Theorem psdmul 22113

Description: Product rule for power series. An outline is available at https://github.com/icecream17/Stuff/blob/main/math/psdmul.pdf. (Contributed by SN, 25-Apr-2025.)

Hypotheses

Ref	Expression
psdmul.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
psdmul.b	⊢ 𝐵 = (Base‘𝑆)
psdmul.p	⊢ + = (+g‘𝑆)
psdmul.m	⊢ · = (.r‘𝑆)
psdmul.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
psdmul.r	⊢ (𝜑 → 𝑅 ∈ CRing)
psdmul.x	⊢ (𝜑 → 𝑋 ∈ 𝐼)
psdmul.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
psdmul.g	⊢ (𝜑 → 𝐺 ∈ 𝐵)

Assertion

Ref	Expression
psdmul	⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝐹 · 𝐺)) = (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) · 𝐺) + (𝐹 · (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺))))

Proof of Theorem psdmul

Dummy variables 𝑏 𝑑 𝑖 𝑘 𝑚 𝑛 𝑜 𝑝 𝑞 𝑟 𝑠 𝑢 𝑣 ℎ 𝑙 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2725	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2725	. . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅)
3		psdmul.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ CRing)
4	3	crngringd 20198	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
5	4	ringcmnd 20232	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ CMnd)
6	5	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ CMnd)
7		simpr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
8		psdmul.i	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
9		eqid 2725	. . . . . . . . . . 11 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
10	9	psrbagsn 22029	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝑉 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
11	8, 10	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
12	11	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
13	9	psrbagaddcl 21878	. . . . . . . 8 ⊢ ((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})
14	7, 12, 13	syl2anc 582	. . . . . . 7 ⊢ ((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})
15	9	psrbaglefi 21882	. . . . . . 7 ⊢ ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∈ Fin)
16	14, 15	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∈ Fin)
17		eqid 2725	. . . . . . 7 ⊢ (.g‘𝑅) = (.g‘𝑅)
18	3	crnggrpd 20199	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Grp)
19	18	grpmndd 18911	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Mnd)
20	19	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}) → 𝑅 ∈ Mnd)
21	9	psrbagf 21868	. . . . . . . . . . 11 ⊢ (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑑:𝐼⟶ℕ0)
22	21	adantl 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑:𝐼⟶ℕ0)
23		psdmul.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
24	23	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑋 ∈ 𝐼)
25	22, 24	ffvelcdmd 7094	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑‘𝑋) ∈ ℕ0)
26		peano2nn0 12545	. . . . . . . . 9 ⊢ ((𝑑‘𝑋) ∈ ℕ0 → ((𝑑‘𝑋) + 1) ∈ ℕ0)
27	25, 26	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑑‘𝑋) + 1) ∈ ℕ0)
28	27	adantr 479	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}) → ((𝑑‘𝑋) + 1) ∈ ℕ0)
29		eqid 2725	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
30	4	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}) → 𝑅 ∈ Ring)
31		psdmul.s	. . . . . . . . . . 11 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
32		psdmul.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑆)
33		psdmul.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
34	31, 1, 9, 32, 33	psrelbas 21896	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
35	34	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}) → 𝐹:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
36		elrabi 3673	. . . . . . . . . 10 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} → 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
37	36	adantl 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}) → 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
38	35, 37	ffvelcdmd 7094	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}) → (𝐹‘𝑢) ∈ (Base‘𝑅))
39		psdmul.g	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
40	31, 1, 9, 32, 39	psrelbas 21896	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
41	40	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}) → 𝐺:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
42		eqid 2725	. . . . . . . . . . . 12 ⊢ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} = {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}
43	9, 42	psrbagconcl 21884	. . . . . . . . . . 11 ⊢ (((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}) → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))})
44	14, 43	sylan 578	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}) → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))})
45		elrabi 3673	. . . . . . . . . 10 ⊢ (((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
46	44, 45	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}) → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
47	41, 46	ffvelcdmd 7094	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}) → (𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)) ∈ (Base‘𝑅))
48	1, 29, 30, 38, 47	ringcld 20211	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}) → ((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))) ∈ (Base‘𝑅))
49	1, 17, 20, 28, 48	mulgnn0cld 19058	. . . . . 6 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}) → (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) ∈ (Base‘𝑅))
50		disjdifr 4474	. . . . . . 7 ⊢ (({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∩ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) = ∅
51	50	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∩ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) = ∅)
52		1nn0 12521	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ0
53		0nn0 12520	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℕ0
54	52, 53	ifcli 4577	. . . . . . . . . . . . . . 15 ⊢ if(𝑖 = 𝑋, 1, 0) ∈ ℕ0
55	54	nn0ge0i 12532	. . . . . . . . . . . . . 14 ⊢ 0 ≤ if(𝑖 = 𝑋, 1, 0)
56	22	ffvelcdmda 7093	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈ ℕ0)
57	56	nn0red 12566	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈ ℝ)
58	54	nn0rei 12516	. . . . . . . . . . . . . . . 16 ⊢ if(𝑖 = 𝑋, 1, 0) ∈ ℝ
59	58	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → if(𝑖 = 𝑋, 1, 0) ∈ ℝ)
60	57, 59	addge01d 11834	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (0 ≤ if(𝑖 = 𝑋, 1, 0) ↔ (𝑑‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0))))
61	55, 60	mpbii 232	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)))
62	61	ralrimiva 3135	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ∀𝑖 ∈ 𝐼 (𝑑‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)))
63	22	ffnd 6724	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 Fn 𝐼)
64	52, 53	ifcli 4577	. . . . . . . . . . . . . . . . 17 ⊢ if(𝑦 = 𝑋, 1, 0) ∈ ℕ0
65	64	elexi 3482	. . . . . . . . . . . . . . . 16 ⊢ if(𝑦 = 𝑋, 1, 0) ∈ V
66		eqid 2725	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))
67	65, 66	fnmpti 6699	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼
68	67	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼)
69	8	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑉)
70		inidm 4217	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∩ 𝐼) = 𝐼
71	63, 68, 69, 69, 70	offn 7698	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) Fn 𝐼)
72		eqidd 2726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) = (𝑑‘𝑖))
73		eqeq1 2729	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑖 → (𝑦 = 𝑋 ↔ 𝑖 = 𝑋))
74	73	ifbid 4553	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑖 → if(𝑦 = 𝑋, 1, 0) = if(𝑖 = 𝑋, 1, 0))
75	54	elexi 3482	. . . . . . . . . . . . . . . 16 ⊢ if(𝑖 = 𝑋, 1, 0) ∈ V
76	74, 66, 75	fvmpt 7004	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝐼 → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑖) = if(𝑖 = 𝑋, 1, 0))
77	76	adantl 480	. . . . . . . . . . . . . 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))
78	63, 68, 69, 69, 70, 72, 77	ofval 7696	. . . . . . . . . . . . 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)))
79	63, 71, 69, 69, 70, 72, 78	ofrfval 7695	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ↔ ∀𝑖 ∈ 𝐼 (𝑑‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0))))
80	62, 79	mpbird 256	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
81	80	adantr 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
82	8	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑉)
83	9	psrbagf 21868	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑘:𝐼⟶ℕ0)
84	83	adantl 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑘:𝐼⟶ℕ0)
85	22	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑:𝐼⟶ℕ0)
86	9	psrbagf 21868	. . . . . . . . . . . . 13 ⊢ ((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)
87	14, 86	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))):𝐼⟶ℕ0)
88	87	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))):𝐼⟶ℕ0)
89		nn0re 12514	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℕ0 → 𝑞 ∈ ℝ)
90		nn0re 12514	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ℕ0 → 𝑟 ∈ ℝ)
91		nn0re 12514	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℕ0 → 𝑠 ∈ ℝ)
92		letr 11340	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ ℝ ∧ 𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ) → ((𝑞 ≤ 𝑟 ∧ 𝑟 ≤ 𝑠) → 𝑞 ≤ 𝑠))
93	89, 90, 91, 92	syl3an 1157	. . . . . . . . . . . 12 ⊢ ((𝑞 ∈ ℕ0 ∧ 𝑟 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) → ((𝑞 ≤ 𝑟 ∧ 𝑟 ≤ 𝑠) → 𝑞 ≤ 𝑠))
94	93	adantl 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑞 ∈ ℕ0 ∧ 𝑟 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0)) → ((𝑞 ≤ 𝑟 ∧ 𝑟 ≤ 𝑠) → 𝑞 ≤ 𝑠))
95	82, 84, 85, 88, 94	caoftrn 7724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑘 ∘r ≤ 𝑑 ∧ 𝑑 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) → 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
96	81, 95	mpan2d 692	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑘 ∘r ≤ 𝑑 → 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
97	96	ss2rabdv 4069	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ⊆ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))})
98		undifr 4484	. . . . . . . 8 ⊢ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ⊆ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ↔ (({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∪ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) = {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))})
99	97, 98	sylib 217	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∪ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) = {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))})
100	99	eqcomd 2731	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} = (({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∪ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}))
101	1, 2, 6, 16, 49, 51, 100	gsummptfidmsplit 19897	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))) = ((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))))
102		eqid 2725	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
103		ovex 7452	. . . . . . . . 9 ⊢ (ℕ0 ↑m 𝐼) ∈ V
104	103	rabex 5335	. . . . . . . 8 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V
105	104	rabex 5335	. . . . . . 7 ⊢ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∈ V
106	105	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∈ V)
107		ovex 7452	. . . . . . . . 9 ⊢ ((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))) ∈ V
108		eqid 2725	. . . . . . . . 9 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ↦ ((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) = (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ↦ ((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))
109	107, 108	fnmpti 6699	. . . . . . . 8 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ↦ ((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) Fn {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}
110	109	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ↦ ((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) Fn {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))})
111		fvexd 6911	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (0g‘𝑅) ∈ V)
112	110, 16, 111	fndmfifsupp 9403	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ↦ ((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) finSupp (0g‘𝑅))
113	1, 102, 17, 106, 48, 112, 6, 27	gsummulg 19909	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))) = (((𝑑‘𝑋) + 1)(.g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ↦ ((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))))
114		difrab 4307	. . . . . . . . . . 11 ⊢ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) = {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ 𝑘 ∘r ≤ 𝑑)}
115	114	eleq2i 2817	. . . . . . . . . 10 ⊢ (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↔ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ 𝑘 ∘r ≤ 𝑑)})
116		breq1 5152	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑢 → (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ↔ 𝑢 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
117		breq1 5152	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑢 → (𝑘 ∘r ≤ 𝑑 ↔ 𝑢 ∘r ≤ 𝑑))
118	117	notbid 317	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑢 → (¬ 𝑘 ∘r ≤ 𝑑 ↔ ¬ 𝑢 ∘r ≤ 𝑑))
119	116, 118	anbi12d 630	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑢 → ((𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ 𝑘 ∘r ≤ 𝑑) ↔ (𝑢 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ 𝑢 ∘r ≤ 𝑑)))
120	119	elrab 3679	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ 𝑘 ∘r ≤ 𝑑)} ↔ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ (𝑢 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ 𝑢 ∘r ≤ 𝑑)))
121	9	psrbagf 21868	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑢:𝐼⟶ℕ0)
122	121	ffnd 6724	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑢 Fn 𝐼)
123	122	adantl 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑢 Fn 𝐼)
124	71	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) Fn 𝐼)
125	8	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑉)
126		eqidd 2726	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑢‘𝑖) = (𝑢‘𝑖))
127	63	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 Fn 𝐼)
128	64	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ 𝐼 → if(𝑦 = 𝑋, 1, 0) ∈ ℕ0)
129	66, 128	fmpti 7121	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)):𝐼⟶ℕ0
130	129	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)):𝐼⟶ℕ0)
131	130	ffnd 6724	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼)
132	131	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼)
133		eqidd 2726	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) = (𝑑‘𝑖))
134	76	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑖) = if(𝑖 = 𝑋, 1, 0))
135	127, 132, 125, 125, 70, 133, 134	ofval 7696	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {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)))
136	123, 124, 125, 125, 70, 126, 135	ofrfval 7695	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ↔ ∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0))))
137	123, 127, 125, 125, 70, 126, 133	ofrfval 7695	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∘r ≤ 𝑑 ↔ ∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ (𝑑‘𝑖)))
138	137	notbid 317	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (¬ 𝑢 ∘r ≤ 𝑑 ↔ ¬ ∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ (𝑑‘𝑖)))
139		rexnal 3089	. . . . . . . . . . . . . . 15 ⊢ (∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖) ↔ ¬ ∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ (𝑑‘𝑖))
140	138, 139	bitr4di 288	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (¬ 𝑢 ∘r ≤ 𝑑 ↔ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖)))
141	136, 140	anbi12d 630	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑢 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ 𝑢 ∘r ≤ 𝑑) ↔ (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))))
142	25	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → (𝑑‘𝑋) ∈ ℕ0)
143	121	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑢:𝐼⟶ℕ0)
144	23	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑋 ∈ 𝐼)
145	143, 144	ffvelcdmd 7094	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢‘𝑋) ∈ ℕ0)
146	145	adantlr 713	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢‘𝑋) ∈ ℕ0)
147	146	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → (𝑢‘𝑋) ∈ ℕ0)
148		nn0nlt0 12531	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑‘𝑋) ∈ ℕ0 → ¬ (𝑑‘𝑋) < 0)
149	142, 148	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → ¬ (𝑑‘𝑋) < 0)
150	22	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑:𝐼⟶ℕ0)
151	150	ffvelcdmda 7093	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈ ℕ0)
152	151	nn0cnd 12567	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈ ℂ)
153	152	addridd 11446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → ((𝑑‘𝑖) + 0) = (𝑑‘𝑖))
154	153	breq2d 5161	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → ((𝑢‘𝑖) ≤ ((𝑑‘𝑖) + 0) ↔ (𝑢‘𝑖) ≤ (𝑑‘𝑖)))
155	154	biimpd 228	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → ((𝑢‘𝑖) ≤ ((𝑑‘𝑖) + 0) → (𝑢‘𝑖) ≤ (𝑑‘𝑖)))
156		ifnefalse 4542	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑖 ≠ 𝑋 → if(𝑖 = 𝑋, 1, 0) = 0)
157	156	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑖 ≠ 𝑋 → ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) = ((𝑑‘𝑖) + 0))
158	157	breq2d 5161	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑖 ≠ 𝑋 → ((𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ↔ (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + 0)))
159	158	imbi1d 340	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑖 ≠ 𝑋 → (((𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) → (𝑢‘𝑖) ≤ (𝑑‘𝑖)) ↔ ((𝑢‘𝑖) ≤ ((𝑑‘𝑖) + 0) → (𝑢‘𝑖) ≤ (𝑑‘𝑖))))
160	155, 159	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑖 ≠ 𝑋 → ((𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) → (𝑢‘𝑖) ≤ (𝑑‘𝑖))))
161	160	imp 405	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ≠ 𝑋) → ((𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) → (𝑢‘𝑖) ≤ (𝑑‘𝑖)))
162	161	impancom 450	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) ∧ (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0))) → (𝑖 ≠ 𝑋 → (𝑢‘𝑖) ≤ (𝑑‘𝑖)))
163	162	necon1bd 2947	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) ∧ (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0))) → (¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖) → 𝑖 = 𝑋))
164	163	ancrd 550	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) ∧ (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0))) → (¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖) → (𝑖 = 𝑋 ∧ ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))))
165	164	ex 411	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → ((𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) → (¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖) → (𝑖 = 𝑋 ∧ ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖)))))
166	165	ralimdva 3156	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) → ∀𝑖 ∈ 𝐼 (¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖) → (𝑖 = 𝑋 ∧ ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖)))))
167	166	anim1d 609	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖)) → (∀𝑖 ∈ 𝐼 (¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖) → (𝑖 = 𝑋 ∧ ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))))
168	167	imp 405	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → (∀𝑖 ∈ 𝐼 (¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖) → (𝑖 = 𝑋 ∧ ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖)))
169		rexim 3076	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑖 ∈ 𝐼 (¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖) → (𝑖 = 𝑋 ∧ ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → (∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖) → ∃𝑖 ∈ 𝐼 (𝑖 = 𝑋 ∧ ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))))
170	169	imp 405	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((∀𝑖 ∈ 𝐼 (¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖) → (𝑖 = 𝑋 ∧ ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖)) → ∃𝑖 ∈ 𝐼 (𝑖 = 𝑋 ∧ ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖)))
171		fveq2 6896	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 = 𝑋 → (𝑢‘𝑖) = (𝑢‘𝑋))
172		fveq2 6896	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 = 𝑋 → (𝑑‘𝑖) = (𝑑‘𝑋))
173	171, 172	breq12d 5162	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 = 𝑋 → ((𝑢‘𝑖) ≤ (𝑑‘𝑖) ↔ (𝑢‘𝑋) ≤ (𝑑‘𝑋)))
174	173	notbid 317	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 𝑋 → (¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖) ↔ ¬ (𝑢‘𝑋) ≤ (𝑑‘𝑋)))
175	174	ceqsrexbv 3639	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃𝑖 ∈ 𝐼 (𝑖 = 𝑋 ∧ ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖)) ↔ (𝑋 ∈ 𝐼 ∧ ¬ (𝑢‘𝑋) ≤ (𝑑‘𝑋)))
176	175	simprbi 495	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃𝑖 ∈ 𝐼 (𝑖 = 𝑋 ∧ ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖)) → ¬ (𝑢‘𝑋) ≤ (𝑑‘𝑋))
177	170, 176	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∀𝑖 ∈ 𝐼 (¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖) → (𝑖 = 𝑋 ∧ ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖)) → ¬ (𝑢‘𝑋) ≤ (𝑑‘𝑋))
178	25	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑‘𝑋) ∈ ℕ0)
179	178	nn0red 12566	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑‘𝑋) ∈ ℝ)
180	146	nn0red 12566	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢‘𝑋) ∈ ℝ)
181	179, 180	ltnled 11393	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑑‘𝑋) < (𝑢‘𝑋) ↔ ¬ (𝑢‘𝑋) ≤ (𝑑‘𝑋)))
182	181	biimpar 476	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ¬ (𝑢‘𝑋) ≤ (𝑑‘𝑋)) → (𝑑‘𝑋) < (𝑢‘𝑋))
183	177, 182	sylan2 591	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (∀𝑖 ∈ 𝐼 (¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖) → (𝑖 = 𝑋 ∧ ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → (𝑑‘𝑋) < (𝑢‘𝑋))
184	168, 183	syldan 589	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → (𝑑‘𝑋) < (𝑢‘𝑋))
185		breq2 5153	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑢‘𝑋) = 0 → ((𝑑‘𝑋) < (𝑢‘𝑋) ↔ (𝑑‘𝑋) < 0))
186	184, 185	syl5ibcom 244	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → ((𝑢‘𝑋) = 0 → (𝑑‘𝑋) < 0))
187	149, 186	mtod 197	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → ¬ (𝑢‘𝑋) = 0)
188	187	neqned 2936	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → (𝑢‘𝑋) ≠ 0)
189		elnnne0 12519	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢‘𝑋) ∈ ℕ ↔ ((𝑢‘𝑋) ∈ ℕ0 ∧ (𝑢‘𝑋) ≠ 0))
190	147, 188, 189	sylanbrc 581	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → (𝑢‘𝑋) ∈ ℕ)
191		elfzo0 13708	. . . . . . . . . . . . . . . 16 ⊢ ((𝑑‘𝑋) ∈ (0..^(𝑢‘𝑋)) ↔ ((𝑑‘𝑋) ∈ ℕ0 ∧ (𝑢‘𝑋) ∈ ℕ ∧ (𝑑‘𝑋) < (𝑢‘𝑋)))
192	142, 190, 184, 191	syl3anbrc 1340	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → (𝑑‘𝑋) ∈ (0..^(𝑢‘𝑋)))
193		fzostep1 13784	. . . . . . . . . . . . . . 15 ⊢ ((𝑑‘𝑋) ∈ (0..^(𝑢‘𝑋)) → (((𝑑‘𝑋) + 1) ∈ (0..^(𝑢‘𝑋)) ∨ ((𝑑‘𝑋) + 1) = (𝑢‘𝑋)))
194	192, 193	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → (((𝑑‘𝑋) + 1) ∈ (0..^(𝑢‘𝑋)) ∨ ((𝑑‘𝑋) + 1) = (𝑢‘𝑋)))
195	147	nn0red 12566	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → (𝑢‘𝑋) ∈ ℝ)
196	27	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → ((𝑑‘𝑋) + 1) ∈ ℕ0)
197	196	nn0red 12566	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → ((𝑑‘𝑋) + 1) ∈ ℝ)
198	23	ad2antrr 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑋 ∈ 𝐼)
199		iftrue 4536	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑋 → if(𝑖 = 𝑋, 1, 0) = 1)
200	172, 199	oveq12d 7437	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑋 → ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) = ((𝑑‘𝑋) + 1))
201	171, 200	breq12d 5162	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑋 → ((𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ↔ (𝑢‘𝑋) ≤ ((𝑑‘𝑋) + 1)))
202	201	rspcv 3602	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑋 ∈ 𝐼 → (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) → (𝑢‘𝑋) ≤ ((𝑑‘𝑋) + 1)))
203	198, 202	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) → (𝑢‘𝑋) ≤ ((𝑑‘𝑋) + 1)))
204	203	imp 405	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0))) → (𝑢‘𝑋) ≤ ((𝑑‘𝑋) + 1))
205	204	adantrr 715	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → (𝑢‘𝑋) ≤ ((𝑑‘𝑋) + 1))
206	195, 197, 205	lensymd 11397	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → ¬ ((𝑑‘𝑋) + 1) < (𝑢‘𝑋))
207	206	intn3an3d 1477	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → ¬ (((𝑑‘𝑋) + 1) ∈ ℕ0 ∧ (𝑢‘𝑋) ∈ ℕ ∧ ((𝑑‘𝑋) + 1) < (𝑢‘𝑋)))
208		elfzo0 13708	. . . . . . . . . . . . . . 15 ⊢ (((𝑑‘𝑋) + 1) ∈ (0..^(𝑢‘𝑋)) ↔ (((𝑑‘𝑋) + 1) ∈ ℕ0 ∧ (𝑢‘𝑋) ∈ ℕ ∧ ((𝑑‘𝑋) + 1) < (𝑢‘𝑋)))
209	207, 208	sylnibr 328	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → ¬ ((𝑑‘𝑋) + 1) ∈ (0..^(𝑢‘𝑋)))
210	194, 209	orcnd 876	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ∧ ∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖))) → ((𝑑‘𝑋) + 1) = (𝑢‘𝑋))
211	141, 210	sylbida 590	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ 𝑢 ∘r ≤ 𝑑)) → ((𝑑‘𝑋) + 1) = (𝑢‘𝑋))
212	211	anasss 465	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ (𝑢 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ 𝑢 ∘r ≤ 𝑑))) → ((𝑑‘𝑋) + 1) = (𝑢‘𝑋))
213	120, 212	sylan2b 592	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ 𝑘 ∘r ≤ 𝑑)}) → ((𝑑‘𝑋) + 1) = (𝑢‘𝑋))
214	115, 213	sylan2b 592	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → ((𝑑‘𝑋) + 1) = (𝑢‘𝑋))
215	214	oveq1d 7434	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) = ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))
216	215	mpteq2dva 5249	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) = (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))
217	216	oveq2d 7435	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))) = (𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))))
218	9	psrbaglefi 21882	. . . . . . . . 9 ⊢ (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∈ Fin)
219	218	adantl 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∈ Fin)
220	19	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑅 ∈ Mnd)
221	27	adantr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝑑‘𝑋) + 1) ∈ ℕ0)
222	4	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑅 ∈ Ring)
223		elrabi 3673	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
224	34	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐹:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
225	224	ffvelcdmda 7093	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝐹‘𝑢) ∈ (Base‘𝑅))
226	223, 225	sylan2 591	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝐹‘𝑢) ∈ (Base‘𝑅))
227	40	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝐺:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
228	22	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑑:𝐼⟶ℕ0)
229	228	ffvelcdmda 7093	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈ ℕ0)
230	229	nn0cnd 12567	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈ ℂ)
231	223, 121	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → 𝑢:𝐼⟶ℕ0)
232	231	adantl 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑢:𝐼⟶ℕ0)
233	232	ffvelcdmda 7093	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑖 ∈ 𝐼) → (𝑢‘𝑖) ∈ ℕ0)
234	233	nn0cnd 12567	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑖 ∈ 𝐼) → (𝑢‘𝑖) ∈ ℂ)
235	54	nn0cni 12517	. . . . . . . . . . . . . . . . 17 ⊢ if(𝑖 = 𝑋, 1, 0) ∈ ℂ
236	235	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑖 ∈ 𝐼) → if(𝑖 = 𝑋, 1, 0) ∈ ℂ)
237	230, 234, 236	subadd23d 11625	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑖 ∈ 𝐼) → (((𝑑‘𝑖) − (𝑢‘𝑖)) + if(𝑖 = 𝑋, 1, 0)) = ((𝑑‘𝑖) + (if(𝑖 = 𝑋, 1, 0) − (𝑢‘𝑖))))
238	230, 236, 234	addsubassd 11623	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑖 ∈ 𝐼) → (((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) − (𝑢‘𝑖)) = ((𝑑‘𝑖) + (if(𝑖 = 𝑋, 1, 0) − (𝑢‘𝑖))))
239	237, 238	eqtr4d 2768	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑖 ∈ 𝐼) → (((𝑑‘𝑖) − (𝑢‘𝑖)) + if(𝑖 = 𝑋, 1, 0)) = (((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) − (𝑢‘𝑖)))
240	239	mpteq2dva 5249	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑖 ∈ 𝐼 ↦ (((𝑑‘𝑖) − (𝑢‘𝑖)) + if(𝑖 = 𝑋, 1, 0))) = (𝑖 ∈ 𝐼 ↦ (((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) − (𝑢‘𝑖))))
241		eqid 2725	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} = {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}
242	9, 241	psrbagconcl 21884	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑑 ∘f − 𝑢) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
243		elrabi 3673	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∘f − 𝑢) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → (𝑑 ∘f − 𝑢) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
244	242, 243	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑑 ∘f − 𝑢) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
245	244	adantll 712	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑑 ∘f − 𝑢) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
246	9	psrbagf 21868	. . . . . . . . . . . . . . . 16 ⊢ ((𝑑 ∘f − 𝑢) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → (𝑑 ∘f − 𝑢):𝐼⟶ℕ0)
247	245, 246	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑑 ∘f − 𝑢):𝐼⟶ℕ0)
248	247	ffnd 6724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑑 ∘f − 𝑢) Fn 𝐼)
249	67	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼)
250	8	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝐼 ∈ 𝑉)
251	228	ffnd 6724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑑 Fn 𝐼)
252	232	ffnd 6724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑢 Fn 𝐼)
253		eqidd 2726	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) = (𝑑‘𝑖))
254		eqidd 2726	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑖 ∈ 𝐼) → (𝑢‘𝑖) = (𝑢‘𝑖))
255	251, 252, 250, 250, 70, 253, 254	ofval 7696	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑖 ∈ 𝐼) → ((𝑑 ∘f − 𝑢)‘𝑖) = ((𝑑‘𝑖) − (𝑢‘𝑖)))
256	76	adantl 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑖 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑖) = if(𝑖 = 𝑋, 1, 0))
257	248, 249, 250, 250, 70, 255, 256	offval 7694	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝑑 ∘f − 𝑢) ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑖 ∈ 𝐼 ↦ (((𝑑‘𝑖) − (𝑢‘𝑖)) + if(𝑖 = 𝑋, 1, 0))))
258		simplr 767	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
259	11	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
260	258, 259, 13	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
261	260, 86	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))):𝐼⟶ℕ0)
262	261	ffnd 6724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) Fn 𝐼)
263	251, 249, 250, 250, 70, 253, 256	ofval 7696	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑖 ∈ 𝐼) → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑖) = ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)))
264	262, 252, 250, 250, 70, 263, 254	offval 7694	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢) = (𝑖 ∈ 𝐼 ↦ (((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) − (𝑢‘𝑖))))
265	240, 257, 264	3eqtr4d 2775	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝑑 ∘f − 𝑢) ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))
266	9	psrbagaddcl 21878	. . . . . . . . . . . . 13 ⊢ (((𝑑 ∘f − 𝑢) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑑 ∘f − 𝑢) ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
267	245, 259, 266	syl2anc 582	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝑑 ∘f − 𝑢) ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
268	265, 267	eqeltrrd 2826	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
269	227, 268	ffvelcdmd 7094	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)) ∈ (Base‘𝑅))
270	1, 29, 222, 226, 269	ringcld 20211	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))) ∈ (Base‘𝑅))
271	1, 17, 220, 221, 270	mulgnn0cld 19058	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) ∈ (Base‘𝑅))
272		disjdifr 4474	. . . . . . . . 9 ⊢ (({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ∩ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) = ∅
273	272	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ∩ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) = ∅)
274		simpl 481	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0) → 𝑘 ∘r ≤ 𝑑)
275	274	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → ((𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0) → 𝑘 ∘r ≤ 𝑑))
276	275	ss2rabi 4070	. . . . . . . . . . 11 ⊢ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ⊆ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}
277	276	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ⊆ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
278		undifr 4484	. . . . . . . . . 10 ⊢ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ⊆ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↔ (({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ∪ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) = {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
279	277, 278	sylib 217	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ∪ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) = {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
280	279	eqcomd 2731	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} = (({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ∪ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}))
281	1, 2, 6, 219, 271, 273, 280	gsummptfidmsplit 19897	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))) = ((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))))
282		eldifi 4123	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) → 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
283	23	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑋 ∈ 𝐼)
284		eqidd 2726	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑋 ∈ 𝐼) → (𝑑‘𝑋) = (𝑑‘𝑋))
285		eqidd 2726	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑋 ∈ 𝐼) → (𝑢‘𝑋) = (𝑢‘𝑋))
286	251, 252, 250, 250, 70, 284, 285	ofval 7696	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑋 ∈ 𝐼) → ((𝑑 ∘f − 𝑢)‘𝑋) = ((𝑑‘𝑋) − (𝑢‘𝑋)))
287	283, 286	mpdan 685	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝑑 ∘f − 𝑢)‘𝑋) = ((𝑑‘𝑋) − (𝑢‘𝑋)))
288	282, 287	sylan2 591	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → ((𝑑 ∘f − 𝑢)‘𝑋) = ((𝑑‘𝑋) − (𝑢‘𝑋)))
289	288	oveq2d 7435	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → ((𝑢‘𝑋) + ((𝑑 ∘f − 𝑢)‘𝑋)) = ((𝑢‘𝑋) + ((𝑑‘𝑋) − (𝑢‘𝑋))))
290	232, 283	ffvelcdmd 7094	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑢‘𝑋) ∈ ℕ0)
291	282, 290	sylan2 591	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → (𝑢‘𝑋) ∈ ℕ0)
292	291	nn0cnd 12567	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → (𝑢‘𝑋) ∈ ℂ)
293	25	nn0cnd 12567	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑‘𝑋) ∈ ℂ)
294	293	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → (𝑑‘𝑋) ∈ ℂ)
295	292, 294	pncan3d 11606	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → ((𝑢‘𝑋) + ((𝑑‘𝑋) − (𝑢‘𝑋))) = (𝑑‘𝑋))
296	289, 295	eqtrd 2765	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → ((𝑢‘𝑋) + ((𝑑 ∘f − 𝑢)‘𝑋)) = (𝑑‘𝑋))
297	296	oveq1d 7434	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → (((𝑢‘𝑋) + ((𝑑 ∘f − 𝑢)‘𝑋)) + 1) = ((𝑑‘𝑋) + 1))
298	247, 283	ffvelcdmd 7094	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝑑 ∘f − 𝑢)‘𝑋) ∈ ℕ0)
299	282, 298	sylan2 591	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → ((𝑑 ∘f − 𝑢)‘𝑋) ∈ ℕ0)
300	299	nn0cnd 12567	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → ((𝑑 ∘f − 𝑢)‘𝑋) ∈ ℂ)
301		1cnd 11241	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → 1 ∈ ℂ)
302	292, 300, 301	addassd 11268	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → (((𝑢‘𝑋) + ((𝑑 ∘f − 𝑢)‘𝑋)) + 1) = ((𝑢‘𝑋) + (((𝑑 ∘f − 𝑢)‘𝑋) + 1)))
303	297, 302	eqtr3d 2767	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → ((𝑑‘𝑋) + 1) = ((𝑢‘𝑋) + (((𝑑 ∘f − 𝑢)‘𝑋) + 1)))
304	303	oveq1d 7434	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) = (((𝑢‘𝑋) + (((𝑑 ∘f − 𝑢)‘𝑋) + 1))(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))
305	19	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → 𝑅 ∈ Mnd)
306		peano2nn0 12545	. . . . . . . . . . . . . . 15 ⊢ (((𝑑 ∘f − 𝑢)‘𝑋) ∈ ℕ0 → (((𝑑 ∘f − 𝑢)‘𝑋) + 1) ∈ ℕ0)
307	298, 306	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (((𝑑 ∘f − 𝑢)‘𝑋) + 1) ∈ ℕ0)
308	282, 307	sylan2 591	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → (((𝑑 ∘f − 𝑢)‘𝑋) + 1) ∈ ℕ0)
309	282, 270	sylan2 591	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → ((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))) ∈ (Base‘𝑅))
310	1, 17, 2	mulgnn0dir 19067	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Mnd ∧ ((𝑢‘𝑋) ∈ ℕ0 ∧ (((𝑑 ∘f − 𝑢)‘𝑋) + 1) ∈ ℕ0 ∧ ((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))) ∈ (Base‘𝑅))) → (((𝑢‘𝑋) + (((𝑑 ∘f − 𝑢)‘𝑋) + 1))(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) = (((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))(+g‘𝑅)((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))
311	305, 291, 308, 309, 310	syl13anc 1369	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → (((𝑢‘𝑋) + (((𝑑 ∘f − 𝑢)‘𝑋) + 1))(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) = (((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))(+g‘𝑅)((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))
312	304, 311	eqtrd 2765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) = (((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))(+g‘𝑅)((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))
313	312	mpteq2dva 5249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) = (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ (((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))(+g‘𝑅)((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))))
314	313	oveq2d 7435	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))) = (𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ (((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))(+g‘𝑅)((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))))
315		difssd 4129	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ⊆ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
316	219, 315	ssfid 9292	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ∈ Fin)
317	1, 17, 220, 290, 270	mulgnn0cld 19058	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) ∈ (Base‘𝑅))
318	282, 317	sylan2 591	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) ∈ (Base‘𝑅))
319	1, 17, 220, 307, 270	mulgnn0cld 19058	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) ∈ (Base‘𝑅))
320	282, 319	sylan2 591	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) → ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) ∈ (Base‘𝑅))
321		eqid 2725	. . . . . . . . . 10 ⊢ (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) = (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))
322		eqid 2725	. . . . . . . . . 10 ⊢ (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) = (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))
323	1, 2, 6, 316, 318, 320, 321, 322	gsummptfidmadd 19892	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ (((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))(+g‘𝑅)((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))) = ((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))))
324	314, 323	eqtrd 2765	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))) = ((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))))
325	23	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) → 𝑋 ∈ 𝐼)
326	63	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) → 𝑑 Fn 𝐼)
327		elrabi 3673	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} → 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
328	327, 122	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} → 𝑢 Fn 𝐼)
329	328	adantl 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) → 𝑢 Fn 𝐼)
330	8	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) → 𝐼 ∈ 𝑉)
331		eqidd 2726	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ∧ 𝑋 ∈ 𝐼) → (𝑑‘𝑋) = (𝑑‘𝑋))
332		eqidd 2726	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ∧ 𝑋 ∈ 𝐼) → (𝑢‘𝑋) = (𝑢‘𝑋))
333	326, 329, 330, 330, 70, 331, 332	ofval 7696	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ∧ 𝑋 ∈ 𝐼) → ((𝑑 ∘f − 𝑢)‘𝑋) = ((𝑑‘𝑋) − (𝑢‘𝑋)))
334	325, 333	mpdan 685	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) → ((𝑑 ∘f − 𝑢)‘𝑋) = ((𝑑‘𝑋) − (𝑢‘𝑋)))
335		fveq1 6895	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑢 → (𝑘‘𝑋) = (𝑢‘𝑋))
336	335	eqeq1d 2727	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑢 → ((𝑘‘𝑋) = 0 ↔ (𝑢‘𝑋) = 0))
337	117, 336	anbi12d 630	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑢 → ((𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0) ↔ (𝑢 ∘r ≤ 𝑑 ∧ (𝑢‘𝑋) = 0)))
338	337	elrab 3679	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↔ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ (𝑢 ∘r ≤ 𝑑 ∧ (𝑢‘𝑋) = 0)))
339	338	simprbi 495	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} → (𝑢 ∘r ≤ 𝑑 ∧ (𝑢‘𝑋) = 0))
340	339	simprd 494	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} → (𝑢‘𝑋) = 0)
341	340	adantl 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) → (𝑢‘𝑋) = 0)
342	341	oveq2d 7435	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) → ((𝑑‘𝑋) − (𝑢‘𝑋)) = ((𝑑‘𝑋) − 0))
343	25	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) → (𝑑‘𝑋) ∈ ℕ0)
344	343	nn0cnd 12567	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) → (𝑑‘𝑋) ∈ ℂ)
345	344	subid1d 11592	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) → ((𝑑‘𝑋) − 0) = (𝑑‘𝑋))
346	334, 342, 345	3eqtrrd 2770	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) → (𝑑‘𝑋) = ((𝑑 ∘f − 𝑢)‘𝑋))
347	346	oveq1d 7434	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) → ((𝑑‘𝑋) + 1) = (((𝑑 ∘f − 𝑢)‘𝑋) + 1))
348	347	oveq1d 7434	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) → (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) = ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))
349	348	mpteq2dva 5249	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) = (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))
350	349	oveq2d 7435	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))) = (𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))))
351	324, 350	oveq12d 7437	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))) = (((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))))
352	18	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ Grp)
353	104	rabex 5335	. . . . . . . . . . 11 ⊢ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∈ V
354	353	difexi 5331	. . . . . . . . . 10 ⊢ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ∈ V
355	354	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ∈ V)
356	318	fmpttd 7124	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))):({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})⟶(Base‘𝑅))
357		ovex 7452	. . . . . . . . . . . 12 ⊢ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) ∈ V
358	357, 321	fnmpti 6699	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) Fn ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})
359	358	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) Fn ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}))
360	359, 316, 111	fndmfifsupp 9403	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) finSupp (0g‘𝑅))
361	1, 102, 6, 355, 356, 360	gsumcl 19882	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))) ∈ (Base‘𝑅))
362	320	fmpttd 7124	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))):({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})⟶(Base‘𝑅))
363		ovex 7452	. . . . . . . . . . . 12 ⊢ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) ∈ V
364	363, 322	fnmpti 6699	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) Fn ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})
365	364	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) Fn ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}))
366	365, 316, 111	fndmfifsupp 9403	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) finSupp (0g‘𝑅))
367	1, 102, 6, 355, 362, 366	gsumcl 19882	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))) ∈ (Base‘𝑅))
368	104	rabex 5335	. . . . . . . . . 10 ⊢ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ∈ V
369	368	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ∈ V)
370	276	sseli 3972	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} → 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
371	370, 319	sylan2 591	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) → ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) ∈ (Base‘𝑅))
372	371	fmpttd 7124	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))):{𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}⟶(Base‘𝑅))
373		eqid 2725	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) = (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))
374	363, 373	fnmpti 6699	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) Fn {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}
375	374	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) Fn {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})
376	219, 277	ssfid 9292	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ∈ Fin)
377	375, 376, 111	fndmfifsupp 9403	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) finSupp (0g‘𝑅))
378	1, 102, 6, 369, 372, 377	gsumcl 19882	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))) ∈ (Base‘𝑅))
379	1, 2, 352, 361, 367, 378	grpassd 18910	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))) = ((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))))))
380	281, 351, 379	3eqtrd 2769	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))) = ((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))))))
381	217, 380	oveq12d 7437	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))) = ((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))))))
382	101, 113, 381	3eqtr3d 2773	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝑑‘𝑋) + 1)(.g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ↦ ((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))) = ((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))))))
383		psdmul.m	. . . . . 6 ⊢ · = (.r‘𝑆)
384	33	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐹 ∈ 𝐵)
385	39	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐺 ∈ 𝐵)
386	31, 32, 29, 383, 9, 384, 385, 14	psrmulval 21906	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝐹 · 𝐺)‘(𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ↦ ((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))
387	386	oveq2d 7435	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹 · 𝐺)‘(𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) = (((𝑑‘𝑋) + 1)(.g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ↦ ((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))))
388	105	difexi 5331	. . . . . . 7 ⊢ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∈ V
389	388	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∈ V)
390		eldifi 4123	. . . . . . . 8 ⊢ (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))})
391	36, 121	syl 17	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} → 𝑢:𝐼⟶ℕ0)
392	391	adantl 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}) → 𝑢:𝐼⟶ℕ0)
393	23	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}) → 𝑋 ∈ 𝐼)
394	392, 393	ffvelcdmd 7094	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}) → (𝑢‘𝑋) ∈ ℕ0)
395	1, 17, 20, 394, 48	mulgnn0cld 19058	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}) → ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) ∈ (Base‘𝑅))
396	390, 395	sylan2 591	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) ∈ (Base‘𝑅))
397	396	fmpttd 7124	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))):({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})⟶(Base‘𝑅))
398		eqid 2725	. . . . . . . . 9 ⊢ (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) = (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))
399	357, 398	fnmpti 6699	. . . . . . . 8 ⊢ (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) Fn ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
400	399	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) Fn ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}))
401		difssd 4129	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ⊆ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))})
402	16, 401	ssfid 9292	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∈ Fin)
403	400, 402, 111	fndmfifsupp 9403	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) finSupp (0g‘𝑅))
404	1, 102, 6, 389, 397, 403	gsumcl 19882	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))) ∈ (Base‘𝑅))
405	1, 2, 352, 367, 378	grpcld 18912	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))) ∈ (Base‘𝑅))
406	1, 2, 352, 404, 361, 405	grpassd 18910	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))))(+g‘𝑅)((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))))) = ((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))))))
407	382, 387, 406	3eqtr4d 2775	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹 · 𝐺)‘(𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) = (((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))))(+g‘𝑅)((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))))))
408	407	mpteq2dva 5249	. 2 ⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹 · 𝐺)‘(𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))) = (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))))(+g‘𝑅)((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))))))
409	31, 32, 383, 4, 33, 39	psrmulcl 21908	. . 3 ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐵)
410	31, 32, 9, 8, 3, 23, 409	psdval 22106	. 2 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝐹 · 𝐺)) = (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑑‘𝑋) + 1)(.g‘𝑅)((𝐹 · 𝐺)‘(𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))))
411		psdmul.p	. . . 4 ⊢ + = (+g‘𝑆)
412	18	grpmgmd 18926	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Mgm)
413	31, 32, 8, 412, 23, 33	psdcl 22108	. . . . 5 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ 𝐵)
414	31, 32, 383, 4, 413, 39	psrmulcl 21908	. . . 4 ⊢ (𝜑 → ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) · 𝐺) ∈ 𝐵)
415	31, 32, 8, 412, 23, 39	psdcl 22108	. . . . 5 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺) ∈ 𝐵)
416	31, 32, 383, 4, 33, 415	psrmulcl 21908	. . . 4 ⊢ (𝜑 → (𝐹 · (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺)) ∈ 𝐵)
417	31, 32, 2, 411, 414, 416	psradd 21899	. . 3 ⊢ (𝜑 → (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) · 𝐺) + (𝐹 · (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺))) = (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) · 𝐺) ∘f (+g‘𝑅)(𝐹 · (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺))))
418	31, 1, 9, 32, 414	psrelbas 21896	. . . . 5 ⊢ (𝜑 → ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) · 𝐺):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
419	418	ffnd 6724	. . . 4 ⊢ (𝜑 → ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) · 𝐺) Fn {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
420	31, 1, 9, 32, 416	psrelbas 21896	. . . . 5 ⊢ (𝜑 → (𝐹 · (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺)):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
421	420	ffnd 6724	. . . 4 ⊢ (𝜑 → (𝐹 · (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺)) Fn {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
422	104	a1i 11	. . . 4 ⊢ (𝜑 → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V)
423		inidm 4217	. . . 4 ⊢ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∩ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
424	413	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ 𝐵)
425	31, 32, 29, 383, 9, 424, 385, 7	psrmulval 21906	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) · 𝐺)‘𝑑) = (𝑅 Σg (𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏)(.r‘𝑅)(𝐺‘(𝑑 ∘f − 𝑏))))))
426	353	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∈ V)
427	4	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑅 ∈ Ring)
428		elrabi 3673	. . . . . . . . 9 ⊢ (𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
429	31, 1, 9, 32, 413	psrelbas 21896	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
430	429	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
431	430	ffvelcdmda 7093	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏) ∈ (Base‘𝑅))
432	428, 431	sylan2 591	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏) ∈ (Base‘𝑅))
433	40	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝐺:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
434	9, 241	psrbagconcl 21884	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ 𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑑 ∘f − 𝑏) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
435	434	adantll 712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑑 ∘f − 𝑏) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
436		elrabi 3673	. . . . . . . . . 10 ⊢ ((𝑑 ∘f − 𝑏) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → (𝑑 ∘f − 𝑏) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
437	435, 436	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑑 ∘f − 𝑏) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
438	433, 437	ffvelcdmd 7094	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝐺‘(𝑑 ∘f − 𝑏)) ∈ (Base‘𝑅))
439	1, 29, 427, 432, 438	ringcld 20211	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏)(.r‘𝑅)(𝐺‘(𝑑 ∘f − 𝑏))) ∈ (Base‘𝑅))
440	439	fmpttd 7124	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏)(.r‘𝑅)(𝐺‘(𝑑 ∘f − 𝑏)))):{𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}⟶(Base‘𝑅))
441		ovex 7452	. . . . . . . . 9 ⊢ (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏)(.r‘𝑅)(𝐺‘(𝑑 ∘f − 𝑏))) ∈ V
442		eqid 2725	. . . . . . . . 9 ⊢ (𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏)(.r‘𝑅)(𝐺‘(𝑑 ∘f − 𝑏)))) = (𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏)(.r‘𝑅)(𝐺‘(𝑑 ∘f − 𝑏))))
443	441, 442	fnmpti 6699	. . . . . . . 8 ⊢ (𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏)(.r‘𝑅)(𝐺‘(𝑑 ∘f − 𝑏)))) Fn {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}
444	443	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏)(.r‘𝑅)(𝐺‘(𝑑 ∘f − 𝑏)))) Fn {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
445	444, 219, 111	fndmfifsupp 9403	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏)(.r‘𝑅)(𝐺‘(𝑑 ∘f − 𝑏)))) finSupp (0g‘𝑅))
446		eqid 2725	. . . . . . 7 ⊢ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ↦ (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ↦ (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
447		df-of 7685	. . . . . . . . . 10 ⊢ ∘f + = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))))
448		vex 3465	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
449	448	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑢 ∈ V)
450		ssv 4001	. . . . . . . . . . 11 ⊢ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ⊆ V
451	450	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ⊆ V)
452		ssv 4001	. . . . . . . . . . 11 ⊢ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ⊆ V
453	452	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ⊆ V)
454	447, 449, 451, 453	elimampo 7558	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ↔ ∃𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}∃𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}𝑢 = (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜)))))
455	454	biimpa 475	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → ∃𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}∃𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}𝑢 = (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))))
456		elrabi 3673	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → 𝑚 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
457	9	psrbagf 21868	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑚:𝐼⟶ℕ0)
458	457	ffund 6727	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → Fun 𝑚)
459	456, 458	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → Fun 𝑚)
460	459	funfnd 6585	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → 𝑚 Fn dom 𝑚)
461	460	ad2antrl 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑚 Fn dom 𝑚)
462		velsn 4646	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ↔ 𝑛 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
463		funmpt 6592	. . . . . . . . . . . . . . . 16 ⊢ Fun (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))
464		funeq 6574	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) → (Fun 𝑛 ↔ Fun (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
465	463, 464	mpbiri 257	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) → Fun 𝑛)
466	465	funfnd 6585	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) → 𝑛 Fn dom 𝑛)
467	462, 466	sylbi 216	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} → 𝑛 Fn dom 𝑛)
468	467	ad2antll 727	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑛 Fn dom 𝑛)
469		vex 3465	. . . . . . . . . . . . . 14 ⊢ 𝑚 ∈ V
470	469	dmex 7917	. . . . . . . . . . . . 13 ⊢ dom 𝑚 ∈ V
471	470	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → dom 𝑚 ∈ V)
472		vex 3465	. . . . . . . . . . . . . 14 ⊢ 𝑛 ∈ V
473	472	dmex 7917	. . . . . . . . . . . . 13 ⊢ dom 𝑛 ∈ V
474	473	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → dom 𝑛 ∈ V)
475		eqid 2725	. . . . . . . . . . . 12 ⊢ (dom 𝑚 ∩ dom 𝑛) = (dom 𝑚 ∩ dom 𝑛)
476		eqidd 2726	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑜 ∈ dom 𝑚) → (𝑚‘𝑜) = (𝑚‘𝑜))
477		eqidd 2726	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑜 ∈ dom 𝑛) → (𝑛‘𝑜) = (𝑛‘𝑜))
478	461, 468, 471, 474, 475, 476, 477	offval 7694	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑚 ∘f + 𝑛) = (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))))
479	478	eqeq2d 2736	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑢 = (𝑚 ∘f + 𝑛) ↔ 𝑢 = (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜)))))
480		elsni 4647	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} → 𝑛 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
481	480	oveq2d 7435	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} → (𝑚 ∘f + 𝑛) = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
482	481	eqeq2d 2736	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} → (𝑢 = (𝑚 ∘f + 𝑛) ↔ 𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
483	482	ad2antll 727	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑢 = (𝑚 ∘f + 𝑛) ↔ 𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
484	8	ad3antrrr 728	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝐼 ∈ 𝑉)
485	456, 457	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → 𝑚:𝐼⟶ℕ0)
486	485	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑚:𝐼⟶ℕ0)
487	129	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)):𝐼⟶ℕ0)
488		nn0cn 12515	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 ∈ ℕ0 → 𝑞 ∈ ℂ)
489		nn0cn 12515	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ∈ ℕ0 → 𝑟 ∈ ℂ)
490		nn0cn 12515	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ℕ0 → 𝑠 ∈ ℂ)
491		addsubass 11502	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑞 ∈ ℂ ∧ 𝑟 ∈ ℂ ∧ 𝑠 ∈ ℂ) → ((𝑞 + 𝑟) − 𝑠) = (𝑞 + (𝑟 − 𝑠)))
492	488, 489, 490, 491	syl3an 1157	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑞 ∈ ℕ0 ∧ 𝑟 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) → ((𝑞 + 𝑟) − 𝑠) = (𝑞 + (𝑟 − 𝑠)))
493	492	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ (𝑞 ∈ ℕ0 ∧ 𝑟 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0)) → ((𝑞 + 𝑟) − 𝑠) = (𝑞 + (𝑟 − 𝑠)))
494	484, 486, 487, 487, 493	caofass 7723	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑚 ∘f + ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
495		simpr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
496	54	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → if(𝑖 = 𝑋, 1, 0) ∈ ℕ0)
497	66, 74, 495, 496	fvmptd3 7027	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑖) = if(𝑖 = 𝑋, 1, 0))
498	131, 131, 8, 8, 70, 497, 497	offval 7694	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑖 ∈ 𝐼 ↦ (if(𝑖 = 𝑋, 1, 0) − if(𝑖 = 𝑋, 1, 0))))
499	498	oveq2d 7435	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑚 ∘f + ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝑚 ∘f + (𝑖 ∈ 𝐼 ↦ (if(𝑖 = 𝑋, 1, 0) − if(𝑖 = 𝑋, 1, 0)))))
500	499	ad3antrrr 728	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑚 ∘f + ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝑚 ∘f + (𝑖 ∈ 𝐼 ↦ (if(𝑖 = 𝑋, 1, 0) − if(𝑖 = 𝑋, 1, 0)))))
501	235	subidi 11563	. . . . . . . . . . . . . . . . . . 19 ⊢ (if(𝑖 = 𝑋, 1, 0) − if(𝑖 = 𝑋, 1, 0)) = 0
502	501	mpteq2i 5254	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ 𝐼 ↦ (if(𝑖 = 𝑋, 1, 0) − if(𝑖 = 𝑋, 1, 0))) = (𝑖 ∈ 𝐼 ↦ 0)
503		fconstmpt 5740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 × {0}) = (𝑖 ∈ 𝐼 ↦ 0)
504	502, 503	eqtr4i 2756	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ 𝐼 ↦ (if(𝑖 = 𝑋, 1, 0) − if(𝑖 = 𝑋, 1, 0))) = (𝐼 × {0})
505	504	oveq2i 7430	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∘f + (𝑖 ∈ 𝐼 ↦ (if(𝑖 = 𝑋, 1, 0) − if(𝑖 = 𝑋, 1, 0)))) = (𝑚 ∘f + (𝐼 × {0}))
506		0zd 12603	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 0 ∈ ℤ)
507	488	addridd 11446	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 ∈ ℕ0 → (𝑞 + 0) = 𝑞)
508	507	adantl 480	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑞 ∈ ℕ0) → (𝑞 + 0) = 𝑞)
509	484, 486, 506, 508	caofid0r 7718	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑚 ∘f + (𝐼 × {0})) = 𝑚)
510	505, 509	eqtrid 2777	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑚 ∘f + (𝑖 ∈ 𝐼 ↦ (if(𝑖 = 𝑋, 1, 0) − if(𝑖 = 𝑋, 1, 0)))) = 𝑚)
511	494, 500, 510	3eqtrd 2769	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = 𝑚)
512		simpr 483	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
513	511, 512	eqeltrd 2825	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
514		oveq1 7426	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
515	514	eleq1d 2810	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → ((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↔ ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}))
516	513, 515	syl5ibrcom 246	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}))
517	516	adantrr 715	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}))
518	483, 517	sylbid 239	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑢 = (𝑚 ∘f + 𝑛) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}))
519	479, 518	sylbird 259	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑢 = (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}))
520	519	rexlimdvva 3201	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (∃𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}∃𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}𝑢 = (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}))
521	455, 520	mpd 15	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
522		simpr 483	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
523	8	mptexd 7236	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ V)
524		elsng 4644	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ V → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ↔ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
525	523, 524	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ↔ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
526	66, 525	mpbiri 257	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})
527	526	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})
528	447	mpofun 7544	. . . . . . . . 9 ⊢ Fun ∘f +
529	528	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → Fun ∘f + )
530		xpss 5694	. . . . . . . . 9 ⊢ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) ⊆ (V × V)
531	470	inex1 5318	. . . . . . . . . . . 12 ⊢ (dom 𝑚 ∩ dom 𝑛) ∈ V
532	531	mptex 7235	. . . . . . . . . . 11 ⊢ (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))) ∈ V
533	532	rgen2w 3055	. . . . . . . . . 10 ⊢ ∀𝑚 ∈ V ∀𝑛 ∈ V (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))) ∈ V
534	447	dmmpoga 8078	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ V ∀𝑛 ∈ V (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))) ∈ V → dom ∘f + = (V × V))
535	533, 534	mp1i 13	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → dom ∘f + = (V × V))
536	530, 535	sseqtrrid 4030	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) ⊆ dom ∘f + )
537	522, 527, 529, 536	elovimad 7468	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑣 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})))
538	8	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → 𝐼 ∈ 𝑉)
539		elrabi 3673	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → 𝑣 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
540	9	psrbagf 21868	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑣:𝐼⟶ℕ0)
541	539, 540	syl 17	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → 𝑣:𝐼⟶ℕ0)
542	541	ad2antll 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → 𝑣:𝐼⟶ℕ0)
543	129	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)):𝐼⟶ℕ0)
544	492	adantl 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) ∧ (𝑞 ∈ ℕ0 ∧ 𝑟 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0)) → ((𝑞 + 𝑟) − 𝑠) = (𝑞 + (𝑟 − 𝑠)))
545	538, 542, 543, 543, 544	caofass 7723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → ((𝑣 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑣 ∘f + ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
546	131	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼)
547	76	adantl 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) ∧ 𝑖 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑖) = if(𝑖 = 𝑋, 1, 0))
548	546, 546, 538, 538, 70, 547, 547	offval 7694	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑖 ∈ 𝐼 ↦ (if(𝑖 = 𝑋, 1, 0) − if(𝑖 = 𝑋, 1, 0))))
549	548	oveq2d 7435	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → (𝑣 ∘f + ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝑣 ∘f + (𝑖 ∈ 𝐼 ↦ (if(𝑖 = 𝑋, 1, 0) − if(𝑖 = 𝑋, 1, 0)))))
550	504	oveq2i 7430	. . . . . . . . . . 11 ⊢ (𝑣 ∘f + (𝑖 ∈ 𝐼 ↦ (if(𝑖 = 𝑋, 1, 0) − if(𝑖 = 𝑋, 1, 0)))) = (𝑣 ∘f + (𝐼 × {0}))
551		0zd 12603	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → 0 ∈ ℤ)
552		nn0cn 12515	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ ℕ0 → 𝑝 ∈ ℂ)
553	552	addridd 11446	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℕ0 → (𝑝 + 0) = 𝑝)
554	553	adantl 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) ∧ 𝑝 ∈ ℕ0) → (𝑝 + 0) = 𝑝)
555	538, 542, 551, 554	caofid0r 7718	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → (𝑣 ∘f + (𝐼 × {0})) = 𝑣)
556	550, 555	eqtrid 2777	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → (𝑣 ∘f + (𝑖 ∈ 𝐼 ↦ (if(𝑖 = 𝑋, 1, 0) − if(𝑖 = 𝑋, 1, 0)))) = 𝑣)
557	545, 549, 556	3eqtrrd 2770	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → 𝑣 = ((𝑣 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
558		oveq1 7426	. . . . . . . . . 10 ⊢ (𝑢 = (𝑣 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = ((𝑣 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
559	558	eqeq2d 2736	. . . . . . . . 9 ⊢ (𝑢 = (𝑣 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑣 = (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ↔ 𝑣 = ((𝑣 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
560	557, 559	syl5ibrcom 246	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → (𝑢 = (𝑣 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → 𝑣 = (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
561	11	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
562	9	psrbagaddcl 21878	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
563	456, 561, 562	syl2an2 684	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
564	9	psrbagf 21868	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))):𝐼⟶ℕ0)
565	563, 564	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))):𝐼⟶ℕ0)
566	565	adantrr 715	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))):𝐼⟶ℕ0)
567		feq1 6704	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑢:𝐼⟶ℕ0 ↔ (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))):𝐼⟶ℕ0))
568	566, 567	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → 𝑢:𝐼⟶ℕ0))
569	483, 568	sylbid 239	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑢 = (𝑚 ∘f + 𝑛) → 𝑢:𝐼⟶ℕ0))
570	479, 569	sylbird 259	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑢 = (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))) → 𝑢:𝐼⟶ℕ0))
571	570	rexlimdvva 3201	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (∃𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}∃𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}𝑢 = (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))) → 𝑢:𝐼⟶ℕ0))
572	455, 571	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → 𝑢:𝐼⟶ℕ0)
573	572	adantrr 715	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → 𝑢:𝐼⟶ℕ0)
574	573	ffvelcdmda 7093	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) ∧ 𝑖 ∈ 𝐼) → (𝑢‘𝑖) ∈ ℕ0)
575	574	nn0cnd 12567	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) ∧ 𝑖 ∈ 𝐼) → (𝑢‘𝑖) ∈ ℂ)
576	235	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) ∧ 𝑖 ∈ 𝐼) → if(𝑖 = 𝑋, 1, 0) ∈ ℂ)
577	575, 576	npcand 11607	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) ∧ 𝑖 ∈ 𝐼) → (((𝑢‘𝑖) − if(𝑖 = 𝑋, 1, 0)) + if(𝑖 = 𝑋, 1, 0)) = (𝑢‘𝑖))
578	577	mpteq2dva 5249	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → (𝑖 ∈ 𝐼 ↦ (((𝑢‘𝑖) − if(𝑖 = 𝑋, 1, 0)) + if(𝑖 = 𝑋, 1, 0))) = (𝑖 ∈ 𝐼 ↦ (𝑢‘𝑖)))
579	573	ffnd 6724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → 𝑢 Fn 𝐼)
580	579, 546, 538, 538, 70	offn 7698	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) Fn 𝐼)
581		eqidd 2726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) ∧ 𝑖 ∈ 𝐼) → (𝑢‘𝑖) = (𝑢‘𝑖))
582	579, 546, 538, 538, 70, 581, 547	ofval 7696	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) ∧ 𝑖 ∈ 𝐼) → ((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑖) = ((𝑢‘𝑖) − if(𝑖 = 𝑋, 1, 0)))
583	580, 546, 538, 538, 70, 582, 547	offval 7694	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → ((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑖 ∈ 𝐼 ↦ (((𝑢‘𝑖) − if(𝑖 = 𝑋, 1, 0)) + if(𝑖 = 𝑋, 1, 0))))
584	573	feqmptd 6966	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → 𝑢 = (𝑖 ∈ 𝐼 ↦ (𝑢‘𝑖)))
585	578, 583, 584	3eqtr4rd 2776	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → 𝑢 = ((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
586		oveq1 7426	. . . . . . . . . 10 ⊢ (𝑣 = (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑣 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = ((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
587	586	eqeq2d 2736	. . . . . . . . 9 ⊢ (𝑣 = (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑢 = (𝑣 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ↔ 𝑢 = ((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
588	585, 587	syl5ibrcom 246	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → (𝑣 = (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → 𝑢 = (𝑣 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
589	560, 588	impbid 211	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → (𝑢 = (𝑣 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ↔ 𝑣 = (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
590	446, 521, 537, 589	f1o2d 7675	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ↦ (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))):( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))–1-1-onto→{𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
591	1, 102, 6, 426, 440, 445, 590	gsumf1o 19883	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏)(.r‘𝑅)(𝐺‘(𝑑 ∘f − 𝑏))))) = (𝑅 Σg ((𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏)(.r‘𝑅)(𝐺‘(𝑑 ∘f − 𝑏)))) ∘ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ↦ (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))))
592	553	adantl 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑝 ∈ ℕ0) → (𝑝 + 0) = 𝑝)
593	484, 486, 506, 592	caofid0r 7718	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑚 ∘f + (𝐼 × {0})) = 𝑚)
594	505, 593	eqtrid 2777	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑚 ∘f + (𝑖 ∈ 𝐼 ↦ (if(𝑖 = 𝑋, 1, 0) − if(𝑖 = 𝑋, 1, 0)))) = 𝑚)
595	494, 500, 594	3eqtrd 2769	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = 𝑚)
596	595, 512	eqeltrd 2825	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
597	596, 515	syl5ibrcom 246	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}))
598	597	adantrr 715	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}))
599	483, 598	sylbid 239	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑢 = (𝑚 ∘f + 𝑛) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}))
600	479, 599	sylbird 259	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑢 = (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}))
601	600	rexlimdvva 3201	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (∃𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}∃𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}𝑢 = (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}))
602	455, 601	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
603		eqidd 2726	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ↦ (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ↦ (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
604		eqidd 2726	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏)(.r‘𝑅)(𝐺‘(𝑑 ∘f − 𝑏)))) = (𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏)(.r‘𝑅)(𝐺‘(𝑑 ∘f − 𝑏)))))
605		fveq2 6896	. . . . . . . . . 10 ⊢ (𝑏 = (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏) = ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘(𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
606		oveq2 7427	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑑 ∘f − 𝑏) = (𝑑 ∘f − (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
607	606	fveq2d 6900	. . . . . . . . . 10 ⊢ (𝑏 = (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝐺‘(𝑑 ∘f − 𝑏)) = (𝐺‘(𝑑 ∘f − (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))
608	605, 607	oveq12d 7437	. . . . . . . . 9 ⊢ (𝑏 = (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏)(.r‘𝑅)(𝐺‘(𝑑 ∘f − 𝑏))) = (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘(𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))(.r‘𝑅)(𝐺‘(𝑑 ∘f − (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))))
609	602, 603, 604, 608	fmptco 7138	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏)(.r‘𝑅)(𝐺‘(𝑑 ∘f − 𝑏)))) ∘ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ↦ (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) = (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ↦ (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘(𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))(.r‘𝑅)(𝐺‘(𝑑 ∘f − (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))))
610	8	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → 𝐼 ∈ 𝑉)
611	3	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → 𝑅 ∈ CRing)
612	23	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → 𝑋 ∈ 𝐼)
613	33	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → 𝐹 ∈ 𝐵)
614		elrabi 3673	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
615	602, 614	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
616	31, 32, 9, 610, 611, 612, 613, 615	psdcoef 22107	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘(𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = ((((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) + 1)(.g‘𝑅)(𝐹‘((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))
617	572	ffnd 6724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → 𝑢 Fn 𝐼)
618	129	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)):𝐼⟶ℕ0)
619	618	ffnd 6724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼)
620		eqidd 2726	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑋 ∈ 𝐼) → (𝑢‘𝑋) = (𝑢‘𝑋))
621		iftrue 4536	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, 1, 0) = 1)
622		1ex 11242	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ V
623	621, 66, 622	fvmpt 7004	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ 𝐼 → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑋) = 1)
624	623	adantl 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑋 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑋) = 1)
625	617, 619, 610, 610, 70, 620, 624	ofval 7696	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑋 ∈ 𝐼) → ((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) = ((𝑢‘𝑋) − 1))
626	612, 625	mpdan 685	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → ((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) = ((𝑢‘𝑋) − 1))
627	626	oveq1d 7434	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) + 1) = (((𝑢‘𝑋) − 1) + 1))
628		nn0sscn 12510	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ0 ⊆ ℂ
629	628	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → ℕ0 ⊆ ℂ)
630	572, 629	fssd 6740	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → 𝑢:𝐼⟶ℂ)
631	630, 612	ffvelcdmd 7094	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (𝑢‘𝑋) ∈ ℂ)
632		1cnd 11241	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → 1 ∈ ℂ)
633	631, 632	npcand 11607	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (((𝑢‘𝑋) − 1) + 1) = (𝑢‘𝑋))
634	627, 633	eqtrd 2765	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) + 1) = (𝑢‘𝑋))
635	617, 619, 610, 610, 70	offn 7698	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) Fn 𝐼)
636		eqidd 2726	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑖 ∈ 𝐼) → (𝑢‘𝑖) = (𝑢‘𝑖))
637	76	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑖 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑖) = if(𝑖 = 𝑋, 1, 0))
638	617, 619, 610, 610, 70, 636, 637	ofval 7696	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑖 ∈ 𝐼) → ((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑖) = ((𝑢‘𝑖) − if(𝑖 = 𝑋, 1, 0)))
639	572	ffvelcdmda 7093	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑖 ∈ 𝐼) → (𝑢‘𝑖) ∈ ℕ0)
640	639	nn0cnd 12567	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑖 ∈ 𝐼) → (𝑢‘𝑖) ∈ ℂ)
641	235	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑖 ∈ 𝐼) → if(𝑖 = 𝑋, 1, 0) ∈ ℂ)
642	640, 641	npcand 11607	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑖 ∈ 𝐼) → (((𝑢‘𝑖) − if(𝑖 = 𝑋, 1, 0)) + if(𝑖 = 𝑋, 1, 0)) = (𝑢‘𝑖))
643	610, 635, 619, 617, 638, 637, 642	offveq 7710	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → ((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = 𝑢)
644	643	fveq2d 6900	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (𝐹‘((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝐹‘𝑢))
645	634, 644	oveq12d 7437	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → ((((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) + 1)(.g‘𝑅)(𝐹‘((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) = ((𝑢‘𝑋)(.g‘𝑅)(𝐹‘𝑢)))
646	616, 645	eqtrd 2765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘(𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = ((𝑢‘𝑋)(.g‘𝑅)(𝐹‘𝑢)))
647	21	ad2antlr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → 𝑑:𝐼⟶ℕ0)
648	647	ffvelcdmda 7093	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈ ℕ0)
649	648	nn0cnd 12567	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈ ℂ)
650	649, 640, 641	subsub3d 11633	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑖 ∈ 𝐼) → ((𝑑‘𝑖) − ((𝑢‘𝑖) − if(𝑖 = 𝑋, 1, 0))) = (((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) − (𝑢‘𝑖)))
651	650	mpteq2dva 5249	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖) − ((𝑢‘𝑖) − if(𝑖 = 𝑋, 1, 0)))) = (𝑖 ∈ 𝐼 ↦ (((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) − (𝑢‘𝑖))))
652	63	adantr 479	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → 𝑑 Fn 𝐼)
653		eqidd 2726	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) = (𝑑‘𝑖))
654	652, 635, 610, 610, 70, 653, 638	offval 7694	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (𝑑 ∘f − (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖) − ((𝑢‘𝑖) − if(𝑖 = 𝑋, 1, 0)))))
655	652, 619, 610, 610, 70	offn 7698	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) Fn 𝐼)
656	652, 619, 610, 610, 70, 653, 637	ofval 7696	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ 𝑖 ∈ 𝐼) → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑖) = ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)))
657	655, 617, 610, 610, 70, 656, 636	offval 7694	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢) = (𝑖 ∈ 𝐼 ↦ (((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) − (𝑢‘𝑖))))
658	651, 654, 657	3eqtr4d 2775	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (𝑑 ∘f − (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))
659	658	fveq2d 6900	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (𝐺‘(𝑑 ∘f − (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) = (𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))
660	646, 659	oveq12d 7437	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘(𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))(.r‘𝑅)(𝐺‘(𝑑 ∘f − (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))) = (((𝑢‘𝑋)(.g‘𝑅)(𝐹‘𝑢))(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))
661	4	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → 𝑅 ∈ Ring)
662	572, 612	ffvelcdmd 7094	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (𝑢‘𝑋) ∈ ℕ0)
663	662	nn0zd 12617	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (𝑢‘𝑋) ∈ ℤ)
664	34	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → 𝐹:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
665		simpllr 774	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
666	11	ad3antrrr 728	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
667		simprl 769	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
668		eqid 2725	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑙 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑙 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} = {𝑙 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑙 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}
669	9, 241, 668	psrbagleadd1 21886	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑙 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑙 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))})
670	665, 666, 667, 669	syl3anc 1368	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑙 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑙 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))})
671		eleq1 2813	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑢 ∈ {𝑙 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑙 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ↔ (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑙 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑙 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}))
672	670, 671	syl5ibrcom 246	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → 𝑢 ∈ {𝑙 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑙 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}))
673	483, 672	sylbid 239	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑢 = (𝑚 ∘f + 𝑛) → 𝑢 ∈ {𝑙 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑙 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}))
674	479, 673	sylbird 259	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) ∧ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑢 = (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))) → 𝑢 ∈ {𝑙 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑙 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}))
675	674	rexlimdvva 3201	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (∃𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}∃𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}𝑢 = (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))) → 𝑢 ∈ {𝑙 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑙 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}))
676	455, 675	mpd 15	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → 𝑢 ∈ {𝑙 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑙 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))})
677		elrabi 3673	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ {𝑙 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑙 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} → 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
678	676, 677	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
679	664, 678	ffvelcdmd 7094	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (𝐹‘𝑢) ∈ (Base‘𝑅))
680	40	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → 𝐺:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
681	14	adantr 479	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
682	9, 668	psrbagconcl 21884	. . . . . . . . . . . . . 14 ⊢ (((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ 𝑢 ∈ {𝑙 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑙 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}) → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢) ∈ {𝑙 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑙 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))})
683	681, 676, 682	syl2anc 582	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢) ∈ {𝑙 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑙 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))})
684		elrabi 3673	. . . . . . . . . . . . 13 ⊢ (((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢) ∈ {𝑙 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑙 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
685	683, 684	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
686	680, 685	ffvelcdmd 7094	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)) ∈ (Base‘𝑅))
687	1, 17, 29	mulgass2 20257	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ ((𝑢‘𝑋) ∈ ℤ ∧ (𝐹‘𝑢) ∈ (Base‘𝑅) ∧ (𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)) ∈ (Base‘𝑅))) → (((𝑢‘𝑋)(.g‘𝑅)(𝐹‘𝑢))(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))) = ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))
688	661, 663, 679, 686, 687	syl13anc 1369	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (((𝑢‘𝑋)(.g‘𝑅)(𝐹‘𝑢))(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))) = ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))
689	660, 688	eqtrd 2765	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘(𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))(.r‘𝑅)(𝐺‘(𝑑 ∘f − (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))) = ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))
690	689	mpteq2dva 5249	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ↦ (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘(𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))(.r‘𝑅)(𝐺‘(𝑑 ∘f − (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))) = (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))
691	609, 690	eqtrd 2765	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏)(.r‘𝑅)(𝐺‘(𝑑 ∘f − 𝑏)))) ∘ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ↦ (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) = (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))
692	691	oveq2d 7435	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σg ((𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏)(.r‘𝑅)(𝐺‘(𝑑 ∘f − 𝑏)))) ∘ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ↦ (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))) = (𝑅 Σg (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))))
693		snex 5433	. . . . . . . . . 10 ⊢ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ∈ V
694	353, 693	xpex 7756	. . . . . . . . 9 ⊢ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) ∈ V
695	694	funimaex 6642	. . . . . . . 8 ⊢ (Fun ∘f + → ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∈ V)
696	528, 695	mp1i 13	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∈ V)
697	19	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → 𝑅 ∈ Mnd)
698	1, 29, 661, 679, 686	ringcld 20211	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → ((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))) ∈ (Base‘𝑅))
699	1, 17, 697, 662, 698	mulgnn0cld 19058	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}))) → ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) ∈ (Base‘𝑅))
700		eqid 2725	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) = (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))
701	357, 700	fnmpti 6699	. . . . . . . . . 10 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) Fn {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))}
702	701	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) Fn {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))})
703	702, 16, 111	fndmfifsupp 9403	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) finSupp (0g‘𝑅))
704	460	ad2antlr 725	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) → 𝑚 Fn dom 𝑚)
705	467	adantl 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) → 𝑛 Fn dom 𝑛)
706	470	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) → dom 𝑚 ∈ V)
707	473	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) → dom 𝑛 ∈ V)
708		eqidd 2726	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) ∧ 𝑜 ∈ dom 𝑚) → (𝑚‘𝑜) = (𝑚‘𝑜))
709		eqidd 2726	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) ∧ 𝑜 ∈ dom 𝑛) → (𝑛‘𝑜) = (𝑛‘𝑜))
710	704, 705, 706, 707, 475, 708, 709	offval 7694	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) → (𝑚 ∘f + 𝑛) = (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))))
711	710	eqeq2d 2736	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) → (𝑢 = (𝑚 ∘f + 𝑛) ↔ 𝑢 = (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜)))))
712	711	rexbidva 3166	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (∃𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}𝑢 = (𝑚 ∘f + 𝑛) ↔ ∃𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}𝑢 = (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜)))))
713	11	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
714		oveq2 7427	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) → (𝑚 ∘f + 𝑛) = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
715	714	eqeq2d 2736	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) → (𝑢 = (𝑚 ∘f + 𝑛) ↔ 𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
716	715	rexsng 4680	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → (∃𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}𝑢 = (𝑚 ∘f + 𝑛) ↔ 𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
717	713, 716	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (∃𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}𝑢 = (𝑚 ∘f + 𝑛) ↔ 𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
718	712, 717	bitr3d 280	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (∃𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}𝑢 = (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))) ↔ 𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
719	718	rexbidva 3166	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (∃𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}∃𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}𝑢 = (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))) ↔ ∃𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
720		breq1 5152	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ↔ (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
721		breq1 5152	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑘 ∘r ≤ 𝑑 ↔ (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘r ≤ 𝑑))
722		fveq1 6895	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑘‘𝑋) = ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋))
723	722	eqeq1d 2727	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → ((𝑘‘𝑋) = 0 ↔ ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) = 0))
724	721, 723	anbi12d 630	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → ((𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0) ↔ ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘r ≤ 𝑑 ∧ ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) = 0)))
725	724	notbid 317	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0) ↔ ¬ ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘r ≤ 𝑑 ∧ ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) = 0)))
726	720, 725	anbi12d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → ((𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)) ↔ ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘r ≤ 𝑑 ∧ ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) = 0))))
727	456, 713, 562	syl2an2 684	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
728		simplr 767	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
729		simpr 483	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
730	9, 241, 42	psrbagleadd1 21886	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))})
731	728, 713, 729, 730	syl3anc 1368	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))})
732	720	elrab 3679	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ↔ ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
733	732	simprbi 495	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} → (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
734	731, 733	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
735	23	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑋 ∈ 𝐼)
736	485	adantl 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑚:𝐼⟶ℕ0)
737	736	ffnd 6724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑚 Fn 𝐼)
738	131	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼)
739	8	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝐼 ∈ 𝑉)
740		eqidd 2726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑋 ∈ 𝐼) → (𝑚‘𝑋) = (𝑚‘𝑋))
741	623	adantl 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑋 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑋) = 1)
742	737, 738, 739, 739, 70, 740, 741	ofval 7696	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∧ 𝑋 ∈ 𝐼) → ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) = ((𝑚‘𝑋) + 1))
743	735, 742	mpdan 685	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) = ((𝑚‘𝑋) + 1))
744	736, 735	ffvelcdmd 7094	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑚‘𝑋) ∈ ℕ0)
745		nn0p1nn 12544	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚‘𝑋) ∈ ℕ0 → ((𝑚‘𝑋) + 1) ∈ ℕ)
746	744, 745	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝑚‘𝑋) + 1) ∈ ℕ)
747	743, 746	eqeltrd 2825	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) ∈ ℕ)
748	747	nnne0d 12295	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) ≠ 0)
749	748	neneqd 2934	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ¬ ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) = 0)
750	749	intnand 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ¬ ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘r ≤ 𝑑 ∧ ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) = 0))
751	734, 750	jca 510	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘r ≤ 𝑑 ∧ ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) = 0)))
752	726, 727, 751	elrabd 3681	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))})
753		eleq1 2813	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))} ↔ (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}))
754	752, 753	syl5ibrcom 246	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}))
755		breq1 5152	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑘 ∘r ≤ 𝑑 ↔ (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘r ≤ 𝑑))
756		elrabi 3673	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))} → 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
757	756	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
758	129	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)):𝐼⟶ℕ0)
759	756, 121	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))} → 𝑢:𝐼⟶ℕ0)
760	759	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → 𝑢:𝐼⟶ℕ0)
761	23	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → 𝑋 ∈ 𝐼)
762	760, 761	ffvelcdmd 7094	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → (𝑢‘𝑋) ∈ ℕ0)
763	337	notbid 317	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑘 = 𝑢 → (¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0) ↔ ¬ (𝑢 ∘r ≤ 𝑑 ∧ (𝑢‘𝑋) = 0)))
764	116, 763	anbi12d 630	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑘 = 𝑢 → ((𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)) ↔ (𝑢 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑢 ∘r ≤ 𝑑 ∧ (𝑢‘𝑋) = 0))))
765	764	elrab 3679	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))} ↔ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ (𝑢 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑢 ∘r ≤ 𝑑 ∧ (𝑢‘𝑋) = 0))))
766	765	simprbi 495	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))} → (𝑢 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑢 ∘r ≤ 𝑑 ∧ (𝑢‘𝑋) = 0)))
767	766	simpld 493	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))} → 𝑢 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
768	767	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → 𝑢 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
769	768	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) → 𝑢 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
770	756, 122	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))} → 𝑢 Fn 𝐼)
771	770	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → 𝑢 Fn 𝐼)
772	771	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) → 𝑢 Fn 𝐼)
773	14	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
774	86	ffnd 6724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((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)
775	773, 774	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) Fn 𝐼)
776	775	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) Fn 𝐼)
777	8	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) → 𝐼 ∈ 𝑉)
778		eqidd 2726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ∈ 𝐼) → (𝑢‘𝑖) = (𝑢‘𝑖))
779		eqidd 2726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ∈ 𝐼) → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑖) = ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑖))
780	772, 776, 777, 777, 70, 778, 779	ofrfval 7695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) → (𝑢 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ↔ ∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑖)))
781	769, 780	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) → ∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑖))
782	781	r19.21bi 3238	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ∈ 𝐼) → (𝑢‘𝑖) ≤ ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑖))
783	782	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ≠ 𝑋) → (𝑢‘𝑖) ≤ ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑖))
784	63	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ≠ 𝑋) → 𝑑 Fn 𝐼)
785	67	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ≠ 𝑋) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼)
786	8	ad4antr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ≠ 𝑋) → 𝐼 ∈ 𝑉)
787		eqidd 2726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ≠ 𝑋) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) = (𝑑‘𝑖))
788	76	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ≠ 𝑋) ∧ 𝑖 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑖) = if(𝑖 = 𝑋, 1, 0))
789	784, 785, 786, 786, 70, 787, 788	ofval 7696	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ≠ 𝑋) ∧ 𝑖 ∈ 𝐼) → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑖) = ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)))
790	789	an32s 650	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ≠ 𝑋) → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑖) = ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)))
791	156	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ≠ 𝑋) → if(𝑖 = 𝑋, 1, 0) = 0)
792	791	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ≠ 𝑋) → ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) = ((𝑑‘𝑖) + 0))
793	22	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) → 𝑑:𝐼⟶ℕ0)
794	793	ffvelcdmda 7093	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈ ℕ0)
795	794	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ≠ 𝑋) → (𝑑‘𝑖) ∈ ℕ0)
796	795	nn0cnd 12567	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ≠ 𝑋) → (𝑑‘𝑖) ∈ ℂ)
797	796	addridd 11446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ≠ 𝑋) → ((𝑑‘𝑖) + 0) = (𝑑‘𝑖))
798	790, 792, 797	3eqtrd 2769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ≠ 𝑋) → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑖) = (𝑑‘𝑖))
799	783, 798	breqtrd 5175	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ≠ 𝑋) → (𝑢‘𝑖) ≤ (𝑑‘𝑖))
800		simpr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) → (𝑢‘𝑋) = 0)
801	22	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → 𝑑:𝐼⟶ℕ0)
802	801, 761	ffvelcdmd 7094	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → (𝑑‘𝑋) ∈ ℕ0)
803	802	nn0ge0d 12568	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → 0 ≤ (𝑑‘𝑋))
804	803	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) → 0 ≤ (𝑑‘𝑋))
805	800, 804	eqbrtrd 5171	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) → (𝑢‘𝑋) ≤ (𝑑‘𝑋))
806	805	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ∈ 𝐼) → (𝑢‘𝑋) ≤ (𝑑‘𝑋))
807	173, 799, 806	pm2.61ne 3016	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ∈ 𝐼) → (𝑢‘𝑖) ≤ (𝑑‘𝑖))
808	807	ralrimiva 3135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) → ∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ (𝑑‘𝑖))
809	63	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → 𝑑 Fn 𝐼)
810	809	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) → 𝑑 Fn 𝐼)
811		eqidd 2726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) = (𝑑‘𝑖))
812	772, 810, 777, 777, 70, 778, 811	ofrfval 7695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) → (𝑢 ∘r ≤ 𝑑 ↔ ∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ (𝑑‘𝑖)))
813	808, 812	mpbird 256	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ (𝑢‘𝑋) = 0) → 𝑢 ∘r ≤ 𝑑)
814	813	ex 411	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → ((𝑢‘𝑋) = 0 → 𝑢 ∘r ≤ 𝑑))
815	766	simprd 494	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))} → ¬ (𝑢 ∘r ≤ 𝑑 ∧ (𝑢‘𝑋) = 0))
816	815	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → ¬ (𝑢 ∘r ≤ 𝑑 ∧ (𝑢‘𝑋) = 0))
817		imnan 398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑢 ∘r ≤ 𝑑 → ¬ (𝑢‘𝑋) = 0) ↔ ¬ (𝑢 ∘r ≤ 𝑑 ∧ (𝑢‘𝑋) = 0))
818	816, 817	sylibr 233	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → (𝑢 ∘r ≤ 𝑑 → ¬ (𝑢‘𝑋) = 0))
819	818	con2d 134	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → ((𝑢‘𝑋) = 0 → ¬ 𝑢 ∘r ≤ 𝑑))
820	814, 819	pm2.65d 195	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → ¬ (𝑢‘𝑋) = 0)
821	820	neqned 2936	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → (𝑢‘𝑋) ≠ 0)
822	762, 821, 189	sylanbrc 581	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → (𝑢‘𝑋) ∈ ℕ)
823	822	nnge1d 12293	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → 1 ≤ (𝑢‘𝑋))
824	823	adantr 479	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → 1 ≤ (𝑢‘𝑋))
825	171	breq2d 5161	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑋 → (1 ≤ (𝑢‘𝑖) ↔ 1 ≤ (𝑢‘𝑋)))
826	824, 825	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → (𝑖 = 𝑋 → 1 ≤ (𝑢‘𝑖)))
827	826	imp 405	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 = 𝑋) → 1 ≤ (𝑢‘𝑖))
828	760	ffvelcdmda 7093	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → (𝑢‘𝑖) ∈ ℕ0)
829	828	nn0ge0d 12568	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → 0 ≤ (𝑢‘𝑖))
830	829	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) ∧ ¬ 𝑖 = 𝑋) → 0 ≤ (𝑢‘𝑖))
831	827, 830	ifpimpda 1078	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → if-(𝑖 = 𝑋, 1 ≤ (𝑢‘𝑖), 0 ≤ (𝑢‘𝑖)))
832		brif1 7517	. . . . . . . . . . . . . . . . . . 19 ⊢ (if(𝑖 = 𝑋, 1, 0) ≤ (𝑢‘𝑖) ↔ if-(𝑖 = 𝑋, 1 ≤ (𝑢‘𝑖), 0 ≤ (𝑢‘𝑖)))
833	831, 832	sylibr 233	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → if(𝑖 = 𝑋, 1, 0) ≤ (𝑢‘𝑖))
834	833	ralrimiva 3135	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → ∀𝑖 ∈ 𝐼 if(𝑖 = 𝑋, 1, 0) ≤ (𝑢‘𝑖))
835	67	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼)
836	8	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → 𝐼 ∈ 𝑉)
837	76	adantl 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑖) = if(𝑖 = 𝑋, 1, 0))
838		eqidd 2726	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → (𝑢‘𝑖) = (𝑢‘𝑖))
839	835, 771, 836, 836, 70, 837, 838	ofrfval 7695	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∘r ≤ 𝑢 ↔ ∀𝑖 ∈ 𝐼 if(𝑖 = 𝑋, 1, 0) ≤ (𝑢‘𝑖)))
840	834, 839	mpbird 256	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∘r ≤ 𝑢)
841	9	psrbagcon 21880	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)):𝐼⟶ℕ0 ∧ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∘r ≤ 𝑢) → ((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘r ≤ 𝑢))
842	757, 758, 840, 841	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → ((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘r ≤ 𝑢))
843	842	simpld 493	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
844		eqidd 2726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) = (𝑑‘𝑖))
845	809, 835, 836, 836, 70, 844, 837	ofval 7696	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → ((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑖) = ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)))
846	771, 775, 836, 836, 70, 838, 845	ofrfval 7695	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → (𝑢 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ↔ ∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0))))
847	768, 846	mpbid 231	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → ∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)))
848	847	r19.21bi 3238	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)))
849	828	nn0red 12566	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → (𝑢‘𝑖) ∈ ℝ)
850	58	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → if(𝑖 = 𝑋, 1, 0) ∈ ℝ)
851	801	ffvelcdmda 7093	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈ ℕ0)
852	851	nn0red 12566	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈ ℝ)
853	849, 850, 852	lesubaddd 11843	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → (((𝑢‘𝑖) − if(𝑖 = 𝑋, 1, 0)) ≤ (𝑑‘𝑖) ↔ (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0))))
854	848, 853	mpbird 256	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → ((𝑢‘𝑖) − if(𝑖 = 𝑋, 1, 0)) ≤ (𝑑‘𝑖))
855	854	ralrimiva 3135	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → ∀𝑖 ∈ 𝐼 ((𝑢‘𝑖) − if(𝑖 = 𝑋, 1, 0)) ≤ (𝑑‘𝑖))
856	771, 835, 836, 836, 70	offn 7698	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) Fn 𝐼)
857	771, 835, 836, 836, 70, 838, 837	ofval 7696	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → ((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑖) = ((𝑢‘𝑖) − if(𝑖 = 𝑋, 1, 0)))
858	856, 809, 836, 836, 70, 857, 844	ofrfval 7695	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → ((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘r ≤ 𝑑 ↔ ∀𝑖 ∈ 𝐼 ((𝑢‘𝑖) − if(𝑖 = 𝑋, 1, 0)) ≤ (𝑑‘𝑖)))
859	855, 858	mpbird 256	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘r ≤ 𝑑)
860	755, 843, 859	elrabd 3681	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})
861	828	nn0cnd 12567	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → (𝑢‘𝑖) ∈ ℂ)
862	235	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → if(𝑖 = 𝑋, 1, 0) ∈ ℂ)
863	861, 862	npcand 11607	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) ∧ 𝑖 ∈ 𝐼) → (((𝑢‘𝑖) − if(𝑖 = 𝑋, 1, 0)) + if(𝑖 = 𝑋, 1, 0)) = (𝑢‘𝑖))
864	863	mpteq2dva 5249	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → (𝑖 ∈ 𝐼 ↦ (((𝑢‘𝑖) − if(𝑖 = 𝑋, 1, 0)) + if(𝑖 = 𝑋, 1, 0))) = (𝑖 ∈ 𝐼 ↦ (𝑢‘𝑖)))
865	856, 835, 836, 836, 70, 857, 837	offval 7694	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → ((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑖 ∈ 𝐼 ↦ (((𝑢‘𝑖) − if(𝑖 = 𝑋, 1, 0)) + if(𝑖 = 𝑋, 1, 0))))
866	760	feqmptd 6966	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → 𝑢 = (𝑖 ∈ 𝐼 ↦ (𝑢‘𝑖)))
867	864, 865, 866	3eqtr4rd 2776	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}) → 𝑢 = ((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
868		oveq1 7426	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = ((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
869	868	eqeq2d 2736	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ↔ 𝑢 = ((𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
870	754, 860, 867, 869	rspceb2dv 3610	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (∃𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ↔ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}))
871	454, 719, 870	3bitrd 304	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ↔ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}))
872	871	eqrdv 2723	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) = {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))})
873		difrab 4307	. . . . . . . . . 10 ⊢ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) = {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∧ ¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0))}
874	872, 873	eqtr4di 2783	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) = ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}))
875		difssd 4129	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ⊆ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))})
876	874, 875	eqsstrd 4015	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ⊆ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))})
877	703, 876, 111	fmptssfisupp 9419	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))) finSupp (0g‘𝑅))
878		difss 4128	. . . . . . . . . 10 ⊢ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ⊆ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}
879		disjdif 4473	. . . . . . . . . 10 ⊢ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∩ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) = ∅
880		ssdisj 4461	. . . . . . . . . 10 ⊢ ((({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ⊆ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∧ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∩ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) = ∅) → (({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ∩ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) = ∅)
881	878, 879, 880	mp2an 690	. . . . . . . . 9 ⊢ (({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ∩ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) = ∅
882	881	ineqcomi 4201	. . . . . . . 8 ⊢ (({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∩ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) = ∅
883	882	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∩ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) = ∅)
884	277, 97	psdmullem 22112	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∪ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})) = ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}))
885	874, 884	eqtr4d 2768	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) = (({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ∪ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)})))
886	1, 102, 2, 6, 696, 699, 877, 883, 885	gsumsplit2 19896	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))) = ((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))))
887	692, 886	eqtrd 2765	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σg ((𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏)(.r‘𝑅)(𝐺‘(𝑑 ∘f − 𝑏)))) ∘ (𝑢 ∈ ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ↦ (𝑢 ∘f − (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))) = ((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))))
888	425, 591, 887	3eqtrd 2769	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) · 𝐺)‘𝑑) = ((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))))
889	415	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺) ∈ 𝐵)
890	31, 32, 29, 383, 9, 384, 889, 7	psrmulval 21906	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝐹 · (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺))‘𝑑) = (𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ ((𝐹‘𝑢)(.r‘𝑅)((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺)‘(𝑑 ∘f − 𝑢))))))
891	3	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑅 ∈ CRing)
892	39	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝐺 ∈ 𝐵)
893	31, 32, 9, 250, 891, 283, 892, 245	psdcoef 22107	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺)‘(𝑑 ∘f − 𝑢)) = ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)(𝐺‘((𝑑 ∘f − 𝑢) ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))
894	265	fveq2d 6900	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (𝐺‘((𝑑 ∘f − 𝑢) ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))
895	894	oveq2d 7435	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)(𝐺‘((𝑑 ∘f − 𝑢) ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) = ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))
896	893, 895	eqtrd 2765	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺)‘(𝑑 ∘f − 𝑢)) = ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))
897	896	oveq2d 7435	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝐹‘𝑢)(.r‘𝑅)((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺)‘(𝑑 ∘f − 𝑢))) = ((𝐹‘𝑢)(.r‘𝑅)((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))
898	307	nn0zd 12617	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → (((𝑑 ∘f − 𝑢)‘𝑋) + 1) ∈ ℤ)
899	1, 17, 29	mulgass3 20304	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1) ∈ ℤ ∧ (𝐹‘𝑢) ∈ (Base‘𝑅) ∧ (𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)) ∈ (Base‘𝑅))) → ((𝐹‘𝑢)(.r‘𝑅)((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) = ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))
900	222, 898, 226, 269, 899	syl13anc 1369	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝐹‘𝑢)(.r‘𝑅)((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) = ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))
901	897, 900	eqtrd 2765	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → ((𝐹‘𝑢)(.r‘𝑅)((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺)‘(𝑑 ∘f − 𝑢))) = ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))
902	901	mpteq2dva 5249	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ ((𝐹‘𝑢)(.r‘𝑅)((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺)‘(𝑑 ∘f − 𝑢)))) = (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))
903	902	oveq2d 7435	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ ((𝐹‘𝑢)(.r‘𝑅)((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺)‘(𝑑 ∘f − 𝑢))))) = (𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))))
904	1, 2, 6, 219, 319, 273, 280	gsummptfidmsplit 19897	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))) = ((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))))
905	890, 903, 904	3eqtrd 2769	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝐹 · (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺))‘𝑑) = ((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))))
906	419, 421, 422, 422, 423, 888, 905	offval 7694	. . 3 ⊢ (𝜑 → (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) · 𝐺) ∘f (+g‘𝑅)(𝐹 · (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺))) = (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))))(+g‘𝑅)((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))))))
907	417, 906	eqtrd 2765	. 2 ⊢ (𝜑 → (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) · 𝐺) + (𝐹 · (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺))) = (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))))))(+g‘𝑅)((𝑅 Σg (𝑢 ∈ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∖ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)}) ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))(+g‘𝑅)(𝑅 Σg (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ↦ ((((𝑑 ∘f − 𝑢)‘𝑋) + 1)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))))))))))
908	408, 410, 907	3eqtr4d 2775	1 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝐹 · 𝐺)) = (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) · 𝐺) + (𝐹 · (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845  if-wif 1060   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2929  ∀wral 3050  ∃wrex 3059  {crab 3418  Vcvv 3461   ∖ cdif 3941   ∪ cun 3942   ∩ cin 3943   ⊆ wss 3944  ∅c0 4322  ifcif 4530  {csn 4630   class class class wbr 5149   ↦ cmpt 5232   × cxp 5676  ^◡ccnv 5677  dom cdm 5678   “ cima 5681   ∘ ccom 5682  Fun wfun 6543   Fn wfn 6544  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419   ∘_f cof 7683   ∘_r cofr 7684   ↑_m cmap 8845  Fincfn 8964  ℂcc 11138  ℝcr 11139  0cc0 11140  1c1 11141   + caddc 11143   < clt 11280   ≤ cle 11281   − cmin 11476  ℕcn 12245  ℕ₀cn0 12505  ℤcz 12591  ..^cfzo 13662  Basecbs 17183  +_gcplusg 17236  ._rcmulr 17237  0_gc0g 17424   Σ_g cgsu 17425  Mndcmnd 18697  Grpcgrp 18898  ._gcmg 19031  CMndccmn 19747  Ringcrg 20185  CRingccrg 20186   mPwSer cmps 21854   mPSDer cpsd 22078

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 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-ofr 7686  df-om 7872  df-1st 7994  df-2nd 7995  df-supp 8166  df-tpos 8232  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-pm 8848  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9388  df-oi 9535  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12506  df-z 12592  df-uz 12856  df-fz 13520  df-fzo 13663  df-seq 14003  df-hash 14326  df-struct 17119  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mulr 17250  df-sca 17252  df-vsca 17253  df-tset 17255  df-0g 17426  df-gsum 17427  df-mre 17569  df-mrc 17570  df-acs 17572  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-mhm 18743  df-submnd 18744  df-grp 18901  df-minusg 18902  df-mulg 19032  df-ghm 19176  df-cntz 19280  df-cmn 19749  df-abl 19750  df-mgp 20087  df-rng 20105  df-ur 20134  df-ring 20187  df-cring 20188  df-oppr 20285  df-psr 21859  df-psd 22103

This theorem is referenced by:  psd1  22114