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Theorem psdmplcl 22166
Description: The derivative of a polynomial is a polynomial. (Contributed by SN, 12-Apr-2025.)
Hypotheses
Ref Expression
psdmplcl.p 𝑃 = (𝐼 mPoly 𝑅)
psdmplcl.b 𝐵 = (Base‘𝑃)
psdmplcl.r (𝜑𝑅 ∈ Mnd)
psdmplcl.x (𝜑𝑋𝐼)
psdmplcl.f (𝜑𝐹𝐵)
Assertion
Ref Expression
psdmplcl (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ 𝐵)

Proof of Theorem psdmplcl
Dummy variables 𝑏 𝑑 𝑖 𝑘 𝑦 𝑧 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . 3 (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
2 eqid 2737 . . 3 (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
3 psdmplcl.r . . . 4 (𝜑𝑅 ∈ Mnd)
4 mndmgm 18754 . . . 4 (𝑅 ∈ Mnd → 𝑅 ∈ Mgm)
53, 4syl 17 . . 3 (𝜑𝑅 ∈ Mgm)
6 psdmplcl.x . . 3 (𝜑𝑋𝐼)
7 psdmplcl.p . . . . 5 𝑃 = (𝐼 mPoly 𝑅)
8 psdmplcl.b . . . . 5 𝐵 = (Base‘𝑃)
97, 1, 8, 2mplbasss 22017 . . . 4 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅))
10 psdmplcl.f . . . 4 (𝜑𝐹𝐵)
119, 10sselid 3981 . . 3 (𝜑𝐹 ∈ (Base‘(𝐼 mPwSer 𝑅)))
121, 2, 5, 6, 11psdcl 22165 . 2 (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ (Base‘(𝐼 mPwSer 𝑅)))
13 eqid 2737 . . . 4 { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
141, 2, 13, 6, 11psdval 22163 . . 3 (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) = (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))))
15 ovex 7464 . . . . . . 7 (ℕ0m 𝐼) ∈ V
1615rabex 5339 . . . . . 6 { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∈ V
1716a1i 11 . . . . 5 (𝜑 → { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∈ V)
1817mptexd 7244 . . . 4 (𝜑 → (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))) ∈ V)
19 fvexd 6921 . . . 4 (𝜑 → (0g𝑅) ∈ V)
20 funmpt 6604 . . . . 5 Fun (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))
2120a1i 11 . . . 4 (𝜑 → Fun (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))))
22 simpr 484 . . . . . . . . 9 ((𝜑𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
23 reldmmpl 22008 . . . . . . . . . . . . 13 Rel dom mPoly
247, 8, 23strov2rcl 17255 . . . . . . . . . . . 12 (𝐹𝐵𝐼 ∈ V)
2510, 24syl 17 . . . . . . . . . . 11 (𝜑𝐼 ∈ V)
2613psrbagsn 22087 . . . . . . . . . . 11 (𝐼 ∈ V → (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
2725, 26syl 17 . . . . . . . . . 10 (𝜑 → (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
2827adantr 480 . . . . . . . . 9 ((𝜑𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
2913psrbagaddcl 21944 . . . . . . . . 9 ((𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∧ (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
3022, 28, 29syl2anc 584 . . . . . . . 8 ((𝜑𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
31 eqidd 2738 . . . . . . . 8 (𝜑 → (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
32 eqid 2737 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
337, 32, 8, 13, 10mplelf 22018 . . . . . . . . 9 (𝜑𝐹:{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶(Base‘𝑅))
3433feqmptd 6977 . . . . . . . 8 (𝜑𝐹 = (𝑧 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹𝑧)))
35 fveq2 6906 . . . . . . . 8 (𝑧 = (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝐹𝑧) = (𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
3630, 31, 34, 35fmptco 7149 . . . . . . 7 (𝜑 → (𝐹 ∘ (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) = (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))
37 eqid 2737 . . . . . . . . 9 (0g𝑅) = (0g𝑅)
387, 8, 37, 10mplelsfi 22015 . . . . . . . 8 (𝜑𝐹 finSupp (0g𝑅))
3930fmpttd 7135 . . . . . . . . 9 (𝜑 → (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))):{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
40 ovex 7464 . . . . . . . . . . . . . . 15 (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ V
41 eqid 2737 . . . . . . . . . . . . . . 15 (𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
4240, 41fnmpti 6711 . . . . . . . . . . . . . 14 (𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) Fn { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
4342a1i 11 . . . . . . . . . . . . 13 (𝜑 → (𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) Fn { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
44 dffn3 6748 . . . . . . . . . . . . 13 ((𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) Fn { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↔ (𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))):{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶ran (𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
4543, 44sylib 218 . . . . . . . . . . . 12 (𝜑 → (𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))):{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶ran (𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
4645, 39fcod 6761 . . . . . . . . . . 11 (𝜑 → ((𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) ∘ (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))):{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶ran (𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
4746ffnd 6737 . . . . . . . . . 10 (𝜑 → ((𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) ∘ (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) Fn { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
48 fnresi 6697 . . . . . . . . . . 11 ( I ↾ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) Fn { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
4948a1i 11 . . . . . . . . . 10 (𝜑 → ( I ↾ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) Fn { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
5013psrbagf 21938 . . . . . . . . . . . . . . . . 17 (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → 𝑑:𝐼⟶ℕ0)
5150adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑:𝐼⟶ℕ0)
5251ffvelcdmda 7104 . . . . . . . . . . . . . . 15 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (𝑑𝑖) ∈ ℕ0)
5352nn0cnd 12589 . . . . . . . . . . . . . 14 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (𝑑𝑖) ∈ ℂ)
54 ax-1cn 11213 . . . . . . . . . . . . . . . 16 1 ∈ ℂ
55 0cn 11253 . . . . . . . . . . . . . . . 16 0 ∈ ℂ
5654, 55ifcli 4573 . . . . . . . . . . . . . . 15 if(𝑖 = 𝑋, 1, 0) ∈ ℂ
5756a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → if(𝑖 = 𝑋, 1, 0) ∈ ℂ)
5853, 57pncand 11621 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (((𝑑𝑖) + if(𝑖 = 𝑋, 1, 0)) − if(𝑖 = 𝑋, 1, 0)) = (𝑑𝑖))
5958mpteq2dva 5242 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑖𝐼 ↦ (((𝑑𝑖) + if(𝑖 = 𝑋, 1, 0)) − if(𝑖 = 𝑋, 1, 0))) = (𝑖𝐼 ↦ (𝑑𝑖)))
60 simpr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
6127adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
6213psrbagaddcl 21944 . . . . . . . . . . . . . . 15 ((𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∧ (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
6360, 61, 62syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
6413psrbagf 21938 . . . . . . . . . . . . . . 15 ((𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → (𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))):𝐼⟶ℕ0)
6564ffnd 6737 . . . . . . . . . . . . . 14 ((𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → (𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) Fn 𝐼)
6663, 65syl 17 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) Fn 𝐼)
67 1ex 11257 . . . . . . . . . . . . . . . 16 1 ∈ V
68 c0ex 11255 . . . . . . . . . . . . . . . 16 0 ∈ V
6967, 68ifex 4576 . . . . . . . . . . . . . . 15 if(𝑦 = 𝑋, 1, 0) ∈ V
70 eqid 2737 . . . . . . . . . . . . . . 15 (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))
7169, 70fnmpti 6711 . . . . . . . . . . . . . 14 (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼
7271a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼)
7325adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝐼 ∈ V)
74 inidm 4227 . . . . . . . . . . . . 13 (𝐼𝐼) = 𝐼
7550ffnd 6737 . . . . . . . . . . . . . . 15 (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → 𝑑 Fn 𝐼)
7675adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑 Fn 𝐼)
77 eqidd 2738 . . . . . . . . . . . . . 14 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (𝑑𝑖) = (𝑑𝑖))
78 eqeq1 2741 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑖 → (𝑦 = 𝑋𝑖 = 𝑋))
7978ifbid 4549 . . . . . . . . . . . . . . 15 (𝑦 = 𝑖 → if(𝑦 = 𝑋, 1, 0) = if(𝑖 = 𝑋, 1, 0))
80 simpr 484 . . . . . . . . . . . . . . 15 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → 𝑖𝐼)
8167, 68ifex 4576 . . . . . . . . . . . . . . . 16 if(𝑖 = 𝑋, 1, 0) ∈ V
8281a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → if(𝑖 = 𝑋, 1, 0) ∈ V)
8370, 79, 80, 82fvmptd3 7039 . . . . . . . . . . . . . 14 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → ((𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑖) = if(𝑖 = 𝑋, 1, 0))
8476, 72, 73, 73, 74, 77, 83ofval 7708 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → ((𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑖) = ((𝑑𝑖) + if(𝑖 = 𝑋, 1, 0)))
8566, 72, 73, 73, 74, 84, 83offval 7706 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑖𝐼 ↦ (((𝑑𝑖) + if(𝑖 = 𝑋, 1, 0)) − if(𝑖 = 𝑋, 1, 0))))
8651feqmptd 6977 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑 = (𝑖𝐼 ↦ (𝑑𝑖)))
8759, 85, 863eqtr4d 2787 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = 𝑑)
8830adantlr 715 . . . . . . . . . . . . . 14 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
8988fmpttd 7135 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))):{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
9089, 60fvco3d 7009 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) ∘ (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))‘𝑑) = ((𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))‘((𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))‘𝑑)))
91 eqid 2737 . . . . . . . . . . . . . 14 (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
92 oveq1 7438 . . . . . . . . . . . . . 14 (𝑘 = 𝑑 → (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
93 ovexd 7466 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ V)
9491, 92, 60, 93fvmptd3 7039 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))‘𝑑) = (𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
9594fveq2d 6910 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))‘((𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))‘𝑑)) = ((𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))‘(𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
96 oveq1 7438 . . . . . . . . . . . . 13 (𝑏 = (𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = ((𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
97 ovexd 7466 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ V)
9841, 96, 63, 97fvmptd3 7039 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))‘(𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = ((𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
9990, 95, 983eqtrd 2781 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) ∘ (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))‘𝑑) = ((𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
100 fvresi 7193 . . . . . . . . . . . 12 (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → (( I ↾ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})‘𝑑) = 𝑑)
101100adantl 481 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (( I ↾ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})‘𝑑) = 𝑑)
10287, 99, 1013eqtr4d 2787 . . . . . . . . . 10 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) ∘ (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))‘𝑑) = (( I ↾ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})‘𝑑))
10347, 49, 102eqfnfvd 7054 . . . . . . . . 9 (𝜑 → ((𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) ∘ (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) = ( I ↾ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}))
104 fcof1 7307 . . . . . . . . 9 (((𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))):{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∧ ((𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) ∘ (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) = ( I ↾ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})) → (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))):{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}–1-1→{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
10539, 103, 104syl2anc 584 . . . . . . . 8 (𝜑 → (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))):{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}–1-1→{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
10638, 105, 19, 10fsuppco 9442 . . . . . . 7 (𝜑 → (𝐹 ∘ (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) finSupp (0g𝑅))
10736, 106eqbrtrrd 5167 . . . . . 6 (𝜑 → (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) finSupp (0g𝑅))
108107fsuppimpd 9409 . . . . 5 (𝜑 → ((𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) supp (0g𝑅)) ∈ Fin)
109 ssidd 4007 . . . . . 6 (𝜑 → ((𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) supp (0g𝑅)) ⊆ ((𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) supp (0g𝑅)))
110 eqid 2737 . . . . . . . 8 (.g𝑅) = (.g𝑅)
11132, 110, 37mulgnn0z 19119 . . . . . . 7 ((𝑅 ∈ Mnd ∧ 𝑛 ∈ ℕ0) → (𝑛(.g𝑅)(0g𝑅)) = (0g𝑅))
1123, 111sylan 580 . . . . . 6 ((𝜑𝑛 ∈ ℕ0) → (𝑛(.g𝑅)(0g𝑅)) = (0g𝑅))
11313psrbagf 21938 . . . . . . . . 9 (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → 𝑘:𝐼⟶ℕ0)
114113adantl 481 . . . . . . . 8 ((𝜑𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑘:𝐼⟶ℕ0)
1156adantr 480 . . . . . . . 8 ((𝜑𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑋𝐼)
116114, 115ffvelcdmd 7105 . . . . . . 7 ((𝜑𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘𝑋) ∈ ℕ0)
117 peano2nn0 12566 . . . . . . 7 ((𝑘𝑋) ∈ ℕ0 → ((𝑘𝑋) + 1) ∈ ℕ0)
118116, 117syl 17 . . . . . 6 ((𝜑𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑘𝑋) + 1) ∈ ℕ0)
119 fvexd 6921 . . . . . 6 ((𝜑𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) ∈ V)
120109, 112, 118, 119, 19suppssov2 8223 . . . . 5 (𝜑 → ((𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))) supp (0g𝑅)) ⊆ ((𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) supp (0g𝑅)))
121108, 120ssfid 9301 . . . 4 (𝜑 → ((𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))) supp (0g𝑅)) ∈ Fin)
12218, 19, 21, 121isfsuppd 9406 . . 3 (𝜑 → (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))) finSupp (0g𝑅))
12314, 122eqbrtrd 5165 . 2 (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) finSupp (0g𝑅))
1247, 1, 2, 37, 8mplelbas 22011 . 2 ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ 𝐵 ↔ ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) finSupp (0g𝑅)))
12512, 123, 124sylanbrc 583 1 (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  ifcif 4525   class class class wbr 5143  cmpt 5225   I cid 5577  ccnv 5684  ran crn 5686  cres 5687  cima 5688  ccom 5689  Fun wfun 6555   Fn wfn 6556  wf 6557  1-1wf1 6558  cfv 6561  (class class class)co 7431  f cof 7695   supp csupp 8185  m cmap 8866  Fincfn 8985   finSupp cfsupp 9401  cc 11153  0cc0 11155  1c1 11156   + caddc 11158  cmin 11492  cn 12266  0cn0 12526  Basecbs 17247  0gc0g 17484  Mgmcmgm 18651  Mndcmnd 18747  .gcmg 19085   mPwSer cmps 21924   mPoly cmpl 21926   mPSDer cpsd 22134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-seq 14043  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-tset 17316  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mulg 19086  df-psr 21929  df-mpl 21931  df-psd 22160
This theorem is referenced by: (None)
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