Theorem psdmplcl 22463

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	⊢ (𝜑 → 𝑅 ∈ Mnd)
psdmplcl.x	⊢ (𝜑 → 𝑋 ∈ 𝐼)
psdmplcl.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)

Assertion

Ref	Expression
psdmplcl	⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ 𝐵)

Proof of Theorem psdmplcl

Dummy variables 𝑏 𝑑 ℎ 𝑖 𝑘 𝑦 𝑧 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2761	. . 3 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
2		eqid 2761	. . 3 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
3		psdmplcl.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Mnd)
4		mndmgm 18910	. . . 4 ⊢ (𝑅 ∈ Mnd → 𝑅 ∈ Mgm)
5	3, 4	syl 18	. . 3 ⊢ (𝜑 → 𝑅 ∈ Mgm)
6		psdmplcl.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
7		psdmplcl.p	. . . . 5 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
8		psdmplcl.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑃)
9	7, 1, 8, 2	mplbasss 22284	. . . 4 ⊢ 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅))
10		psdmplcl.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
11	9, 10	sselid 3929	. . 3 ⊢ (𝜑 → 𝐹 ∈ (Base‘(𝐼 mPwSer 𝑅)))
12	1, 2, 5, 6, 11	psdcl 22462	. 2 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ (Base‘(𝐼 mPwSer 𝑅)))
13		eqid 2761	. . . 4 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
14	1, 2, 13, 6, 11	psdval 22460	. . 3 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) = (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑘‘𝑋) + 1)(.g‘𝑅)(𝐹‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))))
15		ovex 7445	. . . . . . 7 ⊢ (ℕ0 ↑m 𝐼) ∈ V
16	15	rabex 5300	. . . . . 6 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V
17	16	a1i 11	. . . . 5 ⊢ (𝜑 → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V)
18	17	mptexd 7222	. . . 4 ⊢ (𝜑 → (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑘‘𝑋) + 1)(.g‘𝑅)(𝐹‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))) ∈ V)
19		fvexd 6892	. . . 4 ⊢ (𝜑 → (0g‘𝑅) ∈ V)
20		funmpt 6570	. . . . 5 ⊢ Fun (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑘‘𝑋) + 1)(.g‘𝑅)(𝐹‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))
21	20	a1i 11	. . . 4 ⊢ (𝜑 → Fun (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑘‘𝑋) + 1)(.g‘𝑅)(𝐹‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))))
22		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
23		reldmmpl 22275	. . . . . . . . . . . . 13 ⊢ Rel dom mPoly
24	7, 8, 23	strov2rcl 17375	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ 𝐵 → 𝐼 ∈ V)
25	10, 24	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ V)
26	13	psrbagsn 22352	. . . . . . . . . . 11 ⊢ (𝐼 ∈ V → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
27	25, 26	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
28	27	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
29	13	psrbagaddcl 22212	. . . . . . . . 9 ⊢ ((𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
30	22, 28, 29	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
31		eqidd 2762	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
32		eqid 2761	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
33	7, 32, 8, 13, 10	mplelf 22285	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
34	33	feqmptd 6945	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘𝑧)))
35		fveq2 6877	. . . . . . . 8 ⊢ (𝑧 = (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝐹‘𝑧) = (𝐹‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
36	30, 31, 34, 35	fmptco 7122	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) = (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))
37		eqid 2761	. . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅)
38	7, 8, 37, 10	mplelsfi 22282	. . . . . . . 8 ⊢ (𝜑 → 𝐹 finSupp (0g‘𝑅))
39	30	fmpttd 7107	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
40		ovex 7445	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ V
41		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
42	40, 41	fnmpti 6674	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) Fn {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
43	42	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) Fn {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
44		dffn3 6714	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) Fn {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↔ (𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶ran (𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
45	43, 44	sylib 221	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶ran (𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
46	45, 39	fcod 6727	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) ∘ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶ran (𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
47	46	ffnd 6702	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) ∘ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) Fn {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
48		fnresi 6660	. . . . . . . . . . 11 ⊢ ( I ↾ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) Fn {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
49	48	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ( I ↾ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) Fn {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
50	13	psrbagf 22206	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑑:𝐼⟶ℕ0)
51	50	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑:𝐼⟶ℕ0)
52	51	ffvelcdmda 7076	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈ ℕ0)
53	52	nn0cnd 12650	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈ ℂ)
54		ax-1cn 11239	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
55		0cn 11279	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℂ
56	54, 55	ifcli 4530	. . . . . . . . . . . . . . 15 ⊢ if(𝑖 = 𝑋, 1, 0) ∈ ℂ
57	56	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → if(𝑖 = 𝑋, 1, 0) ∈ ℂ)
58	53, 57	pncand 11651	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) − if(𝑖 = 𝑋, 1, 0)) = (𝑑‘𝑖))
59	58	mpteq2dva 5198	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑖 ∈ 𝐼 ↦ (((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) − if(𝑖 = 𝑋, 1, 0))) = (𝑖 ∈ 𝐼 ↦ (𝑑‘𝑖)))
60		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
61	27	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
62	13	psrbagaddcl 22212	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
63	60, 61, 62	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
64	13	psrbagf 22206	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))):𝐼⟶ℕ0)
65	64	ffnd 6702	. . . . . . . . . . . . . 14 ⊢ ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) Fn 𝐼)
66	63, 65	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) Fn 𝐼)
67		1ex 11284	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ V
68		c0ex 11281	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ V
69	67, 68	ifex 4533	. . . . . . . . . . . . . . 15 ⊢ if(𝑦 = 𝑋, 1, 0) ∈ V
70		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))
71	69, 70	fnmpti 6674	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼
72	71	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼)
73	25	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ V)
74		inidm 4172	. . . . . . . . . . . . 13 ⊢ (𝐼 ∩ 𝐼) = 𝐼
75	50	ffnd 6702	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑑 Fn 𝐼)
76	75	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 Fn 𝐼)
77		eqidd 2762	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) = (𝑑‘𝑖))
78		eqeq1 2765	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑖 → (𝑦 = 𝑋 ↔ 𝑖 = 𝑋))
79	78	ifbid 4506	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑖 → if(𝑦 = 𝑋, 1, 0) = if(𝑖 = 𝑋, 1, 0))
80		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
81	67, 68	ifex 4533	. . . . . . . . . . . . . . . 16 ⊢ if(𝑖 = 𝑋, 1, 0) ∈ V
82	81	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → if(𝑖 = 𝑋, 1, 0) ∈ V)
83	70, 79, 80, 82	fvmptd3 7009	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑖) = if(𝑖 = 𝑋, 1, 0))
84	76, 72, 73, 73, 74, 77, 83	ofval 7693	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑖) = ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)))
85	66, 72, 73, 73, 74, 84, 83	offval 7691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑖 ∈ 𝐼 ↦ (((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) − if(𝑖 = 𝑋, 1, 0))))
86	51	feqmptd 6945	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 = (𝑖 ∈ 𝐼 ↦ (𝑑‘𝑖)))
87	59, 85, 86	3eqtr4d 2806	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = 𝑑)
88	30	adantlr 728	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
89	88	fmpttd 7107	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
90	89, 60	fvco3d 6978	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) ∘ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))‘𝑑) = ((𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))‘((𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))‘𝑑)))
91		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
92		oveq1 7419	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑑 → (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
93		ovexd 7447	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ V)
94	91, 92, 60, 93	fvmptd3 7009	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))‘𝑑) = (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
95	94	fveq2d 6881	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))‘((𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))‘𝑑)) = ((𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))‘(𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
96		oveq1 7419	. . . . . . . . . . . . 13 ⊢ (𝑏 = (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
97		ovexd 7447	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ V)
98	41, 96, 63, 97	fvmptd3 7009	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))‘(𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
99	90, 95, 98	3eqtrd 2800	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) ∘ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))‘𝑑) = ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
100		fvresi 7170	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → (( I ↾ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})‘𝑑) = 𝑑)
101	100	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (( I ↾ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})‘𝑑) = 𝑑)
102	87, 99, 101	3eqtr4d 2806	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) ∘ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))‘𝑑) = (( I ↾ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})‘𝑑))
103	47, 49, 102	eqfnfvd 7024	. . . . . . . . 9 ⊢ (𝜑 → ((𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) ∘ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) = ( I ↾ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}))
104		fcof1 7287	. . . . . . . . 9 ⊢ (((𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ ((𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) ∘ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) = ( I ↾ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})) → (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}–1-1→{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
105	39, 103, 104	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}–1-1→{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
106	38, 105, 19, 10	fsuppco 9378	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) finSupp (0g‘𝑅))
107	36, 106	eqbrtrrd 5129	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) finSupp (0g‘𝑅))
108	107	fsuppimpd 9345	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) supp (0g‘𝑅)) ∈ Fin)
109		ssidd 3954	. . . . . 6 ⊢ (𝜑 → ((𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) supp (0g‘𝑅)) ⊆ ((𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) supp (0g‘𝑅)))
110		eqid 2761	. . . . . . . 8 ⊢ (.g‘𝑅) = (.g‘𝑅)
111	32, 110, 37	mulgnn0z 19291	. . . . . . 7 ⊢ ((𝑅 ∈ Mnd ∧ 𝑛 ∈ ℕ0) → (𝑛(.g‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
112	3, 111	sylan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑛(.g‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
113	13	psrbagf 22206	. . . . . . . . 9 ⊢ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑘:𝐼⟶ℕ0)
114	113	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑘:𝐼⟶ℕ0)
115	6	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑋 ∈ 𝐼)
116	114, 115	ffvelcdmd 7077	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑘‘𝑋) ∈ ℕ0)
117		peano2nn0 12627	. . . . . . 7 ⊢ ((𝑘‘𝑋) ∈ ℕ0 → ((𝑘‘𝑋) + 1) ∈ ℕ0)
118	116, 117	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑘‘𝑋) + 1) ∈ ℕ0)
119		fvexd 6892	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝐹‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) ∈ V)
120	109, 112, 118, 119, 19	suppssov2 8199	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑘‘𝑋) + 1)(.g‘𝑅)(𝐹‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))) supp (0g‘𝑅)) ⊆ ((𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) supp (0g‘𝑅)))
121	108, 120	ssfid 9244	. . . 4 ⊢ (𝜑 → ((𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑘‘𝑋) + 1)(.g‘𝑅)(𝐹‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))) supp (0g‘𝑅)) ∈ Fin)
122	18, 19, 21, 121	isfsuppd 9342	. . 3 ⊢ (𝜑 → (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑘‘𝑋) + 1)(.g‘𝑅)(𝐹‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))) finSupp (0g‘𝑅))
123	14, 122	eqbrtrd 5127	. 2 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) finSupp (0g‘𝑅))
124	7, 1, 2, 37, 8	mplelbas 22278	. 2 ⊢ ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ 𝐵 ↔ ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) finSupp (0g‘𝑅)))
125	12, 123, 124	sylanbrc 595	1 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3413  Vcvv 3451  ifcif 4482   class class class wbr 5103   ↦ cmpt 5186   I cid 5545  ^◡ccnv 5650  ran crn 5652   ↾ cres 5653   “ cima 5654   ∘ ccom 5655  Fun wfun 6525   Fn wfn 6526  ⟶wf 6527  –1-1→wf1 6528  ‘cfv 6531  (class class class)co 7412   ∘_f cof 7680   supp csupp 8161   ↑_m cmap 8831  Fincfn 8957   finSupp cfsupp 9337  ℂcc 11179  0cc0 11181  1c1 11182   + caddc 11184   − cmin 11522  ℕcn 12316  ℕ₀cn0 12587  Basecbs 17367  0_gc0g 17590  Mgmcmgm 18794  Mndcmnd 18903  ._gcmg 19257   mPwSer cmps 22192   mPoly cmpl 22194   mPSDer cpsd 22435

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7682  df-om 7867  df-1st 7990  df-2nd 7991  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-er 8701  df-map 8833  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9338  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-2 12386  df-3 12387  df-4 12388  df-5 12389  df-6 12390  df-7 12391  df-8 12392  df-9 12393  df-n0 12588  df-z 12675  df-uz 12947  df-fz 13621  df-seq 14125  df-struct 17305  df-sets 17322  df-slot 17340  df-ndx 17352  df-base 17368  df-ress 17389  df-plusg 17421  df-mulr 17422  df-sca 17424  df-vsca 17425  df-tset 17427  df-0g 17592  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-mulg 19258  df-psr 22197  df-mpl 22199  df-psd 22457

This theorem is used by: (None)