psdmplcl - Metamath Proof Explorer

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Theorem psdmplcl 22100

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 2735	. . 3 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
2		eqid 2735	. . 3 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
3		psdmplcl.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Mnd)
4		mndmgm 18719	. . . 4 ⊢ (𝑅 ∈ Mnd → 𝑅 ∈ Mgm)
5	3, 4	syl 17	. . 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 21957	. . . 4 ⊢ 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅))
10		psdmplcl.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
11	9, 10	sselid 3956	. . 3 ⊢ (𝜑 → 𝐹 ∈ (Base‘(𝐼 mPwSer 𝑅)))
12	1, 2, 5, 6, 11	psdcl 22099	. 2 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ (Base‘(𝐼 mPwSer 𝑅)))
13		eqid 2735	. . . 4 ⊢ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
14	1, 2, 13, 6, 11	psdval 22097	. . 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 7438	. . . . . . 7 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
16	15	rabex 5309	. . . . . 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 7216	. . . 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 6891	. . . 4 ⊢ (𝜑 → (0_g‘𝑅) ∈ V)
20		funmpt 6574	. . . . 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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝑘 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin})
23		reldmmpl 21948	. . . . . . . . . . . . 13 ⊢ Rel dom mPoly
24	7, 8, 23	strov2rcl 17236	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ 𝐵 → 𝐼 ∈ V)
25	10, 24	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ V)
26	13	psrbagsn 22021	. . . . . . . . . . 11 ⊢ (𝐼 ∈ V → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin})
27	25, 26	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin})
28	27	adantr 480	. . . . . . . . 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 21884	. . . . . . . . 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 584	. . . . . . . 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 2736	. . . . . . . 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 2735	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
33	7, 32, 8, 13, 10	mplelf 21958	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:{ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
34	33	feqmptd 6947	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘𝑧)))
35		fveq2 6876	. . . . . . . 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 7119	. . . . . . 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 2735	. . . . . . . . 9 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
38	7, 8, 37, 10	mplelsfi 21955	. . . . . . . 8 ⊢ (𝜑 → 𝐹 finSupp (0_g‘𝑅))
39	30	fmpttd 7105	. . . . . . . . 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 7438	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∘_f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ V
41		eqid 2735	. . . . . . . . . . . . . . 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 6681	. . . . . . . . . . . . . 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 6718	. . . . . . . . . . . . 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 218	. . . . . . . . . . . 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 6731	. . . . . . . . . . 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 6707	. . . . . . . . . 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 6667	. . . . . . . . . . 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 21878	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} → 𝑑:𝐼⟶ℕ₀)
51	50	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝑑:𝐼⟶ℕ₀)
52	51	ffvelcdmda 7074	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈ ℕ₀)
53	52	nn0cnd 12564	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈ ℂ)
54		ax-1cn 11187	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
55		0cn 11227	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℂ
56	54, 55	ifcli 4548	. . . . . . . . . . . . . . 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 11595	. . . . . . . . . . . . 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 5214	. . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin})
61	27	adantr 480	. . . . . . . . . . . . . . 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 21884	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 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 21878	. . . . . . . . . . . . . . 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 6707	. . . . . . . . . . . . . 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 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ∘_f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) Fn 𝐼)
67		1ex 11231	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ V
68		c0ex 11229	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ V
69	67, 68	ifex 4551	. . . . . . . . . . . . . . 15 ⊢ if(𝑦 = 𝑋, 1, 0) ∈ V
70		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))
71	69, 70	fnmpti 6681	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ V)
74		inidm 4202	. . . . . . . . . . . . 13 ⊢ (𝐼 ∩ 𝐼) = 𝐼
75	50	ffnd 6707	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} → 𝑑 Fn 𝐼)
76	75	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝑑 Fn 𝐼)
77		eqidd 2736	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) = (𝑑‘𝑖))
78		eqeq1 2739	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑖 → (𝑦 = 𝑋 ↔ 𝑖 = 𝑋))
79	78	ifbid 4524	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑖 → if(𝑦 = 𝑋, 1, 0) = if(𝑖 = 𝑋, 1, 0))
80		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
81	67, 68	ifex 4551	. . . . . . . . . . . . . . . 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 7682	. . . . . . . . . . . . 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 7680	. . . . . . . . . . . 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 6947	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝑑 = (𝑖 ∈ 𝐼 ↦ (𝑑‘𝑖)))
87	59, 85, 86	3eqtr4d 2780	. . . . . . . . . . 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 715	. . . . . . . . . . . . . 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 7105	. . . . . . . . . . . . 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 6979	. . . . . . . . . . . 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 2735	. . . . . . . . . . . . . 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 7412	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑑 → (𝑘 ∘_f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑑 ∘_f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
93		ovexd 7440	. . . . . . . . . . . . . 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 6880	. . . . . . . . . . . 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 7412	. . . . . . . . . . . . 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 7440	. . . . . . . . . . . . 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 2774	. . . . . . . . . . 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 7165	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} → (( I ↾ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin})‘𝑑) = 𝑑)
101	100	adantl 481	. . . . . . . . . . 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 2780	. . . . . . . . . 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 7280	. . . . . . . . 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 584	. . . . . . . 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 9414	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ (𝑘 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ↦ (𝑘 ∘_f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) finSupp (0_g‘𝑅))
107	36, 106	eqbrtrrd 5143	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘 ∘_f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) finSupp (0_g‘𝑅))
108	107	fsuppimpd 9381	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘 ∘_f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) supp (0_g‘𝑅)) ∈ Fin)
109		ssidd 3982	. . . . . 6 ⊢ (𝜑 → ((𝑘 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘 ∘_f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) supp (0_g‘𝑅)) ⊆ ((𝑘 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘 ∘_f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) supp (0_g‘𝑅)))
110		eqid 2735	. . . . . . . 8 ⊢ (._g‘𝑅) = (._g‘𝑅)
111	32, 110, 37	mulgnn0z 19084	. . . . . . 7 ⊢ ((𝑅 ∈ Mnd ∧ 𝑛 ∈ ℕ₀) → (𝑛(._g‘𝑅)(0_g‘𝑅)) = (0_g‘𝑅))
112	3, 111	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝑛(._g‘𝑅)(0_g‘𝑅)) = (0_g‘𝑅))
113	13	psrbagf 21878	. . . . . . . . 9 ⊢ (𝑘 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} → 𝑘:𝐼⟶ℕ₀)
114	113	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝑘:𝐼⟶ℕ₀)
115	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝑋 ∈ 𝐼)
116	114, 115	ffvelcdmd 7075	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → (𝑘‘𝑋) ∈ ℕ₀)
117		peano2nn0 12541	. . . . . . 7 ⊢ ((𝑘‘𝑋) ∈ ℕ₀ → ((𝑘‘𝑋) + 1) ∈ ℕ₀)
118	116, 117	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → ((𝑘‘𝑋) + 1) ∈ ℕ₀)
119		fvexd 6891	. . . . . 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 8197	. . . . 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 (0_g‘𝑅)) ⊆ ((𝑘 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘(𝑘 ∘_f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) supp (0_g‘𝑅)))
121	108, 120	ssfid 9273	. . . 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 (0_g‘𝑅)) ∈ Fin)
122	18, 19, 21, 121	isfsuppd 9378	. . 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 (0_g‘𝑅))
123	14, 122	eqbrtrd 5141	. 2 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) finSupp (0_g‘𝑅))
124	7, 1, 2, 37, 8	mplelbas 21951	. 2 ⊢ ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ 𝐵 ↔ ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) finSupp (0_g‘𝑅)))
125	12, 123, 124	sylanbrc 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 3415 Vcvv 3459 ifcif 4500 class class class wbr 5119 ↦ cmpt 5201 I cid 5547 ^◡ccnv 5653 ran crn 5655 ↾ cres 5656 “ cima 5657 ∘ ccom 5658 Fun wfun 6525 Fn wfn 6526 ⟶wf 6527 –1-1→wf1 6528 ‘cfv 6531 (class class class)co 7405 ∘_f cof 7669 supp csupp 8159 ↑_m cmap 8840 Fincfn 8959 finSupp cfsupp 9373 ℂcc 11127 0cc0 11129 1c1 11130 + caddc 11132 − cmin 11466 ℕcn 12240 ℕ₀cn0 12501 Basecbs 17228 0_gc0g 17453 Mgmcmgm 18616 Mndcmnd 18712 ._gcmg 19050 mPwSer cmps 21864 mPoly cmpl 21866 mPSDer cpsd 22068

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 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-uz 12853 df-fz 13525 df-seq 14020 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-sca 17287 df-vsca 17288 df-tset 17290 df-0g 17455 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-mulg 19051 df-psr 21869 df-mpl 21871 df-psd 22094

This theorem is referenced by: (None)

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