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Theorem psdmplcl 22183
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 2734 . . 3 (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
2 eqid 2734 . . 3 (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
3 psdmplcl.r . . . 4 (𝜑𝑅 ∈ Mnd)
4 mndmgm 18766 . . . 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 22034 . . . 4 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅))
10 psdmplcl.f . . . 4 (𝜑𝐹𝐵)
119, 10sselid 3992 . . 3 (𝜑𝐹 ∈ (Base‘(𝐼 mPwSer 𝑅)))
121, 2, 5, 6, 11psdcl 22182 . 2 (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ (Base‘(𝐼 mPwSer 𝑅)))
13 eqid 2734 . . . 4 { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
141, 2, 13, 6, 11psdval 22180 . . 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 7463 . . . . . . 7 (ℕ0m 𝐼) ∈ V
1615rabex 5344 . . . . . 6 { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∈ V
1716a1i 11 . . . . 5 (𝜑 → { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∈ V)
1817mptexd 7243 . . . 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 6605 . . . . 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 22025 . . . . . . . . . . . . 13 Rel dom mPoly
247, 8, 23strov2rcl 17252 . . . . . . . . . . . 12 (𝐹𝐵𝐼 ∈ V)
2510, 24syl 17 . . . . . . . . . . 11 (𝜑𝐼 ∈ V)
2613psrbagsn 22104 . . . . . . . . . . 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 21961 . . . . . . . . 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 2735 . . . . . . . 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 2734 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
337, 32, 8, 13, 10mplelf 22035 . . . . . . . . 9 (𝜑𝐹:{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶(Base‘𝑅))
3433feqmptd 6976 . . . . . . . 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 7148 . . . . . . 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 2734 . . . . . . . . 9 (0g𝑅) = (0g𝑅)
387, 8, 37, 10mplelsfi 22032 . . . . . . . 8 (𝜑𝐹 finSupp (0g𝑅))
3930fmpttd 7134 . . . . . . . . 9 (𝜑 → (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))):{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
40 ovex 7463 . . . . . . . . . . . . . . 15 (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ V
41 eqid 2734 . . . . . . . . . . . . . . 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 21955 . . . . . . . . . . . . . . . . 17 (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → 𝑑:𝐼⟶ℕ0)
5150adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑:𝐼⟶ℕ0)
5251ffvelcdmda 7103 . . . . . . . . . . . . . . 15 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (𝑑𝑖) ∈ ℕ0)
5352nn0cnd 12586 . . . . . . . . . . . . . 14 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (𝑑𝑖) ∈ ℂ)
54 ax-1cn 11210 . . . . . . . . . . . . . . . 16 1 ∈ ℂ
55 0cn 11250 . . . . . . . . . . . . . . . 16 0 ∈ ℂ
5654, 55ifcli 4577 . . . . . . . . . . . . . . 15 if(𝑖 = 𝑋, 1, 0) ∈ ℂ
5756a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → if(𝑖 = 𝑋, 1, 0) ∈ ℂ)
5853, 57pncand 11618 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (((𝑑𝑖) + if(𝑖 = 𝑋, 1, 0)) − if(𝑖 = 𝑋, 1, 0)) = (𝑑𝑖))
5958mpteq2dva 5247 . . . . . . . . . . . 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 21961 . . . . . . . . . . . . . . 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 21955 . . . . . . . . . . . . . . 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 11254 . . . . . . . . . . . . . . . 16 1 ∈ V
68 c0ex 11252 . . . . . . . . . . . . . . . 16 0 ∈ V
6967, 68ifex 4580 . . . . . . . . . . . . . . 15 if(𝑦 = 𝑋, 1, 0) ∈ V
70 eqid 2734 . . . . . . . . . . . . . . 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 4234 . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . 14 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (𝑑𝑖) = (𝑑𝑖))
78 eqeq1 2738 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑖 → (𝑦 = 𝑋𝑖 = 𝑋))
7978ifbid 4553 . . . . . . . . . . . . . . 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 4580 . . . . . . . . . . . . . . . 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 7038 . . . . . . . . . . . . . 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 7707 . . . . . . . . . . . . 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 7705 . . . . . . . . . . . 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 6976 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑 = (𝑖𝐼 ↦ (𝑑𝑖)))
8759, 85, 863eqtr4d 2784 . . . . . . . . . . 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 7134 . . . . . . . . . . . . 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 7008 . . . . . . . . . . . 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 2734 . . . . . . . . . . . . . 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 7437 . . . . . . . . . . . . . 14 (𝑘 = 𝑑 → (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
93 ovexd 7465 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ V)
9491, 92, 60, 93fvmptd3 7038 . . . . . . . . . . . . 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 7437 . . . . . . . . . . . . 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 7465 . . . . . . . . . . . . 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 7038 . . . . . . . . . . . 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 2778 . . . . . . . . . . 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 7192 . . . . . . . . . . . 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 2784 . . . . . . . . . 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 7053 . . . . . . . . 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 7306 . . . . . . . . 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 9439 . . . . . . 7 (𝜑 → (𝐹 ∘ (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) finSupp (0g𝑅))
10736, 106eqbrtrrd 5171 . . . . . 6 (𝜑 → (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) finSupp (0g𝑅))
108107fsuppimpd 9406 . . . . 5 (𝜑 → ((𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) supp (0g𝑅)) ∈ Fin)
109 ssidd 4018 . . . . . 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 2734 . . . . . . . 8 (.g𝑅) = (.g𝑅)
11132, 110, 37mulgnn0z 19131 . . . . . . 7 ((𝑅 ∈ Mnd ∧ 𝑛 ∈ ℕ0) → (𝑛(.g𝑅)(0g𝑅)) = (0g𝑅))
1123, 111sylan 580 . . . . . 6 ((𝜑𝑛 ∈ ℕ0) → (𝑛(.g𝑅)(0g𝑅)) = (0g𝑅))
11313psrbagf 21955 . . . . . . . . 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 7104 . . . . . . 7 ((𝜑𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘𝑋) ∈ ℕ0)
117 peano2nn0 12563 . . . . . . 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 8221 . . . . 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 9298 . . . 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 9403 . . 3 (𝜑 → (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))) finSupp (0g𝑅))
12314, 122eqbrtrd 5169 . 2 (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) finSupp (0g𝑅))
1247, 1, 2, 37, 8mplelbas 22028 . 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 1536  wcel 2105  {crab 3432  Vcvv 3477  ifcif 4530   class class class wbr 5147  cmpt 5230   I cid 5581  ccnv 5687  ran crn 5689  cres 5690  cima 5691  ccom 5692  Fun wfun 6556   Fn wfn 6557  wf 6558  1-1wf1 6559  cfv 6562  (class class class)co 7430  f cof 7694   supp csupp 8183  m cmap 8864  Fincfn 8983   finSupp cfsupp 9398  cc 11150  0cc0 11152  1c1 11153   + caddc 11155  cmin 11489  cn 12263  0cn0 12523  Basecbs 17244  0gc0g 17485  Mgmcmgm 18663  Mndcmnd 18759  .gcmg 19097   mPwSer cmps 21941   mPoly cmpl 21943   mPSDer cpsd 22151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-seq 14039  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-tset 17316  df-0g 17487  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mulg 19098  df-psr 21946  df-mpl 21948  df-psd 22177
This theorem is referenced by: (None)
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