÿ 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mnringvald.1 . 2 𝐹 = (𝑅 MndRing 𝑀)
2 mnringvald.8 . . . 4 (πœ‘ β†’ 𝑅 ∈ π‘ˆ)
32elexd 3495 . . 3 (πœ‘ β†’ 𝑅 ∈ V)
4 mnringvald.9 . . . 4 (πœ‘ β†’ 𝑀 ∈ π‘Š)
54elexd 3495 . . 3 (πœ‘ β†’ 𝑀 ∈ V)
6 nfv 1918 . . . . 5 Ⅎ𝑣(π‘Ÿ = 𝑅 ∧ π‘š = 𝑀)
7 nfcvd 2905 . . . . 5 ((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) β†’ Ⅎ𝑣(𝑉 sSet ⟨(.rβ€˜ndx), (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (𝑉 Ξ£g (π‘Ž ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (π‘Ž + 𝑏), ((π‘₯β€˜π‘Ž) Β· (π‘¦β€˜π‘)), 0 )))))⟩))
8 ovexd 7444 . . . . 5 ((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) β†’ (π‘Ÿ freeLMod (Baseβ€˜π‘š)) ∈ V)
9 simpr 486 . . . . . . . 8 (((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) ∧ 𝑣 = (π‘Ÿ freeLMod (Baseβ€˜π‘š))) β†’ 𝑣 = (π‘Ÿ freeLMod (Baseβ€˜π‘š)))
10 simpll 766 . . . . . . . . 9 (((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) ∧ 𝑣 = (π‘Ÿ freeLMod (Baseβ€˜π‘š))) β†’ π‘Ÿ = 𝑅)
11 fveq2 6892 . . . . . . . . . . 11 (π‘š = 𝑀 β†’ (Baseβ€˜π‘š) = (Baseβ€˜π‘€))
12 mnringvald.4 . . . . . . . . . . 11 𝐴 = (Baseβ€˜π‘€)
1311, 12eqtr4di 2791 . . . . . . . . . 10 (π‘š = 𝑀 β†’ (Baseβ€˜π‘š) = 𝐴)
1413ad2antlr 726 . . . . . . . . 9 (((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) ∧ 𝑣 = (π‘Ÿ freeLMod (Baseβ€˜π‘š))) β†’ (Baseβ€˜π‘š) = 𝐴)
1510, 14oveq12d 7427 . . . . . . . 8 (((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) ∧ 𝑣 = (π‘Ÿ freeLMod (Baseβ€˜π‘š))) β†’ (π‘Ÿ freeLMod (Baseβ€˜π‘š)) = (𝑅 freeLMod 𝐴))
169, 15eqtrd 2773 . . . . . . 7 (((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) ∧ 𝑣 = (π‘Ÿ freeLMod (Baseβ€˜π‘š))) β†’ 𝑣 = (𝑅 freeLMod 𝐴))
17 mnringvald.6 . . . . . . 7 𝑉 = (𝑅 freeLMod 𝐴)
1816, 17eqtr4di 2791 . . . . . 6 (((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) ∧ 𝑣 = (π‘Ÿ freeLMod (Baseβ€˜π‘š))) β†’ 𝑣 = 𝑉)
1918fveq2d 6896 . . . . . . . . 9 (((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) ∧ 𝑣 = (π‘Ÿ freeLMod (Baseβ€˜π‘š))) β†’ (Baseβ€˜π‘£) = (Baseβ€˜π‘‰))
20 mnringvald.7 . . . . . . . . 9 𝐡 = (Baseβ€˜π‘‰)
2119, 20eqtr4di 2791 . . . . . . . 8 (((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) ∧ 𝑣 = (π‘Ÿ freeLMod (Baseβ€˜π‘š))) β†’ (Baseβ€˜π‘£) = 𝐡)
22 fveq2 6892 . . . . . . . . . . . . . . . 16 (π‘š = 𝑀 β†’ (+gβ€˜π‘š) = (+gβ€˜π‘€))
23 mnringvald.5 . . . . . . . . . . . . . . . 16 + = (+gβ€˜π‘€)
2422, 23eqtr4di 2791 . . . . . . . . . . . . . . 15 (π‘š = 𝑀 β†’ (+gβ€˜π‘š) = + )
2524oveqd 7426 . . . . . . . . . . . . . 14 (π‘š = 𝑀 β†’ (π‘Ž(+gβ€˜π‘š)𝑏) = (π‘Ž + 𝑏))
2625ad2antlr 726 . . . . . . . . . . . . 13 (((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) ∧ 𝑣 = (π‘Ÿ freeLMod (Baseβ€˜π‘š))) β†’ (π‘Ž(+gβ€˜π‘š)𝑏) = (π‘Ž + 𝑏))
2726eqeq2d 2744 . . . . . . . . . . . 12 (((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) ∧ 𝑣 = (π‘Ÿ freeLMod (Baseβ€˜π‘š))) β†’ (𝑖 = (π‘Ž(+gβ€˜π‘š)𝑏) ↔ 𝑖 = (π‘Ž + 𝑏)))
28 fveq2 6892 . . . . . . . . . . . . . . 15 (π‘Ÿ = 𝑅 β†’ (.rβ€˜π‘Ÿ) = (.rβ€˜π‘…))
29 mnringvald.2 . . . . . . . . . . . . . . 15 Β· = (.rβ€˜π‘…)
3028, 29eqtr4di 2791 . . . . . . . . . . . . . 14 (π‘Ÿ = 𝑅 β†’ (.rβ€˜π‘Ÿ) = Β· )
3130oveqd 7426 . . . . . . . . . . . . 13 (π‘Ÿ = 𝑅 β†’ ((π‘₯β€˜π‘Ž)(.rβ€˜π‘Ÿ)(π‘¦β€˜π‘)) = ((π‘₯β€˜π‘Ž) Β· (π‘¦β€˜π‘)))
3231ad2antrr 725 . . . . . . . . . . . 12 (((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) ∧ 𝑣 = (π‘Ÿ freeLMod (Baseβ€˜π‘š))) β†’ ((π‘₯β€˜π‘Ž)(.rβ€˜π‘Ÿ)(π‘¦β€˜π‘)) = ((π‘₯β€˜π‘Ž) Β· (π‘¦β€˜π‘)))
33 fveq2 6892 . . . . . . . . . . . . . 14 (π‘Ÿ = 𝑅 β†’ (0gβ€˜π‘Ÿ) = (0gβ€˜π‘…))
34 mnringvald.3 . . . . . . . . . . . . . 14 0 = (0gβ€˜π‘…)
3533, 34eqtr4di 2791 . . . . . . . . . . . . 13 (π‘Ÿ = 𝑅 β†’ (0gβ€˜π‘Ÿ) = 0 )
3635ad2antrr 725 . . . . . . . . . . . 12 (((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) ∧ 𝑣 = (π‘Ÿ freeLMod (Baseβ€˜π‘š))) β†’ (0gβ€˜π‘Ÿ) = 0 )
3727, 32, 36ifbieq12d 4557 . . . . . . . . . . 11 (((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) ∧ 𝑣 = (π‘Ÿ freeLMod (Baseβ€˜π‘š))) β†’ if(𝑖 = (π‘Ž(+gβ€˜π‘š)𝑏), ((π‘₯β€˜π‘Ž)(.rβ€˜π‘Ÿ)(π‘¦β€˜π‘)), (0gβ€˜π‘Ÿ)) = if(𝑖 = (π‘Ž + 𝑏), ((π‘₯β€˜π‘Ž) Β· (π‘¦β€˜π‘)), 0 ))
3814, 37mpteq12dv 5240 . . . . . . . . . 10 (((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) ∧ 𝑣 = (π‘Ÿ freeLMod (Baseβ€˜π‘š))) β†’ (𝑖 ∈ (Baseβ€˜π‘š) ↦ if(𝑖 = (π‘Ž(+gβ€˜π‘š)𝑏), ((π‘₯β€˜π‘Ž)(.rβ€˜π‘Ÿ)(π‘¦β€˜π‘)), (0gβ€˜π‘Ÿ))) = (𝑖 ∈ 𝐴 ↦ if(𝑖 = (π‘Ž + 𝑏), ((π‘₯β€˜π‘Ž) Β· (π‘¦β€˜π‘)), 0 )))
3914, 14, 38mpoeq123dv 7484 . . . . . . . . 9 (((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) ∧ 𝑣 = (π‘Ÿ freeLMod (Baseβ€˜π‘š))) β†’ (π‘Ž ∈ (Baseβ€˜π‘š), 𝑏 ∈ (Baseβ€˜π‘š) ↦ (𝑖 ∈ (Baseβ€˜π‘š) ↦ if(𝑖 = (π‘Ž(+gβ€˜π‘š)𝑏), ((π‘₯β€˜π‘Ž)(.rβ€˜π‘Ÿ)(π‘¦β€˜π‘)), (0gβ€˜π‘Ÿ)))) = (π‘Ž ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (π‘Ž + 𝑏), ((π‘₯β€˜π‘Ž) Β· (π‘¦β€˜π‘)), 0 ))))
4018, 39oveq12d 7427 . . . . . . . 8 (((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) ∧ 𝑣 = (π‘Ÿ freeLMod (Baseβ€˜π‘š))) β†’ (𝑣 Ξ£g (π‘Ž ∈ (Baseβ€˜π‘š), 𝑏 ∈ (Baseβ€˜π‘š) ↦ (𝑖 ∈ (Baseβ€˜π‘š) ↦ if(𝑖 = (π‘Ž(+gβ€˜π‘š)𝑏), ((π‘₯β€˜π‘Ž)(.rβ€˜π‘Ÿ)(π‘¦β€˜π‘)), (0gβ€˜π‘Ÿ))))) = (𝑉 Ξ£g (π‘Ž ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (π‘Ž + 𝑏), ((π‘₯β€˜π‘Ž) Β· (π‘¦β€˜π‘)), 0 )))))
4121, 21, 40mpoeq123dv 7484 . . . . . . 7 (((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) ∧ 𝑣 = (π‘Ÿ freeLMod (Baseβ€˜π‘š))) β†’ (π‘₯ ∈ (Baseβ€˜π‘£), 𝑦 ∈ (Baseβ€˜π‘£) ↦ (𝑣 Ξ£g (π‘Ž ∈ (Baseβ€˜π‘š), 𝑏 ∈ (Baseβ€˜π‘š) ↦ (𝑖 ∈ (Baseβ€˜π‘š) ↦ if(𝑖 = (π‘Ž(+gβ€˜π‘š)𝑏), ((π‘₯β€˜π‘Ž)(.rβ€˜π‘Ÿ)(π‘¦β€˜π‘)), (0gβ€˜π‘Ÿ)))))) = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (𝑉 Ξ£g (π‘Ž ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (π‘Ž + 𝑏), ((π‘₯β€˜π‘Ž) Β· (π‘¦β€˜π‘)), 0 ))))))
4241opeq2d 4881 . . . . . 6 (((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) ∧ 𝑣 = (π‘Ÿ freeLMod (Baseβ€˜π‘š))) β†’ ⟨(.rβ€˜ndx), (π‘₯ ∈ (Baseβ€˜π‘£), 𝑦 ∈ (Baseβ€˜π‘£) ↦ (𝑣 Ξ£g (π‘Ž ∈ (Baseβ€˜π‘š), 𝑏 ∈ (Baseβ€˜π‘š) ↦ (𝑖 ∈ (Baseβ€˜π‘š) ↦ if(𝑖 = (π‘Ž(+gβ€˜π‘š)𝑏), ((π‘₯β€˜π‘Ž)(.rβ€˜π‘Ÿ)(π‘¦β€˜π‘)), (0gβ€˜π‘Ÿ))))))⟩ = ⟨(.rβ€˜ndx), (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (𝑉 Ξ£g (π‘Ž ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (π‘Ž + 𝑏), ((π‘₯β€˜π‘Ž) Β· (π‘¦β€˜π‘)), 0 )))))⟩)
4318, 42oveq12d 7427 . . . . 5 (((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) ∧ 𝑣 = (π‘Ÿ freeLMod (Baseβ€˜π‘š))) β†’ (𝑣 sSet ⟨(.rβ€˜ndx), (π‘₯ ∈ (Baseβ€˜π‘£), 𝑦 ∈ (Baseβ€˜π‘£) ↦ (𝑣 Ξ£g (π‘Ž ∈ (Baseβ€˜π‘š), 𝑏 ∈ (Baseβ€˜π‘š) ↦ (𝑖 ∈ (Baseβ€˜π‘š) ↦ if(𝑖 = (π‘Ž(+gβ€˜π‘š)𝑏), ((π‘₯β€˜π‘Ž)(.rβ€˜π‘Ÿ)(π‘¦β€˜π‘)), (0gβ€˜π‘Ÿ))))))⟩) = (𝑉 sSet ⟨(.rβ€˜ndx), (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (𝑉 Ξ£g (π‘Ž ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (π‘Ž + 𝑏), ((π‘₯β€˜π‘Ž) Β· (π‘¦β€˜π‘)), 0 )))))⟩))
446, 7, 8, 43csbiedf 3925 . . . 4 ((π‘Ÿ = 𝑅 ∧ π‘š = 𝑀) β†’ ⦋(π‘Ÿ freeLMod (Baseβ€˜π‘š)) / π‘£β¦Œ(𝑣 sSet ⟨(.rβ€˜ndx), (π‘₯ ∈ (Baseβ€˜π‘£), 𝑦 ∈ (Baseβ€˜π‘£) ↦ (𝑣 Ξ£g (π‘Ž ∈ (Baseβ€˜π‘š), 𝑏 ∈ (Baseβ€˜π‘š) ↦ (𝑖 ∈ (Baseβ€˜π‘š) ↦ if(𝑖 = (π‘Ž(+gβ€˜π‘š)𝑏), ((π‘₯β€˜π‘Ž)(.rβ€˜π‘Ÿ)(π‘¦β€˜π‘)), (0gβ€˜π‘Ÿ))))))⟩) = (𝑉 sSet ⟨(.rβ€˜ndx), (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (𝑉 Ξ£g (π‘Ž ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (π‘Ž + 𝑏), ((π‘₯β€˜π‘Ž) Β· (π‘¦β€˜π‘)), 0 )))))⟩))
45 df-mnring 42966 . . . 4 MndRing = (π‘Ÿ ∈ V, π‘š ∈ V ↦ ⦋(π‘Ÿ freeLMod (Baseβ€˜π‘š)) / π‘£β¦Œ(𝑣 sSet ⟨(.rβ€˜ndx), (π‘₯ ∈ (Baseβ€˜π‘£), 𝑦 ∈ (Baseβ€˜π‘£) ↦ (𝑣 Ξ£g (π‘Ž ∈ (Baseβ€˜π‘š), 𝑏 ∈ (Baseβ€˜π‘š) ↦ (𝑖 ∈ (Baseβ€˜π‘š) ↦ if(𝑖 = (π‘Ž(+gβ€˜π‘š)𝑏), ((π‘₯β€˜π‘Ž)(.rβ€˜π‘Ÿ)(π‘¦β€˜π‘)), (0gβ€˜π‘Ÿ))))))⟩))
46 ovex 7442 . . . 4 (𝑉 sSet ⟨(.rβ€˜ndx), (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (𝑉 Ξ£g (π‘Ž ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (π‘Ž + 𝑏), ((π‘₯β€˜π‘Ž) Β· (π‘¦β€˜π‘)), 0 )))))⟩) ∈ V
4744, 45, 46ovmpoa 7563 . . 3 ((𝑅 ∈ V ∧ 𝑀 ∈ V) β†’ (𝑅 MndRing 𝑀) = (𝑉 sSet ⟨(.rβ€˜ndx), (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (𝑉 Ξ£g (π‘Ž ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (π‘Ž + 𝑏), ((π‘₯β€˜π‘Ž) Β· (π‘¦β€˜π‘)), 0 )))))⟩))
483, 5, 47syl2anc 585 . 2 (πœ‘ β†’ (𝑅 MndRing 𝑀) = (𝑉 sSet ⟨(.rβ€˜ndx), (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (𝑉 Ξ£g (π‘Ž ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (π‘Ž + 𝑏), ((π‘₯β€˜π‘Ž) Β· (π‘¦β€˜π‘)), 0 )))))⟩))
491, 48eqtrid 2785 1 (πœ‘ β†’ 𝐹 = (𝑉 sSet ⟨(.rβ€˜ndx), (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (𝑉 Ξ£g (π‘Ž ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (π‘Ž + 𝑏), ((π‘₯β€˜π‘Ž) Β· (π‘¦β€˜π‘)), 0 )))))⟩))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475  β¦‹csb 3894  ifcif 4529  βŸ¨cop 4635   ↦ cmpt 5232  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411   sSet csts 17096  ndxcnx 17126  Basecbs 17144  +gcplusg 17197  .rcmulr 17198  0gc0g 17385   Ξ£g cgsu 17386   freeLMod cfrlm 21301   MndRing cmnring 42965
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 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-mnring 42966
This theorem is referenced by:  mnringnmulrd  42968  mnringnmulrdOLD  42969  mnringmulrd  42980
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