Theorem psdmul 22079

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.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 2731	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2731	. . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅)
3		psdmul.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ CRing)
4	3	crngringd 20162	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
5	4	ringcmnd 20200	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ CMnd)
6	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ CMnd)
7		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
8		psdmul.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
9		psdmul.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
10		psdmul.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑆)
11		reldmpsr 21849	. . . . . . . . . . . 12 ⊢ Rel dom mPwSer
12	9, 10, 11	strov2rcl 17125	. . . . . . . . . . 11 ⊢ (𝐹 ∈ 𝐵 → 𝐼 ∈ V)
13	8, 12	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ V)
14		eqid 2731	. . . . . . . . . . 11 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
15	14	psrbagsn 21996	. . . . . . . . . 10 ⊢ (𝐼 ∈ V → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
16	13, 15	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
17	16	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
18	14	psrbagaddcl 21859	. . . . . . . 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})
19	7, 17, 18	syl2anc 584	. . . . . . 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})
20	14	psrbaglefi 21861	. . . . . . 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)
21	19, 20	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)
22		eqid 2731	. . . . . . 7 ⊢ (.g‘𝑅) = (.g‘𝑅)
23	3	crnggrpd 20163	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Grp)
24	23	grpmndd 18856	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Mnd)
25	24	ad2antrr 726	. . . . . . 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)
26	14	psrbagf 21853	. . . . . . . . . . 11 ⊢ (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑑:𝐼⟶ℕ0)
27	26	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑:𝐼⟶ℕ0)
28		psdmul.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
29	28	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑋 ∈ 𝐼)
30	27, 29	ffvelcdmd 7018	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑‘𝑋) ∈ ℕ0)
31		peano2nn0 12418	. . . . . . . . 9 ⊢ ((𝑑‘𝑋) ∈ ℕ0 → ((𝑑‘𝑋) + 1) ∈ ℕ0)
32	30, 31	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑑‘𝑋) + 1) ∈ ℕ0)
33	32	adantr 480	. . . . . . 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)
34		eqid 2731	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
35	4	ad2antrr 726	. . . . . . . 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)
36	9, 1, 14, 10, 8	psrelbas 21869	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
37	36	ad2antrr 726	. . . . . . . . 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))
38		elrabi 3643	. . . . . . . . . 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})
39	38	adantl 481	. . . . . . . . 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})
40	37, 39	ffvelcdmd 7018	. . . . . . . 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))
41		psdmul.g	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
42	9, 1, 14, 10, 41	psrelbas 21869	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
43	42	ad2antrr 726	. . . . . . . . 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))
44		eqid 2731	. . . . . . . . . . . 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)))}
45	14, 44	psrbagconcl 21862	. . . . . . . . . . 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)))})
46	19, 45	sylan 580	. . . . . . . . . 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)))})
47		elrabi 3643	. . . . . . . . . 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})
48	46, 47	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})
49	43, 48	ffvelcdmd 7018	. . . . . . . 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))
50	1, 34, 35, 40, 49	ringcld 20176	. . . . . . 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))
51	1, 22, 25, 33, 50	mulgnn0cld 19005	. . . . . 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))
52		disjdifr 4423	. . . . . . 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}) = ∅
53	52	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}) = ∅)
54		1nn0 12394	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ0
55		0nn0 12393	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℕ0
56	54, 55	ifcli 4523	. . . . . . . . . . . . . . 15 ⊢ if(𝑖 = 𝑋, 1, 0) ∈ ℕ0
57	56	nn0ge0i 12405	. . . . . . . . . . . . . 14 ⊢ 0 ≤ if(𝑖 = 𝑋, 1, 0)
58	27	ffvelcdmda 7017	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈ ℕ0)
59	58	nn0red 12440	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈ ℝ)
60	56	nn0rei 12389	. . . . . . . . . . . . . . . 16 ⊢ if(𝑖 = 𝑋, 1, 0) ∈ ℝ
61	60	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → if(𝑖 = 𝑋, 1, 0) ∈ ℝ)
62	59, 61	addge01d 11702	. . . . . . . . . . . . . 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))))
63	57, 62	mpbii 233	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)))
64	63	ralrimiva 3124	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ∀𝑖 ∈ 𝐼 (𝑑‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)))
65	27	ffnd 6652	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 Fn 𝐼)
66	54, 55	ifcli 4523	. . . . . . . . . . . . . . . . 17 ⊢ if(𝑦 = 𝑋, 1, 0) ∈ ℕ0
67	66	elexi 3459	. . . . . . . . . . . . . . . 16 ⊢ if(𝑦 = 𝑋, 1, 0) ∈ V
68		eqid 2731	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))
69	67, 68	fnmpti 6624	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼
70	69	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼)
71	13	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ V)
72		inidm 4177	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∩ 𝐼) = 𝐼
73	65, 70, 71, 71, 72	offn 7623	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) Fn 𝐼)
74		eqidd 2732	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) = (𝑑‘𝑖))
75		eqeq1 2735	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑖 → (𝑦 = 𝑋 ↔ 𝑖 = 𝑋))
76	75	ifbid 4499	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑖 → if(𝑦 = 𝑋, 1, 0) = if(𝑖 = 𝑋, 1, 0))
77	56	elexi 3459	. . . . . . . . . . . . . . . 16 ⊢ if(𝑖 = 𝑋, 1, 0) ∈ V
78	76, 68, 77	fvmpt 6929	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝐼 → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑖) = if(𝑖 = 𝑋, 1, 0))
79	78	adantl 481	. . . . . . . . . . . . . 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))
80	65, 70, 71, 71, 72, 74, 79	ofval 7621	. . . . . . . . . . . . 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)))
81	65, 73, 71, 71, 72, 74, 80	ofrfval 7620	. . . . . . . . . . . 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))))
82	64, 81	mpbird 257	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
83	82	adantr 480	. . . . . . . . . 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))))
84	13	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ V)
85	14	psrbagf 21853	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑘:𝐼⟶ℕ0)
86	85	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑘:𝐼⟶ℕ0)
87	27	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑:𝐼⟶ℕ0)
88	14	psrbagf 21853	. . . . . . . . . . . . 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)
89	19, 88	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))):𝐼⟶ℕ0)
90	89	adantr 480	. . . . . . . . . . 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)
91		nn0re 12387	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℕ0 → 𝑞 ∈ ℝ)
92		nn0re 12387	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ℕ0 → 𝑟 ∈ ℝ)
93		nn0re 12387	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℕ0 → 𝑠 ∈ ℝ)
94		letr 11204	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ ℝ ∧ 𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ) → ((𝑞 ≤ 𝑟 ∧ 𝑟 ≤ 𝑠) → 𝑞 ≤ 𝑠))
95	91, 92, 93, 94	syl3an 1160	. . . . . . . . . . . 12 ⊢ ((𝑞 ∈ ℕ0 ∧ 𝑟 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) → ((𝑞 ≤ 𝑟 ∧ 𝑟 ≤ 𝑠) → 𝑞 ≤ 𝑠))
96	95	adantl 481	. . . . . . . . . . 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))
97	84, 86, 87, 90, 96	caoftrn 7651	. . . . . . . . . 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)))))
98	83, 97	mpan2d 694	. . . . . . . . 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)))))
99	98	ss2rabdv 4026	. . . . . . . 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)))})
100		undifr 4433	. . . . . . . 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)))})
101	99, 100	sylib 218	. . . . . . 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)))})
102	101	eqcomd 2737	. . . . . 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}))
103	1, 2, 6, 21, 51, 53, 102	gsummptfidmsplit 19840	. . . . 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))))))))
104		eqid 2731	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
105		ovex 7379	. . . . . . . . 9 ⊢ (ℕ0 ↑m 𝐼) ∈ V
106	105	rabex 5277	. . . . . . . 8 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V
107	106	rabex 5277	. . . . . . 7 ⊢ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))} ∈ V
108	107	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)
109		ovex 7379	. . . . . . . . 9 ⊢ ((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢))) ∈ V
110		eqid 2731	. . . . . . . . 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))))
111	109, 110	fnmpti 6624	. . . . . . . 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)))}
112	111	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)))})
113		fvexd 6837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (0g‘𝑅) ∈ V)
114	112, 21, 113	fndmfifsupp 9262	. . . . . 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))
115	1, 104, 22, 108, 50, 114, 6, 32	gsummulg 19852	. . . . 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)))))))
116		difrab 4268	. . . . . . . . . . 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)}
117	116	eleq2i 2823	. . . . . . . . . 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)})
118		breq1 5094	. . . . . . . . . . . . 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)))))
119		breq1 5094	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑢 → (𝑘 ∘r ≤ 𝑑 ↔ 𝑢 ∘r ≤ 𝑑))
120	119	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑢 → (¬ 𝑘 ∘r ≤ 𝑑 ↔ ¬ 𝑢 ∘r ≤ 𝑑))
121	118, 120	anbi12d 632	. . . . . . . . . . . 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)))
122	121	elrab 3647	. . . . . . . . . . 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)))
123	14	psrbagf 21853	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑢:𝐼⟶ℕ0)
124	123	ffnd 6652	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑢 Fn 𝐼)
125	124	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑢 Fn 𝐼)
126	73	adantr 480	. . . . . . . . . . . . . . 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)
127	13	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ V)
128		eqidd 2732	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑢‘𝑖) = (𝑢‘𝑖))
129	65	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 Fn 𝐼)
130	66	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ 𝐼 → if(𝑦 = 𝑋, 1, 0) ∈ ℕ0)
131	68, 130	fmpti 7045	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)):𝐼⟶ℕ0
132	131	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)):𝐼⟶ℕ0)
133	132	ffnd 6652	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼)
134	133	ad2antrr 726	. . . . . . . . . . . . . . . 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)
135		eqidd 2732	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) = (𝑑‘𝑖))
136	78	adantl 481	. . . . . . . . . . . . . . . 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))
137	129, 134, 127, 127, 72, 135, 136	ofval 7621	. . . . . . . . . . . . . . 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)))
138	125, 126, 127, 127, 72, 128, 137	ofrfval 7620	. . . . . . . . . . . . . 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))))
139	125, 129, 127, 127, 72, 128, 135	ofrfval 7620	. . . . . . . . . . . . . . . 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)))
140	139	notbid 318	. . . . . . . . . . . . . . 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)))
141		rexnal 3084	. . . . . . . . . . . . . . 15 ⊢ (∃𝑖 ∈ 𝐼 ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖) ↔ ¬ ∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ (𝑑‘𝑖))
142	140, 141	bitr4di 289	. . . . . . . . . . . . . 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)))
143	138, 142	anbi12d 632	. . . . . . . . . . . . 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))))
144	30	ad2antrr 726	. . . . . . . . . . . . . . . 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)
145	123	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑢:𝐼⟶ℕ0)
146	28	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑋 ∈ 𝐼)
147	145, 146	ffvelcdmd 7018	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢‘𝑋) ∈ ℕ0)
148	147	adantlr 715	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢‘𝑋) ∈ ℕ0)
149	148	adantr 480	. . . . . . . . . . . . . . . . 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)
150		nn0nlt0 12404	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑‘𝑋) ∈ ℕ0 → ¬ (𝑑‘𝑋) < 0)
151	144, 150	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)
152	27	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑:𝐼⟶ℕ0)
153	152	ffvelcdmda 7017	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈ ℕ0)
154	153	nn0cnd 12441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈ ℂ)
155	154	addridd 11310	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
156	155	breq2d 5103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
157	156	biimpd 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
158		ifnefalse 4487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑖 ≠ 𝑋 → if(𝑖 = 𝑋, 1, 0) = 0)
159	158	oveq2d 7362	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑖 ≠ 𝑋 → ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) = ((𝑑‘𝑖) + 0))
160	159	breq2d 5103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑖 ≠ 𝑋 → ((𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ↔ (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + 0)))
161	160	imbi1d 341	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
162	157, 161	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
163	162	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
164	163	impancom 451	. . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
165	164	necon1bd 2946	. . . . . . . . . . . . . . . . . . . . . . . . . 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))
166	165	ancrd 551	. . . . . . . . . . . . . . . . . . . . . . . . 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))))
167	166	ex 412	. . . . . . . . . . . . . . . . . . . . . . . 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)))))
168	167	ralimdva 3144	. . . . . . . . . . . . . . . . . . . . . . 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)))))
169	168	anim1d 611	. . . . . . . . . . . . . . . . . . . . . 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))))
170	169	imp 406	. . . . . . . . . . . . . . . . . . . . 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)))
171		rexim 3073	. . . . . . . . . . . . . . . . . . . . . . . 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))))
172	171	imp 406	. . . . . . . . . . . . . . . . . . . . . . 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)))
173		fveq2 6822	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 = 𝑋 → (𝑢‘𝑖) = (𝑢‘𝑋))
174		fveq2 6822	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 = 𝑋 → (𝑑‘𝑖) = (𝑑‘𝑋))
175	173, 174	breq12d 5104	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 = 𝑋 → ((𝑢‘𝑖) ≤ (𝑑‘𝑖) ↔ (𝑢‘𝑋) ≤ (𝑑‘𝑋)))
176	175	notbid 318	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 𝑋 → (¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖) ↔ ¬ (𝑢‘𝑋) ≤ (𝑑‘𝑋)))
177	176	ceqsrexbv 3611	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃𝑖 ∈ 𝐼 (𝑖 = 𝑋 ∧ ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖)) ↔ (𝑋 ∈ 𝐼 ∧ ¬ (𝑢‘𝑋) ≤ (𝑑‘𝑋)))
178	177	simprbi 496	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃𝑖 ∈ 𝐼 (𝑖 = 𝑋 ∧ ¬ (𝑢‘𝑖) ≤ (𝑑‘𝑖)) → ¬ (𝑢‘𝑋) ≤ (𝑑‘𝑋))
179	172, 178	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))
180	30	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑‘𝑋) ∈ ℕ0)
181	180	nn0red 12440	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑‘𝑋) ∈ ℝ)
182	148	nn0red 12440	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑢‘𝑋) ∈ ℝ)
183	181, 182	ltnled 11257	. . . . . . . . . . . . . . . . . . . . . . 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)))
184	183	biimpar 477	. . . . . . . . . . . . . . . . . . . . . 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))
185	179, 184	sylan2 593	. . . . . . . . . . . . . . . . . . . . 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))
186	170, 185	syldan 591	. . . . . . . . . . . . . . . . . . . 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))
187		breq2 5095	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑢‘𝑋) = 0 → ((𝑑‘𝑋) < (𝑢‘𝑋) ↔ (𝑑‘𝑋) < 0))
188	186, 187	syl5ibcom 245	. . . . . . . . . . . . . . . . . . 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))
189	151, 188	mtod 198	. . . . . . . . . . . . . . . . . 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)
190	189	neqned 2935	. . . . . . . . . . . . . . . . 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)
191		elnnne0 12392	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢‘𝑋) ∈ ℕ ↔ ((𝑢‘𝑋) ∈ ℕ0 ∧ (𝑢‘𝑋) ≠ 0))
192	149, 190, 191	sylanbrc 583	. . . . . . . . . . . . . . . 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)
193		elfzo0 13597	. . . . . . . . . . . . . . . 16 ⊢ ((𝑑‘𝑋) ∈ (0..^(𝑢‘𝑋)) ↔ ((𝑑‘𝑋) ∈ ℕ0 ∧ (𝑢‘𝑋) ∈ ℕ ∧ (𝑑‘𝑋) < (𝑢‘𝑋)))
194	144, 192, 186, 193	syl3anbrc 1344	. . . . . . . . . . . . . . 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)))
195		fzostep1 13683	. . . . . . . . . . . . . . 15 ⊢ ((𝑑‘𝑋) ∈ (0..^(𝑢‘𝑋)) → (((𝑑‘𝑋) + 1) ∈ (0..^(𝑢‘𝑋)) ∨ ((𝑑‘𝑋) + 1) = (𝑢‘𝑋)))
196	194, 195	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)))
197	149	nn0red 12440	. . . . . . . . . . . . . . . . 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)
198	32	ad2antrr 726	. . . . . . . . . . . . . . . . . 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)
199	198	nn0red 12440	. . . . . . . . . . . . . . . . 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)
200	28	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑋 ∈ 𝐼)
201		iftrue 4481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑋 → if(𝑖 = 𝑋, 1, 0) = 1)
202	174, 201	oveq12d 7364	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑋 → ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) = ((𝑑‘𝑋) + 1))
203	173, 202	breq12d 5104	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑋 → ((𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) ↔ (𝑢‘𝑋) ≤ ((𝑑‘𝑋) + 1)))
204	203	rspcv 3573	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑋 ∈ 𝐼 → (∀𝑖 ∈ 𝐼 (𝑢‘𝑖) ≤ ((𝑑‘𝑖) + if(𝑖 = 𝑋, 1, 0)) → (𝑢‘𝑋) ≤ ((𝑑‘𝑋) + 1)))
205	200, 204	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)))
206	205	imp 406	. . . . . . . . . . . . . . . . . 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))
207	206	adantrr 717	. . . . . . . . . . . . . . . . 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))
208	197, 199, 207	lensymd 11261	. . . . . . . . . . . . . . . 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))
209	208	intn3an3d 1483	. . . . . . . . . . . . . . 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)))
210		elfzo0 13597	. . . . . . . . . . . . . . 15 ⊢ (((𝑑‘𝑋) + 1) ∈ (0..^(𝑢‘𝑋)) ↔ (((𝑑‘𝑋) + 1) ∈ ℕ0 ∧ (𝑢‘𝑋) ∈ ℕ ∧ ((𝑑‘𝑋) + 1) < (𝑢‘𝑋)))
211	209, 210	sylnibr 329	. . . . . . . . . . . . . 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)))
212	196, 211	orcnd 878	. . . . . . . . . . . . 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))
213	143, 212	sylbida 592	. . . . . . . . . . . 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))
214	213	anasss 466	. . . . . . . . . . 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))
215	122, 214	sylan2b 594	. . . . . . . . . 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))
216	117, 215	sylan2b 594	. . . . . . . . 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))
217	216	oveq1d 7361	. . . . . . . 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)))))
218	217	mpteq2dva 5184	. . . . . . 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))))))
219	218	oveq2d 7362	. . . . . 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)))))))
220	14	psrbaglefi 21861	. . . . . . . . 9 ⊢ (𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∈ Fin)
221	220	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∈ Fin)
222	24	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑅 ∈ Mnd)
223	32	adantr 480	. . . . . . . . 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)
224	4	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑅 ∈ Ring)
225		elrabi 3643	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
226	36	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐹:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
227	226	ffvelcdmda 7017	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝐹‘𝑢) ∈ (Base‘𝑅))
228	225, 227	sylan2 593	. . . . . . . . . 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))
229	42	ad2antrr 726	. . . . . . . . . . 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))
230	27	adantr 480	. . . . . . . . . . . . . . . . . 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)
231	230	ffvelcdmda 7017	. . . . . . . . . . . . . . . . 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)
232	231	nn0cnd 12441	. . . . . . . . . . . . . . . 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)
233	225, 123	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → 𝑢:𝐼⟶ℕ0)
234	233	adantl 481	. . . . . . . . . . . . . . . . . 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)
235	234	ffvelcdmda 7017	. . . . . . . . . . . . . . . . 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)
236	235	nn0cnd 12441	. . . . . . . . . . . . . . . 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)
237	56	nn0cni 12390	. . . . . . . . . . . . . . . . 17 ⊢ if(𝑖 = 𝑋, 1, 0) ∈ ℂ
238	237	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)
239	232, 236, 238	subadd23d 11491	. . . . . . . . . . . . . . 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))))
240	232, 238, 236	addsubassd 11489	. . . . . . . . . . . . . . 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))))
241	239, 240	eqtr4d 2769	. . . . . . . . . . . . . 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)))
242	241	mpteq2dva 5184	. . . . . . . . . . . . 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))))
243		eqid 2731	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} = {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}
244	14, 243	psrbagconcl 21862	. . . . . . . . . . . . . . . . . 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})
245		elrabi 3643	. . . . . . . . . . . . . . . . . 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})
246	244, 245	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})
247	246	adantll 714	. . . . . . . . . . . . . . . 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})
248	14	psrbagf 21853	. . . . . . . . . . . . . . . 16 ⊢ ((𝑑 ∘f − 𝑢) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → (𝑑 ∘f − 𝑢):𝐼⟶ℕ0)
249	247, 248	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)
250	249	ffnd 6652	. . . . . . . . . . . . . 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)
251	69	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)
252	13	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝐼 ∈ V)
253	230	ffnd 6652	. . . . . . . . . . . . . . 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)
254	234	ffnd 6652	. . . . . . . . . . . . . . 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)
255		eqidd 2732	. . . . . . . . . . . . . . 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))
256		eqidd 2732	. . . . . . . . . . . . . . 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))
257	253, 254, 252, 252, 72, 255, 256	ofval 7621	. . . . . . . . . . . . . 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)))
258	78	adantl 481	. . . . . . . . . . . . . 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))
259	250, 251, 252, 252, 72, 257, 258	offval 7619	. . . . . . . . . . . . 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))))
260		simplr 768	. . . . . . . . . . . . . . . . 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})
261	16	ad2antrr 726	. . . . . . . . . . . . . . . . 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})
262	260, 261, 18	syl2anc 584	. . . . . . . . . . . . . . . 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})
263	262, 88	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)
264	263	ffnd 6652	. . . . . . . . . . . . . 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)
265	253, 251, 252, 252, 72, 255, 258	ofval 7621	. . . . . . . . . . . . . 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)))
266	264, 254, 252, 252, 72, 265, 256	offval 7619	. . . . . . . . . . . . 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))))
267	242, 259, 266	3eqtr4d 2776	. . . . . . . . . . . 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))
268	14	psrbagaddcl 21859	. . . . . . . . . . . . 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})
269	247, 261, 268	syl2anc 584	. . . . . . . . . . . 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})
270	267, 269	eqeltrrd 2832	. . . . . . . . . . 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})
271	229, 270	ffvelcdmd 7018	. . . . . . . . . 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))
272	1, 34, 224, 228, 271	ringcld 20176	. . . . . . . . 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))
273	1, 22, 222, 223, 272	mulgnn0cld 19005	. . . . . . . 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))
274		disjdifr 4423	. . . . . . . . 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)}) = ∅
275	274	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)}) = ∅)
276		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0) → 𝑘 ∘r ≤ 𝑑)
277	276	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → ((𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0) → 𝑘 ∘r ≤ 𝑑))
278	277	ss2rabi 4027	. . . . . . . . . . 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}
279	278	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})
280		undifr 4433	. . . . . . . . . 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})
281	279, 280	sylib 218	. . . . . . . . 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})
282	281	eqcomd 2737	. . . . . . . 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)}))
283	1, 2, 6, 221, 273, 275, 282	gsummptfidmsplit 19840	. . . . . . 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))))))))
284		eldifi 4081	. . . . . . . . . . . . . . . . . 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})
285	28	ad2antrr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑋 ∈ 𝐼)
286		eqidd 2732	. . . . . . . . . . . . . . . . . . . 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))
287		eqidd 2732	. . . . . . . . . . . . . . . . . . . 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))
288	253, 254, 252, 252, 72, 286, 287	ofval 7621	. . . . . . . . . . . . . . . . . . 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)))
289	285, 288	mpdan 687	. . . . . . . . . . . . . . . . . 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)))
290	284, 289	sylan2 593	. . . . . . . . . . . . . . . . 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)))
291	290	oveq2d 7362	. . . . . . . . . . . . . . . 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))))
292	234, 285	ffvelcdmd 7018	. . . . . . . . . . . . . . . . . . 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)
293	284, 292	sylan2 593	. . . . . . . . . . . . . . . . . 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)
294	293	nn0cnd 12441	. . . . . . . . . . . . . . . . 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)
295	30	nn0cnd 12441	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑‘𝑋) ∈ ℂ)
296	295	adantr 480	. . . . . . . . . . . . . . . . 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)
297	294, 296	pncan3d 11472	. . . . . . . . . . . . . . . 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))
298	291, 297	eqtrd 2766	. . . . . . . . . . . . . . 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))
299	298	oveq1d 7361	. . . . . . . . . . . . . 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))
300	249, 285	ffvelcdmd 7018	. . . . . . . . . . . . . . . . 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)
301	284, 300	sylan2 593	. . . . . . . . . . . . . . . 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)
302	301	nn0cnd 12441	. . . . . . . . . . . . . . 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)
303		1cnd 11104	. . . . . . . . . . . . . . 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)
304	294, 302, 303	addassd 11131	. . . . . . . . . . . . . 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)))
305	299, 304	eqtr3d 2768	. . . . . . . . . . . . 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)))
306	305	oveq1d 7361	. . . . . . . . . . . 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)))))
307	24	ad2antrr 726	. . . . . . . . . . . . 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)
308		peano2nn0 12418	. . . . . . . . . . . . . . 15 ⊢ (((𝑑 ∘f − 𝑢)‘𝑋) ∈ ℕ0 → (((𝑑 ∘f − 𝑢)‘𝑋) + 1) ∈ ℕ0)
309	300, 308	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)
310	284, 309	sylan2 593	. . . . . . . . . . . . 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)
311	284, 272	sylan2 593	. . . . . . . . . . . . 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))
312	1, 22, 2	mulgnn0dir 19014	. . . . . . . . . . . . 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))))))
313	307, 293, 310, 311, 312	syl13anc 1374	. . . . . . . . . . . 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))))))
314	306, 313	eqtrd 2766	. . . . . . . . . . 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))))))
315	314	mpteq2dva 5184	. . . . . . . . . 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)))))))
316	315	oveq2d 7362	. . . . . . . . 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))))))))
317		difssd 4087	. . . . . . . . . . 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})
318	221, 317	ssfid 9153	. . . . . . . . . 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)
319	1, 22, 222, 292, 272	mulgnn0cld 19005	. . . . . . . . . . 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))
320	284, 319	sylan2 593	. . . . . . . . . 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))
321	1, 22, 222, 309, 272	mulgnn0cld 19005	. . . . . . . . . . 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))
322	284, 321	sylan2 593	. . . . . . . . . 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))
323		eqid 2731	. . . . . . . . . 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)))))
324		eqid 2731	. . . . . . . . . 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)))))
325	1, 2, 6, 318, 320, 322, 323, 324	gsummptfidmadd 19835	. . . . . . . . 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))))))))
326	316, 325	eqtrd 2766	. . . . . . . 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))))))))
327	28	ad2antrr 726	. . . . . . . . . . . . . 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)
328	65	adantr 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)}) → 𝑑 Fn 𝐼)
329		elrabi 3643	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} → 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
330	329, 124	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} → 𝑢 Fn 𝐼)
331	330	adantl 481	. . . . . . . . . . . . . . 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)
332	13	ad2antrr 726	. . . . . . . . . . . . . . 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)
333		eqidd 2732	. . . . . . . . . . . . . . 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))
334		eqidd 2732	. . . . . . . . . . . . . . 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))
335	328, 331, 332, 332, 72, 333, 334	ofval 7621	. . . . . . . . . . . . . 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)))
336	327, 335	mpdan 687	. . . . . . . . . . . . 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)))
337		fveq1 6821	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑢 → (𝑘‘𝑋) = (𝑢‘𝑋))
338	337	eqeq1d 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑢 → ((𝑘‘𝑋) = 0 ↔ (𝑢‘𝑋) = 0))
339	119, 338	anbi12d 632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑢 → ((𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0) ↔ (𝑢 ∘r ≤ 𝑑 ∧ (𝑢‘𝑋) = 0)))
340	339	elrab 3647	. . . . . . . . . . . . . . . . 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)))
341	340	simprbi 496	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} → (𝑢 ∘r ≤ 𝑑 ∧ (𝑢‘𝑋) = 0))
342	341	simprd 495	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} → (𝑢‘𝑋) = 0)
343	342	adantl 481	. . . . . . . . . . . . . 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)
344	343	oveq2d 7362	. . . . . . . . . . . . 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))
345	30	adantr 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)}) → (𝑑‘𝑋) ∈ ℕ0)
346	345	nn0cnd 12441	. . . . . . . . . . . . . 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)
347	346	subid1d 11458	. . . . . . . . . . . . 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))
348	336, 344, 347	3eqtrrd 2771	. . . . . . . . . . . 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))
349	348	oveq1d 7361	. . . . . . . . . . 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))
350	349	oveq1d 7361	. . . . . . . . . 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)))))
351	350	mpteq2dva 5184	. . . . . . . . 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))))))
352	351	oveq2d 7362	. . . . . . . 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)))))))
353	326, 352	oveq12d 7364	. . . . . . 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))))))))
354	23	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ Grp)
355	106	rabex 5277	. . . . . . . . . . 11 ⊢ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∈ V
356	355	difexi 5268	. . . . . . . . . 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
357	356	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)
358	320	fmpttd 7048	. . . . . . . . 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))
359		ovex 7379	. . . . . . . . . . . 12 ⊢ ((𝑢‘𝑋)(.g‘𝑅)((𝐹‘𝑢)(.r‘𝑅)(𝐺‘((𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∘f − 𝑢)))) ∈ V
360	359, 323	fnmpti 6624	. . . . . . . . . . 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)})
361	360	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)}))
362	361, 318, 113	fndmfifsupp 9262	. . . . . . . . 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))
363	1, 104, 6, 357, 358, 362	gsumcl 19825	. . . . . . . 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))
364	322	fmpttd 7048	. . . . . . . . 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))
365		ovex 7379	. . . . . . . . . . . 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
366	365, 324	fnmpti 6624	. . . . . . . . . . 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)})
367	366	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)}))
368	367, 318, 113	fndmfifsupp 9262	. . . . . . . . 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))
369	1, 104, 6, 357, 364, 368	gsumcl 19825	. . . . . . . 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))
370	106	rabex 5277	. . . . . . . . . 10 ⊢ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0)} ∈ V
371	370	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)
372	278	sseli 3930	. . . . . . . . . . 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})
373	372, 321	sylan2 593	. . . . . . . . . 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))
374	373	fmpttd 7048	. . . . . . . . 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))
375		eqid 2731	. . . . . . . . . . . 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)))))
376	365, 375	fnmpti 6624	. . . . . . . . . . 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)}
377	376	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)})
378	221, 279	ssfid 9153	. . . . . . . . . 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)
379	377, 378, 113	fndmfifsupp 9262	. . . . . . . . 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))
380	1, 104, 6, 371, 374, 379	gsumcl 19825	. . . . . . . 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))
381	1, 2, 354, 363, 369, 380	grpassd 18855	. . . . . . 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)))))))))
382	283, 353, 381	3eqtrd 2770	. . . . . 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)))))))))
383	219, 382	oveq12d 7364	. . . . 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))))))))))
384	103, 115, 383	3eqtr3d 2774	. . . 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))))))))))
385		psdmul.m	. . . . . 6 ⊢ · = (.r‘𝑆)
386	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐹 ∈ 𝐵)
387	41	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐺 ∈ 𝐵)
388	9, 10, 34, 385, 14, 386, 387, 19	psrmulval 21879	. . . . 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))))))
389	388	oveq2d 7362	. . . 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)))))))
390	107	difexi 5268	. . . . . . 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
391	390	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)
392		eldifi 4081	. . . . . . . 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)))})
393	38, 123	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)
394	393	adantl 481	. . . . . . . . . 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)
395	28	ad2antrr 726	. . . . . . . . . 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)
396	394, 395	ffvelcdmd 7018	. . . . . . . . 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)
397	1, 22, 25, 396, 50	mulgnn0cld 19005	. . . . . . . 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))
398	392, 397	sylan2 593	. . . . . . 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))
399	398	fmpttd 7048	. . . . . 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))
400		eqid 2731	. . . . . . . . 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)))))
401	359, 400	fnmpti 6624	. . . . . . . 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})
402	401	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}))
403		difssd 4087	. . . . . . . 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)))})
404	21, 403	ssfid 9153	. . . . . . 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)
405	402, 404, 113	fndmfifsupp 9262	. . . . . 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))
406	1, 104, 6, 391, 399, 405	gsumcl 19825	. . . . 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))
407	1, 2, 354, 369, 380	grpcld 18857	. . . . 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))
408	1, 2, 354, 406, 363, 407	grpassd 18855	. . . 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))))))))))
409	384, 389, 408	3eqtr4d 2776	. . 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)))))))))
410	409	mpteq2dva 5184	. 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))))))))))
411	9, 10, 385, 4, 8, 41	psrmulcl 21881	. . 3 ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐵)
412	9, 10, 14, 28, 411	psdval 22072	. 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)))))))
413		psdmul.p	. . . 4 ⊢ + = (+g‘𝑆)
414	23	grpmgmd 18871	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Mgm)
415	9, 10, 414, 28, 8	psdcl 22074	. . . . 5 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ 𝐵)
416	9, 10, 385, 4, 415, 41	psrmulcl 21881	. . . 4 ⊢ (𝜑 → ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) · 𝐺) ∈ 𝐵)
417	9, 10, 414, 28, 41	psdcl 22074	. . . . 5 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺) ∈ 𝐵)
418	9, 10, 385, 4, 8, 417	psrmulcl 21881	. . . 4 ⊢ (𝜑 → (𝐹 · (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺)) ∈ 𝐵)
419	9, 10, 2, 413, 416, 418	psradd 21872	. . 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))))
420	9, 1, 14, 10, 416	psrelbas 21869	. . . . 5 ⊢ (𝜑 → ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) · 𝐺):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
421	420	ffnd 6652	. . . 4 ⊢ (𝜑 → ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) · 𝐺) Fn {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
422	9, 1, 14, 10, 418	psrelbas 21869	. . . . 5 ⊢ (𝜑 → (𝐹 · (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺)):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
423	422	ffnd 6652	. . . 4 ⊢ (𝜑 → (𝐹 · (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺)) Fn {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
424	106	a1i 11	. . . 4 ⊢ (𝜑 → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V)
425		inidm 4177	. . . 4 ⊢ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∩ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
426	415	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ 𝐵)
427	9, 10, 34, 385, 14, 426, 387, 7	psrmulval 21879	. . . . 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))))))
428	355	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ∈ V)
429	4	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑅 ∈ Ring)
430		elrabi 3643	. . . . . . . . 9 ⊢ (𝑏 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
431	9, 1, 14, 10, 415	psrelbas 21869	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
432	431	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
433	432	ffvelcdmda 7017	. . . . . . . . 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))
434	430, 433	sylan2 593	. . . . . . . 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))
435	42	ad2antrr 726	. . . . . . . . 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))
436	14, 243	psrbagconcl 21862	. . . . . . . . . . 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})
437	436	adantll 714	. . . . . . . . . 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})
438		elrabi 3643	. . . . . . . . . 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})
439	437, 438	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})
440	435, 439	ffvelcdmd 7018	. . . . . . . 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))
441	1, 34, 429, 434, 440	ringcld 20176	. . . . . . 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))
442	441	fmpttd 7048	. . . . . 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))
443		ovex 7379	. . . . . . . . 9 ⊢ (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝑏)(.r‘𝑅)(𝐺‘(𝑑 ∘f − 𝑏))) ∈ V
444		eqid 2731	. . . . . . . . 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))))
445	443, 444	fnmpti 6624	. . . . . . . 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}
446	445	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})
447	446, 221, 113	fndmfifsupp 9262	. . . . . 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))
448		eqid 2731	. . . . . . 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))))
449		df-of 7610	. . . . . . . . . 10 ⊢ ∘f + = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))))
450		vex 3440	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
451	450	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑢 ∈ V)
452		ssv 3959	. . . . . . . . . . 11 ⊢ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ⊆ V
453	452	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} ⊆ V)
454		ssv 3959	. . . . . . . . . . 11 ⊢ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ⊆ V
455	454	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ⊆ V)
456	449, 451, 453, 455	elimampo 7483	. . . . . . . . 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)))))
457	456	biimpa 476	. . . . . . . 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))))
458		elrabi 3643	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → 𝑚 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
459	14	psrbagf 21853	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑚:𝐼⟶ℕ0)
460	459	ffund 6655	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → Fun 𝑚)
461	458, 460	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → Fun 𝑚)
462	461	funfnd 6512	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → 𝑚 Fn dom 𝑚)
463	462	ad2antrl 728	. . . . . . . . . . . 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)
464		velsn 4592	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ↔ 𝑛 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
465		funmpt 6519	. . . . . . . . . . . . . . . 16 ⊢ Fun (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))
466		funeq 6501	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) → (Fun 𝑛 ↔ Fun (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
467	465, 466	mpbiri 258	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) → Fun 𝑛)
468	467	funfnd 6512	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) → 𝑛 Fn dom 𝑛)
469	464, 468	sylbi 217	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} → 𝑛 Fn dom 𝑛)
470	469	ad2antll 729	. . . . . . . . . . . 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)
471		vex 3440	. . . . . . . . . . . . . 14 ⊢ 𝑚 ∈ V
472	471	dmex 7839	. . . . . . . . . . . . 13 ⊢ dom 𝑚 ∈ V
473	472	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)
474		vex 3440	. . . . . . . . . . . . . 14 ⊢ 𝑛 ∈ V
475	474	dmex 7839	. . . . . . . . . . . . 13 ⊢ dom 𝑛 ∈ V
476	475	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)
477		eqid 2731	. . . . . . . . . . . 12 ⊢ (dom 𝑚 ∩ dom 𝑛) = (dom 𝑚 ∩ dom 𝑛)
478		eqidd 2732	. . . . . . . . . . . 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))
479		eqidd 2732	. . . . . . . . . . . 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))
480	463, 470, 473, 476, 477, 478, 479	offval 7619	. . . . . . . . . . 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))))
481	480	eqeq2d 2742	. . . . . . . . . 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)))))
482		elsni 4593	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} → 𝑛 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
483	482	oveq2d 7362	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} → (𝑚 ∘f + 𝑛) = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
484	483	eqeq2d 2742	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} → (𝑢 = (𝑚 ∘f + 𝑛) ↔ 𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
485	484	ad2antll 729	. . . . . . . . . . 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)))))
486	13	ad3antrrr 730	. . . . . . . . . . . . . . . 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)
487	458, 459	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → 𝑚:𝐼⟶ℕ0)
488	487	adantl 481	. . . . . . . . . . . . . . . 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)
489	131	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)
490		nn0cn 12388	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 ∈ ℕ0 → 𝑞 ∈ ℂ)
491		nn0cn 12388	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ∈ ℕ0 → 𝑟 ∈ ℂ)
492		nn0cn 12388	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ℕ0 → 𝑠 ∈ ℂ)
493		addsubass 11367	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑞 ∈ ℂ ∧ 𝑟 ∈ ℂ ∧ 𝑠 ∈ ℂ) → ((𝑞 + 𝑟) − 𝑠) = (𝑞 + (𝑟 − 𝑠)))
494	490, 491, 492, 493	syl3an 1160	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑞 ∈ ℕ0 ∧ 𝑟 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0) → ((𝑞 + 𝑟) − 𝑠) = (𝑞 + (𝑟 − 𝑠)))
495	494	adantl 481	. . . . . . . . . . . . . . . 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)))
496	486, 488, 489, 489, 495	caofass 7650	. . . . . . . . . . . . . . 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)))))
497		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
498	56	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → if(𝑖 = 𝑋, 1, 0) ∈ ℕ0)
499	68, 76, 497, 498	fvmptd3 6952	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑖) = if(𝑖 = 𝑋, 1, 0))
500	133, 133, 13, 13, 72, 499, 499	offval 7619	. . . . . . . . . . . . . . . . 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))))
501	500	oveq2d 7362	. . . . . . . . . . . . . . . 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)))))
502	501	ad3antrrr 730	. . . . . . . . . . . . . . 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)))))
503	237	subidi 11429	. . . . . . . . . . . . . . . . . . 19 ⊢ (if(𝑖 = 𝑋, 1, 0) − if(𝑖 = 𝑋, 1, 0)) = 0
504	503	mpteq2i 5187	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ 𝐼 ↦ (if(𝑖 = 𝑋, 1, 0) − if(𝑖 = 𝑋, 1, 0))) = (𝑖 ∈ 𝐼 ↦ 0)
505		fconstmpt 5678	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 × {0}) = (𝑖 ∈ 𝐼 ↦ 0)
506	504, 505	eqtr4i 2757	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ 𝐼 ↦ (if(𝑖 = 𝑋, 1, 0) − if(𝑖 = 𝑋, 1, 0))) = (𝐼 × {0})
507	506	oveq2i 7357	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∘f + (𝑖 ∈ 𝐼 ↦ (if(𝑖 = 𝑋, 1, 0) − if(𝑖 = 𝑋, 1, 0)))) = (𝑚 ∘f + (𝐼 × {0}))
508		0zd 12477	. . . . . . . . . . . . . . . . 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)
509	490	addridd 11310	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 ∈ ℕ0 → (𝑞 + 0) = 𝑞)
510	509	adantl 481	. . . . . . . . . . . . . . . . 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)
511	486, 488, 508, 510	caofid0r 7644	. . . . . . . . . . . . . . . 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)
512	507, 511	eqtrid 2778	. . . . . . . . . . . . . . 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)
513	496, 502, 512	3eqtrd 2770	. . . . . . . . . . . . . 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)
514		simpr 484	. . . . . . . . . . . . . 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})
515	513, 514	eqeltrd 2831	. . . . . . . . . . . . 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})
516		oveq1 7353	. . . . . . . . . . . . . 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))))
517	516	eleq1d 2816	. . . . . . . . . . . . 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}))
518	515, 517	syl5ibrcom 247	. . . . . . . . . . . 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}))
519	518	adantrr 717	. . . . . . . . . . 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}))
520	485, 519	sylbid 240	. . . . . . . . . 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}))
521	481, 520	sylbird 260	. . . . . . . . 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}))
522	521	rexlimdvva 3189	. . . . . . . 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}))
523	457, 522	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})
524		simpr 484	. . . . . . . 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})
525	13	mptexd 7158	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ V)
526		elsng 4590	. . . . . . . . . . 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))))
527	525, 526	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))))
528	68, 527	mpbiri 258	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})
529	528	ad2antrr 726	. . . . . . . 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))})
530	449	mpofun 7470	. . . . . . . . 9 ⊢ Fun ∘f +
531	530	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 + )
532		xpss 5632	. . . . . . . . 9 ⊢ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) ⊆ (V × V)
533	472	inex1 5255	. . . . . . . . . . . 12 ⊢ (dom 𝑚 ∩ dom 𝑛) ∈ V
534	533	mptex 7157	. . . . . . . . . . 11 ⊢ (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))) ∈ V
535	534	rgen2w 3052	. . . . . . . . . 10 ⊢ ∀𝑚 ∈ V ∀𝑛 ∈ V (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))) ∈ V
536	449	dmmpoga 8005	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ V ∀𝑛 ∈ V (𝑜 ∈ (dom 𝑚 ∩ dom 𝑛) ↦ ((𝑚‘𝑜) + (𝑛‘𝑜))) ∈ V → dom ∘f + = (V × V))
537	535, 536	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))
538	532, 537	sseqtrrid 3978	. . . . . . . 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 + )
539	524, 529, 531, 538	elovimad 7396	. . . . . . 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))})))
540	13	ad2antrr 726	. . . . . . . . . . 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)
541		elrabi 3643	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → 𝑣 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
542	14	psrbagf 21853	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑣:𝐼⟶ℕ0)
543	541, 542	syl 17	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} → 𝑣:𝐼⟶ℕ0)
544	543	ad2antll 729	. . . . . . . . . . 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)
545	131	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)
546	494	adantl 481	. . . . . . . . . . 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)))
547	540, 544, 545, 545, 546	caofass 7650	. . . . . . . . . 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)))))
548	133	ad2antrr 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))})) ∧ 𝑣 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑})) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼)
549	78	adantl 481	. . . . . . . . . . . 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))
550	548, 548, 540, 540, 72, 549, 549	offval 7619	. . . . . . . . . . 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))))
551	550	oveq2d 7362	. . . . . . . . . 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)))))
552	506	oveq2i 7357	. . . . . . . . . . 11 ⊢ (𝑣 ∘f + (𝑖 ∈ 𝐼 ↦ (if(𝑖 = 𝑋, 1, 0) − if(𝑖 = 𝑋, 1, 0)))) = (𝑣 ∘f + (𝐼 × {0}))
553		0zd 12477	. . . . . . . . . . . 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)
554		nn0cn 12388	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ ℕ0 → 𝑝 ∈ ℂ)
555	554	addridd 11310	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℕ0 → (𝑝 + 0) = 𝑝)
556	555	adantl 481	. . . . . . . . . . . 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)
557	540, 544, 553, 556	caofid0r 7644	. . . . . . . . . . 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)
558	552, 557	eqtrid 2778	. . . . . . . . . 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)
559	547, 551, 558	3eqtrrd 2771	. . . . . . . . 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))))
560		oveq1 7353	. . . . . . . . . 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))))
561	560	eqeq2d 2742	. . . . . . . . 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)))))
562	559, 561	syl5ibrcom 247	. . . . . . . 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)))))
563	16	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . 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})
564	14	psrbagaddcl 21859	. . . . . . . . . . . . . . . . . . . . . . 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})
565	458, 563, 564	syl2an2 686	. . . . . . . . . . . . . . . . . . . . . 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})
566	14	psrbagf 21853	. . . . . . . . . . . . . . . . . . . . . 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)
567	565, 566	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)
568	567	adantrr 717	. . . . . . . . . . . . . . . . . . . 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)
569		feq1 6629	. . . . . . . . . . . . . . . . . . . 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))
570	568, 569	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . 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))
571	485, 570	sylbid 240	. . . . . . . . . . . . . . . . . 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))
572	481, 571	sylbird 260	. . . . . . . . . . . . . . . . 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))
573	572	rexlimdvva 3189	. . . . . . . . . . . . . . . 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))
574	457, 573	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)
575	574	adantrr 717	. . . . . . . . . . . . . 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)
576	575	ffvelcdmda 7017	. . . . . . . . . . . . 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)
577	576	nn0cnd 12441	. . . . . . . . . . . 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)
578	237	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)
579	577, 578	npcand 11473	. . . . . . . . . . 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))
580	579	mpteq2dva 5184	. . . . . . . . . 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)))
581	575	ffnd 6652	. . . . . . . . . . . 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)
582	581, 548, 540, 540, 72	offn 7623	. . . . . . . . . . 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)
583		eqidd 2732	. . . . . . . . . . . 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))
584	581, 548, 540, 540, 72, 583, 549	ofval 7621	. . . . . . . . . . 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)))
585	582, 548, 540, 540, 72, 584, 549	offval 7619	. . . . . . . . . 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))))
586	575	feqmptd 6890	. . . . . . . . . 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)))
587	580, 585, 586	3eqtr4rd 2777	. . . . . . . . 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))))
588		oveq1 7353	. . . . . . . . . 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))))
589	588	eqeq2d 2742	. . . . . . . . 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)))))
590	587, 589	syl5ibrcom 247	. . . . . . . 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)))))
591	562, 590	impbid 212	. . . . . . 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)))))
592	448, 523, 539, 591	f1o2d 7600	. . . . . 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})
593	1, 104, 6, 428, 442, 447, 592	gsumf1o 19826	. . . . 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)))))))
594	555	adantl 481	. . . . . . . . . . . . . . . . . . 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)
595	486, 488, 508, 594	caofid0r 7644	. . . . . . . . . . . . . . . . . 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)
596	507, 595	eqtrid 2778	. . . . . . . . . . . . . . . . 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)
597	496, 502, 596	3eqtrd 2770	. . . . . . . . . . . . . . . 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)
598	597, 514	eqeltrd 2831	. . . . . . . . . . . . . . 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})
599	598, 517	syl5ibrcom 247	. . . . . . . . . . . . . 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}))
600	599	adantrr 717	. . . . . . . . . . . . 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}))
601	485, 600	sylbid 240	. . . . . . . . . . . 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}))
602	481, 601	sylbird 260	. . . . . . . . . . 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}))
603	602	rexlimdvva 3189	. . . . . . . . . 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}))
604	457, 603	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})
605		eqidd 2732	. . . . . . . . 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)))))
606		eqidd 2732	. . . . . . . . 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)))))
607		fveq2 6822	. . . . . . . . . 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)))))
608		oveq2 7354	. . . . . . . . . . 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)))))
609	608	fveq2d 6826	. . . . . . . . . 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))))))
610	607, 609	oveq12d 7364	. . . . . . . . 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)))))))
611	604, 605, 606, 610	fmptco 7062	. . . . . . . 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))))))))
612	28	ad2antrr 726	. . . . . . . . . . . . 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	8	ad2antrr 726	. . . . . . . . . . . . 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 3643	. . . . . . . . . . . . . 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	604, 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	9, 10, 14, 612, 613, 615	psdcoef 22073	. . . . . . . . . . . 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	574	ffnd 6652	. . . . . . . . . . . . . . . . 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	131	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 6652	. . . . . . . . . . . . . . . . 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	13	ad2antrr 726	. . . . . . . . . . . . . . . . 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 ∈ V)
621		eqidd 2732	. . . . . . . . . . . . . . . . 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))
622		iftrue 4481	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, 1, 0) = 1)
623		1ex 11105	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ V
624	622, 68, 623	fvmpt 6929	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ 𝐼 → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑋) = 1)
625	624	adantl 481	. . . . . . . . . . . . . . . . 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)
626	617, 619, 620, 620, 72, 621, 625	ofval 7621	. . . . . . . . . . . . . . . 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))
627	612, 626	mpdan 687	. . . . . . . . . . . . . . 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))
628	627	oveq1d 7361	. . . . . . . . . . . . . 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))
629		nn0sscn 12383	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ0 ⊆ ℂ
630	629	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)
631	574, 630	fssd 6668	. . . . . . . . . . . . . . . 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)
632	631, 612	ffvelcdmd 7018	. . . . . . . . . . . . . . 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)
633		1cnd 11104	. . . . . . . . . . . . . . 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)
634	632, 633	npcand 11473	. . . . . . . . . . . . . 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))
635	628, 634	eqtrd 2766	. . . . . . . . . . . . 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))
636	617, 619, 620, 620, 72	offn 7623	. . . . . . . . . . . . . . 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)
637		eqidd 2732	. . . . . . . . . . . . . . . 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))
638	78	adantl 481	. . . . . . . . . . . . . . . 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))
639	617, 619, 620, 620, 72, 637, 638	ofval 7621	. . . . . . . . . . . . . . 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)))
640	574	ffvelcdmda 7017	. . . . . . . . . . . . . . . . 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)
641	640	nn0cnd 12441	. . . . . . . . . . . . . . . 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)
642	237	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)
643	641, 642	npcand 11473	. . . . . . . . . . . . . . 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))
644	620, 636, 619, 617, 639, 638, 643	offveq 7636	. . . . . . . . . . . . . 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)
645	644	fveq2d 6826	. . . . . . . . . . . . 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))
646	635, 645	oveq12d 7364	. . . . . . . . . . . 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)))
647	616, 646	eqtrd 2766	. . . . . . . . . . 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)))
648	26	ad2antlr 727	. . . . . . . . . . . . . . . . 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)
649	648	ffvelcdmda 7017	. . . . . . . . . . . . . . . 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)
650	649	nn0cnd 12441	. . . . . . . . . . . . . . 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)
651	650, 641, 642	subsub3d 11499	. . . . . . . . . . . . . 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)))
652	651	mpteq2dva 5184	. . . . . . . . . . . . 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))))
653	65	adantr 480	. . . . . . . . . . . . . 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)
654		eqidd 2732	. . . . . . . . . . . . . 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))
655	653, 636, 620, 620, 72, 654, 639	offval 7619	. . . . . . . . . . . . 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)))))
656	653, 619, 620, 620, 72	offn 7623	. . . . . . . . . . . . . 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)
657	653, 619, 620, 620, 72, 654, 638	ofval 7621	. . . . . . . . . . . . . 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)))
658	656, 617, 620, 620, 72, 657, 637	offval 7619	. . . . . . . . . . . . 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))))
659	652, 655, 658	3eqtr4d 2776	. . . . . . . . . . . 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))
660	659	fveq2d 6826	. . . . . . . . . . 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)))
661	647, 660	oveq12d 7364	. . . . . . . . . 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))))
662	4	ad2antrr 726	. . . . . . . . . . 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)
663	574, 612	ffvelcdmd 7018	. . . . . . . . . . . 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)
664	663	nn0zd 12491	. . . . . . . . . . 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)
665	36	ad2antrr 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))}))) → 𝐹:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
666		simpllr 775	. . . . . . . . . . . . . . . . . . 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})
667	16	ad3antrrr 730	. . . . . . . . . . . . . . . . . . 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})
668		simprl 770	. . . . . . . . . . . . . . . . . . 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})
669		eqid 2731	. . . . . . . . . . . . . . . . . . . 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)))}
670	14, 243, 669	psrbagleadd1 21863	. . . . . . . . . . . . . . . . . . 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)))})
671	666, 667, 668, 670	syl3anc 1373	. . . . . . . . . . . . . . . . . 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)))})
672		eleq1 2819	. . . . . . . . . . . . . . . . . 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)))}))
673	671, 672	syl5ibrcom 247	. . . . . . . . . . . . . . . . 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)))}))
674	485, 673	sylbid 240	. . . . . . . . . . . . . . . 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)))}))
675	481, 674	sylbird 260	. . . . . . . . . . . . . . 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)))}))
676	675	rexlimdvva 3189	. . . . . . . . . . . . . 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)))}))
677	457, 676	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)))})
678		elrabi 3643	. . . . . . . . . . . . 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})
679	677, 678	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})
680	665, 679	ffvelcdmd 7018	. . . . . . . . . . 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))
681	42	ad2antrr 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))}))) → 𝐺:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅))
682	19	adantr 480	. . . . . . . . . . . . . 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})
683	14, 669	psrbagconcl 21862	. . . . . . . . . . . . . 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)))})
684	682, 677, 683	syl2anc 584	. . . . . . . . . . . . 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)))})
685		elrabi 3643	. . . . . . . . . . . . 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})
686	684, 685	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})
687	681, 686	ffvelcdmd 7018	. . . . . . . . . . 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))
688	1, 22, 34	mulgass2 20225	. . . . . . . . . . 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)))))
689	662, 664, 680, 687, 688	syl13anc 1374	. . . . . . . . . 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)))))
690	661, 689	eqtrd 2766	. . . . . . . . 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)))))
691	690	mpteq2dva 5184	. . . . . . . 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))))))
692	611, 691	eqtrd 2766	. . . . . . 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))))))
693	692	oveq2d 7362	. . . . . 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)))))))
694		snex 5374	. . . . . . . . . 10 ⊢ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ∈ V
695	355, 694	xpex 7686	. . . . . . . . 9 ⊢ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) ∈ V
696	695	funimaex 6569	. . . . . . . 8 ⊢ (Fun ∘f + → ( ∘f + “ ({𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑} × {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) ∈ V)
697	530, 696	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)
698	24	ad2antrr 726	. . . . . . . 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)
699	1, 34, 662, 680, 687	ringcld 20176	. . . . . . . 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))
700	1, 22, 698, 663, 699	mulgnn0cld 19005	. . . . . . 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))
701		eqid 2731	. . . . . . . . . . 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)))))
702	359, 701	fnmpti 6624	. . . . . . . . . 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)))}
703	702	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)))})
704	703, 21, 113	fndmfifsupp 9262	. . . . . . . 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))
705	462	ad2antlr 727	. . . . . . . . . . . . . . . . 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)
706	469	adantl 481	. . . . . . . . . . . . . . . . 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)
707	472	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)
708	475	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)
709		eqidd 2732	. . . . . . . . . . . . . . . . 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))
710		eqidd 2732	. . . . . . . . . . . . . . . . 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))
711	705, 706, 707, 708, 477, 709, 710	offval 7619	. . . . . . . . . . . . . . . 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))))
712	711	eqeq2d 2742	. . . . . . . . . . . . . . 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)))))
713	712	rexbidva 3154	. . . . . . . . . . . . . 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)))))
714	16	ad2antrr 726	. . . . . . . . . . . . . . 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})
715		oveq2 7354	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) → (𝑚 ∘f + 𝑛) = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
716	715	eqeq2d 2742	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) → (𝑢 = (𝑚 ∘f + 𝑛) ↔ 𝑢 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
717	716	rexsng 4629	. . . . . . . . . . . . . . 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)))))
718	714, 717	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)))))
719	713, 718	bitr3d 281	. . . . . . . . . . . . 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)))))
720	719	rexbidva 3154	. . . . . . . . . . . 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)))))
721		breq1 5094	. . . . . . . . . . . . . . . 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)))))
722		breq1 5094	. . . . . . . . . . . . . . . . . 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))
723		fveq1 6821	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) → (𝑘‘𝑋) = ((𝑚 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋))
724	723	eqeq1d 2733	. . . . . . . . . . . . . . . . . 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))
725	722, 724	anbi12d 632	. . . . . . . . . . . . . . . . 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)))
726	725	notbid 318	. . . . . . . . . . . . . . . 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)))
727	721, 726	anbi12d 632	. . . . . . . . . . . . . . 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))))
728	458, 714, 564	syl2an2 686	. . . . . . . . . . . . . . 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})
729		simplr 768	. . . . . . . . . . . . . . . . . 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})
730		simpr 484	. . . . . . . . . . . . . . . . . 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})
731	14, 243, 44	psrbagleadd1 21863	. . . . . . . . . . . . . . . . . 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)))})
732	729, 714, 730, 731	syl3anc 1373	. . . . . . . . . . . . . . . . 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)))})
733	721	elrab 3647	. . . . . . . . . . . . . . . . . 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)))))
734	733	simprbi 496	. . . . . . . . . . . . . . . . 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))))
735	732, 734	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))))
736	28	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝑋 ∈ 𝐼)
737	487	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 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)
738	737	ffnd 6652	. . . . . . . . . . . . . . . . . . . . . 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)
739	133	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . 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)
740	13	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑚 ∈ {𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑘 ∘r ≤ 𝑑}) → 𝐼 ∈ V)
741		eqidd 2732	. . . . . . . . . . . . . . . . . . . . . 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))
742	624	adantl 481	. . . . . . . . . . . . . . . . . . . . . 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)
743	738, 739, 740, 740, 72, 741, 742	ofval 7621	. . . . . . . . . . . . . . . . . . . . 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))
744	736, 743	mpdan 687	. . . . . . . . . . . . . . . . . . . 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))
745	737, 736	ffvelcdmd 7018	. . . . . . . . . . . . . . . . . . . . 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)
746		nn0p1nn 12417	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚‘𝑋) ∈ ℕ0 → ((𝑚‘𝑋) + 1) ∈ ℕ)
747	745, 746	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)
748	744, 747	eqeltrd 2831	. . . . . . . . . . . . . . . . . . 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)
749	748	nnne0d 12172	. . . . . . . . . . . . . . . . . 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)
750	749	neneqd 2933	. . . . . . . . . . . . . . . . 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)
751	750	intnand 488	. . . . . . . . . . . . . . . 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))
752	735, 751	jca 511	. . . . . . . . . . . . . . 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)))
753	727, 728, 752	elrabd 3649	. . . . . . . . . . . . . 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))})
754		eleq1 2819	. . . . . . . . . . . . . 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))}))
755	753, 754	syl5ibrcom 247	. . . . . . . . . . . . 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))}))
756		breq1 5094	. . . . . . . . . . . . . 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))
757		elrabi 3643	. . . . . . . . . . . . . . . . 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})
758	757	adantl 481	. . . . . . . . . . . . . . . 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})
759	131	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)
760	757, 123	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)
761	760	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 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)
762	28	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . 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)
763	761, 762	ffvelcdmd 7018	. . . . . . . . . . . . . . . . . . . . . . . . 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)
764	339	notbid 318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑘 = 𝑢 → (¬ (𝑘 ∘r ≤ 𝑑 ∧ (𝑘‘𝑋) = 0) ↔ ¬ (𝑢 ∘r ≤ 𝑑 ∧ (𝑢‘𝑋) = 0)))
765	118, 764	anbi12d 632	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
766	765	elrab 3647	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
767	766	simprbi 496	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
768	767	simpld 494	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
769	768	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
770	769	adantr 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) → 𝑢 ∘r ≤ (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
771	757, 124	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)
772	771	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
773	772	adantr 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‘𝑋) = 0) → 𝑢 Fn 𝐼)
774	19	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})
775	88	ffnd 6652	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
776	774, 775	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)
777	776	adantr 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‘𝑋) = 0) → (𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) Fn 𝐼)
778	13	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
779		eqidd 2732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
780		eqidd 2732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
781	773, 777, 778, 778, 72, 779, 780	ofrfval 7620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
782	770, 781	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
783	782	r19.21bi 3224	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
784	783	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
785	65	ad3antrrr 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) → 𝑑 Fn 𝐼)
786	69	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)
787	13	ad4antr 732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
788		eqidd 2732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
789	78	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
790	785, 786, 787, 787, 72, 788, 789	ofval 7621	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
791	790	an32s 652	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
792	158	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
793	792	oveq2d 7362	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
794	27	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
795	794	ffvelcdmda 7017	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
796	795	adantr 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 ∈ 𝐼) ∧ 𝑖 ≠ 𝑋) → (𝑑‘𝑖) ∈ ℕ0)
797	796	nn0cnd 12441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
798	797	addridd 11310	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
799	791, 793, 798	3eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
800	784, 799	breqtrd 5117	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
801		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
802	27	adantr 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))}) → 𝑑:𝐼⟶ℕ0)
803	802, 762	ffvelcdmd 7018	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
804	803	nn0ge0d 12442	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
805	804	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
806	801, 805	eqbrtrd 5113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
807	806	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
808	175, 800, 807	pm2.61ne 3013	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
809	808	ralrimiva 3124	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
810	65	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
811	810	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
812		eqidd 2732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
813	773, 811, 778, 778, 72, 779, 812	ofrfval 7620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
814	809, 813	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
815	814	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . 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))
816	767	simprd 495	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
817	816	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
818		imnan 399	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑢 ∘r ≤ 𝑑 → ¬ (𝑢‘𝑋) = 0) ↔ ¬ (𝑢 ∘r ≤ 𝑑 ∧ (𝑢‘𝑋) = 0))
819	817, 818	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
820	819	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))
821	815, 820	pm2.65d 196	. . . . . . . . . . . . . . . . . . . . . . . . . 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)
822	821	neqned 2935	. . . . . . . . . . . . . . . . . . . . . . . . 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)
823	763, 822, 191	sylanbrc 583	. . . . . . . . . . . . . . . . . . . . . . . 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)
824	823	nnge1d 12170	. . . . . . . . . . . . . . . . . . . . . . 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))
825	824	adantr 480	. . . . . . . . . . . . . . . . . . . . . 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))
826	173	breq2d 5103	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑋 → (1 ≤ (𝑢‘𝑖) ↔ 1 ≤ (𝑢‘𝑋)))
827	825, 826	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . 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)))
828	827	imp 406	. . . . . . . . . . . . . . . . . . . 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))
829	761	ffvelcdmda 7017	. . . . . . . . . . . . . . . . . . . . . 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)
830	829	nn0ge0d 12442	. . . . . . . . . . . . . . . . . . . . 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))
831	830	adantr 480	. . . . . . . . . . . . . . . . . . . 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))
832	828, 831	ifpimpda 1080	. . . . . . . . . . . . . . . . . . 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)))
833		brif1 7443	. . . . . . . . . . . . . . . . . . 19 ⊢ (if(𝑖 = 𝑋, 1, 0) ≤ (𝑢‘𝑖) ↔ if-(𝑖 = 𝑋, 1 ≤ (𝑢‘𝑖), 0 ≤ (𝑢‘𝑖)))
834	832, 833	sylibr 234	. . . . . . . . . . . . . . . . . 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))
835	834	ralrimiva 3124	. . . . . . . . . . . . . . . . 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))
836	69	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)
837	13	ad2antrr 726	. . . . . . . . . . . . . . . . . 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)
838	78	adantl 481	. . . . . . . . . . . . . . . . . 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))
839		eqidd 2732	. . . . . . . . . . . . . . . . . 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))
840	836, 772, 837, 837, 72, 838, 839	ofrfval 7620	. . . . . . . . . . . . . . . . 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)))
841	835, 840	mpbird 257	. . . . . . . . . . . . . . . 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)
842	14	psrbagcon 21860	. . . . . . . . . . . . . . . 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))
843	758, 759, 841, 842	syl3anc 1373	. . . . . . . . . . . . . . 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))
844	843	simpld 494	. . . . . . . . . . . . . 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})
845		eqidd 2732	. . . . . . . . . . . . . . . . . . . . 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))
846	810, 836, 837, 837, 72, 845, 838	ofval 7621	. . . . . . . . . . . . . . . . . . . 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)))
847	772, 776, 837, 837, 72, 839, 846	ofrfval 7620	. . . . . . . . . . . . . . . . . . 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))))
848	769, 847	mpbid 232	. . . . . . . . . . . . . . . . . 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)))
849	848	r19.21bi 3224	. . . . . . . . . . . . . . . . 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)))
850	829	nn0red 12440	. . . . . . . . . . . . . . . . . 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)
851	60	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)
852	802	ffvelcdmda 7017	. . . . . . . . . . . . . . . . . . 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)
853	852	nn0red 12440	. . . . . . . . . . . . . . . . . 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)
854	850, 851, 853	lesubaddd 11711	. . . . . . . . . . . . . . . . 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))))
855	849, 854	mpbird 257	. . . . . . . . . . . . . . . 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))
856	855	ralrimiva 3124	. . . . . . . . . . . . . . 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))
857	772, 836, 837, 837, 72	offn 7623	. . . . . . . . . . . . . . . 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)
858	772, 836, 837, 837, 72, 839, 838	ofval 7621	. . . . . . . . . . . . . . . 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)))
859	857, 810, 837, 837, 72, 858, 845	ofrfval 7620	. . . . . . . . . . . . . . 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)))
860	856, 859	mpbird 257	. . . . . . . . . . . . . 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)
861	756, 844, 860	elrabd 3649	. . . . . . . . . . . . 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})
862	829	nn0cnd 12441	. . . . . . . . . . . . . . . 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)
863	237	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)
864	862, 863	npcand 11473	. . . . . . . . . . . . . . 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))
865	864	mpteq2dva 5184	. . . . . . . . . . . . . 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)))
866	857, 836, 837, 837, 72, 858, 838	offval 7619	. . . . . . . . . . . . . 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))))
867	761	feqmptd 6890	. . . . . . . . . . . . . 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)))
868	865, 866, 867	3eqtr4rd 2777	. . . . . . . . . . . . 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))))
869		oveq1 7353	. . . . . . . . . . . . . 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))))
870	869	eqeq2d 2742	. . . . . . . . . . . . 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)))))
871	755, 861, 868, 870	rspceb2dv 3581	. . . . . . . . . . . 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))}))
872	456, 720, 871	3bitrd 305	. . . . . . . . . . 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))}))
873	872	eqrdv 2729	. . . . . . . . . 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))})
874		difrab 4268	. . . . . . . . . 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))}
875	873, 874	eqtr4di 2784	. . . . . . . . 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)}))
876		difssd 4087	. . . . . . . . 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)))})
877	875, 876	eqsstrd 3969	. . . . . . . 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)))})
878	704, 877, 113	fmptssfisupp 9278	. . . . . . 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))
879		difss 4086	. . . . . . . . . 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}
880		disjdif 4422	. . . . . . . . . 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})) = ∅
881		ssdisj 4410	. . . . . . . . . 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})) = ∅)
882	879, 880, 881	mp2an 692	. . . . . . . . 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})) = ∅
883	882	ineqcomi 4161	. . . . . . . 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)})) = ∅
884	883	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)})) = ∅)
885	279, 99	psdmullem 22078	. . . . . . . 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)}))
886	875, 885	eqtr4d 2769	. . . . . . 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)})))
887	1, 104, 2, 6, 697, 700, 878, 884, 886	gsumsplit2 19839	. . . . . 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))))))))
888	693, 887	eqtrd 2766	. . . . 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))))))))
889	427, 593, 888	3eqtrd 2770	. . . 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))))))))
890	417	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺) ∈ 𝐵)
891	9, 10, 34, 385, 14, 386, 890, 7	psrmulval 21879	. . . . 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))))))
892	41	ad2antrr 726	. . . . . . . . . . 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	9, 10, 14, 285, 892, 247	psdcoef 22073	. . . . . . . . . 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	267	fveq2d 6826	. . . . . . . . . . 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 7362	. . . . . . . . . 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 2766	. . . . . . . . 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 7362	. . . . . . . 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	309	nn0zd 12491	. . . . . . . . 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, 22, 34	mulgass3 20269	. . . . . . . . 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	224, 898, 228, 271, 899	syl13anc 1374	. . . . . . . 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 2766	. . . . . . 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 5184	. . . . . 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 7362	. . . . 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, 221, 321, 275, 282	gsummptfidmsplit 19840	. . . . 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	891, 903, 904	3eqtrd 2770	. . . 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	421, 423, 424, 424, 425, 889, 905	offval 7619	. . 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	419, 906	eqtrd 2766	. 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	410, 412, 907	3eqtr4d 2776	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 206   ∧ wa 395   ∨ wo 847  if-wif 1062   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  ∃wrex 3056  {crab 3395  Vcvv 3436   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4283  ifcif 4475  {csn 4576   class class class wbr 5091   ↦ cmpt 5172   × cxp 5614  ^◡ccnv 5615  dom cdm 5616   “ cima 5619   ∘ ccom 5620  Fun wfun 6475   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ∘_f cof 7608   ∘_r cofr 7609   ↑_m cmap 8750  Fincfn 8869  ℂcc 11001  ℝcr 11002  0cc0 11003  1c1 11004   + caddc 11006   < clt 11143   ≤ cle 11144   − cmin 11341  ℕcn 12122  ℕ₀cn0 12378  ℤcz 12465  ..^cfzo 13551  Basecbs 17117  +_gcplusg 17158  ._rcmulr 17159  0_gc0g 17340   Σ_g cgsu 17341  Mndcmnd 18639  Grpcgrp 18843  ._gcmg 18977  CMndccmn 19690  Ringcrg 20149  CRingccrg 20150   mPwSer cmps 21839   mPSDer cpsd 22043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-iin 4944  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-se 5570  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-of 7610  df-ofr 7611  df-om 7797  df-1st 7921  df-2nd 7922  df-supp 8091  df-tpos 8156  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-er 8622  df-map 8752  df-pm 8753  df-ixp 8822  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-fsupp 9246  df-oi 9396  df-card 9829  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-nn 12123  df-2 12185  df-3 12186  df-4 12187  df-5 12188  df-6 12189  df-7 12190  df-8 12191  df-9 12192  df-n0 12379  df-z 12466  df-uz 12730  df-fz 13405  df-fzo 13552  df-seq 13906  df-hash 14235  df-struct 17055  df-sets 17072  df-slot 17090  df-ndx 17102  df-base 17118  df-ress 17139  df-plusg 17171  df-mulr 17172  df-sca 17174  df-vsca 17175  df-tset 17177  df-0g 17342  df-gsum 17343  df-mre 17485  df-mrc 17486  df-acs 17488  df-mgm 18545  df-sgrp 18624  df-mnd 18640  df-mhm 18688  df-submnd 18689  df-grp 18846  df-minusg 18847  df-mulg 18978  df-ghm 19123  df-cntz 19227  df-cmn 19692  df-abl 19693  df-mgp 20057  df-rng 20069  df-ur 20098  df-ring 20151  df-cring 20152  df-oppr 20253  df-psr 21844  df-psd 22069

This theorem is referenced by:  psd1  22080  psdpw  22083