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Theorem psdmplcl 22093
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.i (𝜑𝐼𝑉)
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 2728 . . 3 (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
2 eqid 2728 . . 3 (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
3 psdmplcl.i . . 3 (𝜑𝐼𝑉)
4 psdmplcl.r . . . 4 (𝜑𝑅 ∈ Mnd)
5 mndmgm 18708 . . . 4 (𝑅 ∈ Mnd → 𝑅 ∈ Mgm)
64, 5syl 17 . . 3 (𝜑𝑅 ∈ Mgm)
7 psdmplcl.x . . 3 (𝜑𝑋𝐼)
8 psdmplcl.p . . . . 5 𝑃 = (𝐼 mPoly 𝑅)
9 psdmplcl.b . . . . 5 𝐵 = (Base‘𝑃)
108, 1, 9, 2mplbasss 21946 . . . 4 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅))
11 psdmplcl.f . . . 4 (𝜑𝐹𝐵)
1210, 11sselid 3980 . . 3 (𝜑𝐹 ∈ (Base‘(𝐼 mPwSer 𝑅)))
131, 2, 3, 6, 7, 12psdcl 22092 . 2 (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ (Base‘(𝐼 mPwSer 𝑅)))
14 eqid 2728 . . . 4 { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
151, 2, 14, 3, 6, 7, 12psdval 22090 . . 3 (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) = (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))))
16 ovex 7459 . . . . . . 7 (ℕ0m 𝐼) ∈ V
1716rabex 5338 . . . . . 6 { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∈ V
1817a1i 11 . . . . 5 (𝜑 → { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∈ V)
1918mptexd 7242 . . . 4 (𝜑 → (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))) ∈ V)
20 fvexd 6917 . . . 4 (𝜑 → (0g𝑅) ∈ V)
21 funmpt 6596 . . . . 5 Fun (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))
2221a1i 11 . . . 4 (𝜑 → Fun (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))))
23 simpr 483 . . . . . . . . 9 ((𝜑𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
2414psrbagsn 22014 . . . . . . . . . . 11 (𝐼𝑉 → (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
253, 24syl 17 . . . . . . . . . 10 (𝜑 → (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
2625adantr 479 . . . . . . . . 9 ((𝜑𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
2714psrbagaddcl 21868 . . . . . . . . 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})
2823, 26, 27syl2anc 582 . . . . . . . 8 ((𝜑𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
29 eqidd 2729 . . . . . . . 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)))))
30 eqid 2728 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
318, 30, 9, 14, 11mplelf 21947 . . . . . . . . 9 (𝜑𝐹:{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶(Base‘𝑅))
3231feqmptd 6972 . . . . . . . 8 (𝜑𝐹 = (𝑧 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹𝑧)))
33 fveq2 6902 . . . . . . . 8 (𝑧 = (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝐹𝑧) = (𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
3428, 29, 32, 33fmptco 7144 . . . . . . 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))))))
35 eqid 2728 . . . . . . . . 9 (0g𝑅) = (0g𝑅)
368, 9, 35, 11, 4mplelsfi 21944 . . . . . . . 8 (𝜑𝐹 finSupp (0g𝑅))
3728fmpttd 7130 . . . . . . . . 9 (𝜑 → (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))):{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
38 ovex 7459 . . . . . . . . . . . . . . 15 (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ V
39 eqid 2728 . . . . . . . . . . . . . . 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))))
4038, 39fnmpti 6703 . . . . . . . . . . . . . 14 (𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) Fn { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
4140a1i 11 . . . . . . . . . . . . 13 (𝜑 → (𝑏 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑏f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) Fn { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
42 dffn3 6740 . . . . . . . . . . . . 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)))))
4341, 42sylib 217 . . . . . . . . . . . 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)))))
4443, 37fcod 6754 . . . . . . . . . . 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)))))
4544ffnd 6728 . . . . . . . . . 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})
46 fnresi 6689 . . . . . . . . . . 11 ( I ↾ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) Fn { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
4746a1i 11 . . . . . . . . . 10 (𝜑 → ( I ↾ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) Fn { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
4814psrbagf 21858 . . . . . . . . . . . . . . . . 17 (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → 𝑑:𝐼⟶ℕ0)
4948adantl 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑:𝐼⟶ℕ0)
5049ffvelcdmda 7099 . . . . . . . . . . . . . . 15 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (𝑑𝑖) ∈ ℕ0)
5150nn0cnd 12572 . . . . . . . . . . . . . 14 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (𝑑𝑖) ∈ ℂ)
52 ax-1cn 11204 . . . . . . . . . . . . . . . 16 1 ∈ ℂ
53 0cn 11244 . . . . . . . . . . . . . . . 16 0 ∈ ℂ
5452, 53ifcli 4579 . . . . . . . . . . . . . . 15 if(𝑖 = 𝑋, 1, 0) ∈ ℂ
5554a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → if(𝑖 = 𝑋, 1, 0) ∈ ℂ)
5651, 55pncand 11610 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (((𝑑𝑖) + if(𝑖 = 𝑋, 1, 0)) − if(𝑖 = 𝑋, 1, 0)) = (𝑑𝑖))
5756mpteq2dva 5252 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑖𝐼 ↦ (((𝑑𝑖) + if(𝑖 = 𝑋, 1, 0)) − if(𝑖 = 𝑋, 1, 0))) = (𝑖𝐼 ↦ (𝑑𝑖)))
58 simpr 483 . . . . . . . . . . . . . . 15 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
5925adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
6014psrbagaddcl 21868 . . . . . . . . . . . . . . 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})
6158, 59, 60syl2anc 582 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
6214psrbagf 21858 . . . . . . . . . . . . . . 15 ((𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → (𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))):𝐼⟶ℕ0)
6362ffnd 6728 . . . . . . . . . . . . . 14 ((𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → (𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) Fn 𝐼)
6461, 63syl 17 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) Fn 𝐼)
65 1ex 11248 . . . . . . . . . . . . . . . 16 1 ∈ V
66 c0ex 11246 . . . . . . . . . . . . . . . 16 0 ∈ V
6765, 66ifex 4582 . . . . . . . . . . . . . . 15 if(𝑦 = 𝑋, 1, 0) ∈ V
68 eqid 2728 . . . . . . . . . . . . . . 15 (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))
6967, 68fnmpti 6703 . . . . . . . . . . . . . 14 (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼
7069a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼)
713adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝐼𝑉)
72 inidm 4221 . . . . . . . . . . . . 13 (𝐼𝐼) = 𝐼
7348ffnd 6728 . . . . . . . . . . . . . . 15 (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → 𝑑 Fn 𝐼)
7473adantl 480 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑 Fn 𝐼)
75 eqidd 2729 . . . . . . . . . . . . . 14 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → (𝑑𝑖) = (𝑑𝑖))
76 eqeq1 2732 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑖 → (𝑦 = 𝑋𝑖 = 𝑋))
7776ifbid 4555 . . . . . . . . . . . . . . 15 (𝑦 = 𝑖 → if(𝑦 = 𝑋, 1, 0) = if(𝑖 = 𝑋, 1, 0))
78 simpr 483 . . . . . . . . . . . . . . 15 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → 𝑖𝐼)
7965, 66ifex 4582 . . . . . . . . . . . . . . . 16 if(𝑖 = 𝑋, 1, 0) ∈ V
8079a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → if(𝑖 = 𝑋, 1, 0) ∈ V)
8168, 77, 78, 80fvmptd3 7033 . . . . . . . . . . . . . 14 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → ((𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑖) = if(𝑖 = 𝑋, 1, 0))
8274, 70, 71, 71, 72, 75, 81ofval 7702 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑖𝐼) → ((𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑖) = ((𝑑𝑖) + if(𝑖 = 𝑋, 1, 0)))
8364, 70, 71, 71, 72, 82, 81offval 7700 . . . . . . . . . . . 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))))
8449feqmptd 6972 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑑 = (𝑖𝐼 ↦ (𝑑𝑖)))
8557, 83, 843eqtr4d 2778 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = 𝑑)
8628adantlr 713 . . . . . . . . . . . . . 14 (((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
8786fmpttd 7130 . . . . . . . . . . . . 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})
8887, 58fvco3d 7003 . . . . . . . . . . . 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)))
89 eqid 2728 . . . . . . . . . . . . . 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))))
90 oveq1 7433 . . . . . . . . . . . . . 14 (𝑘 = 𝑑 → (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
91 ovexd 7461 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ V)
9289, 90, 58, 91fvmptd3 7033 . . . . . . . . . . . . 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))))
9392fveq2d 6906 . . . . . . . . . . . 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)))))
94 oveq1 7433 . . . . . . . . . . . . 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))))
95 ovexd 7461 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑑f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ V)
9639, 94, 61, 95fvmptd3 7033 . . . . . . . . . . . 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))))
9788, 93, 963eqtrd 2772 . . . . . . . . . . 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))))
98 fvresi 7188 . . . . . . . . . . . 12 (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → (( I ↾ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})‘𝑑) = 𝑑)
9998adantl 480 . . . . . . . . . . 11 ((𝜑𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (( I ↾ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})‘𝑑) = 𝑑)
10085, 97, 993eqtr4d 2778 . . . . . . . . . 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))
10145, 47, 100eqfnfvd 7048 . . . . . . . . 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}))
102 fcof1 7302 . . . . . . . . 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})
10337, 101, 102syl2anc 582 . . . . . . . 8 (𝜑 → (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))):{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}–1-1→{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
10436, 103, 20, 11fsuppco 9433 . . . . . . 7 (𝜑 → (𝐹 ∘ (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) finSupp (0g𝑅))
10534, 104eqbrtrrd 5176 . . . . . 6 (𝜑 → (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) finSupp (0g𝑅))
106105fsuppimpd 9401 . . . . 5 (𝜑 → ((𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) supp (0g𝑅)) ∈ Fin)
107 ssidd 4005 . . . . . 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)))
108 eqid 2728 . . . . . . . 8 (.g𝑅) = (.g𝑅)
10930, 108, 35mulgnn0z 19063 . . . . . . 7 ((𝑅 ∈ Mnd ∧ 𝑛 ∈ ℕ0) → (𝑛(.g𝑅)(0g𝑅)) = (0g𝑅))
1104, 109sylan 578 . . . . . 6 ((𝜑𝑛 ∈ ℕ0) → (𝑛(.g𝑅)(0g𝑅)) = (0g𝑅))
11114psrbagf 21858 . . . . . . . . 9 (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → 𝑘:𝐼⟶ℕ0)
112111adantl 480 . . . . . . . 8 ((𝜑𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑘:𝐼⟶ℕ0)
1137adantr 479 . . . . . . . 8 ((𝜑𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑋𝐼)
114112, 113ffvelcdmd 7100 . . . . . . 7 ((𝜑𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑘𝑋) ∈ ℕ0)
115 peano2nn0 12550 . . . . . . 7 ((𝑘𝑋) ∈ ℕ0 → ((𝑘𝑋) + 1) ∈ ℕ0)
116114, 115syl 17 . . . . . 6 ((𝜑𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑘𝑋) + 1) ∈ ℕ0)
117 fvexd 6917 . . . . . 6 ((𝜑𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) ∈ V)
118107, 110, 116, 117, 20suppssov2 8210 . . . . 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)))
119106, 118ssfid 9298 . . . 4 (𝜑 → ((𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))) supp (0g𝑅)) ∈ Fin)
12019, 20, 22, 119isfsuppd 9398 . . 3 (𝜑 → (𝑘 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))) finSupp (0g𝑅))
12115, 120eqbrtrd 5174 . 2 (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) finSupp (0g𝑅))
1228, 1, 2, 35, 9mplelbas 21940 . 2 ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ 𝐵 ↔ ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) finSupp (0g𝑅)))
12313, 121, 122sylanbrc 581 1 (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  {crab 3430  Vcvv 3473  ifcif 4532   class class class wbr 5152  cmpt 5235   I cid 5579  ccnv 5681  ran crn 5683  cres 5684  cima 5685  ccom 5686  Fun wfun 6547   Fn wfn 6548  wf 6549  1-1wf1 6550  cfv 6553  (class class class)co 7426  f cof 7689   supp csupp 8171  m cmap 8851  Fincfn 8970   finSupp cfsupp 9393  cc 11144  0cc0 11146  1c1 11147   + caddc 11149  cmin 11482  cn 12250  0cn0 12510  Basecbs 17187  0gc0g 17428  Mgmcmgm 18605  Mndcmnd 18701  .gcmg 19030   mPwSer cmps 21844   mPoly cmpl 21846   mPSDer cpsd 22063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-of 7691  df-om 7877  df-1st 7999  df-2nd 8000  df-supp 8172  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-fsupp 9394  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-uz 12861  df-fz 13525  df-seq 14007  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-mulr 17254  df-sca 17256  df-vsca 17257  df-tset 17259  df-0g 17430  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-mulg 19031  df-psr 21849  df-mpl 21851  df-psd 22087
This theorem is referenced by: (None)
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