ovnsubaddlem1 - Mathbox for Glauco Siliprandi

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Theorem ovnsubaddlem1 45221

Description: The Lebesgue outer measure is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)

**Hypotheses**
Ref	Expression
ovnsubaddlem1.x	⊢ (𝜑 → 𝑋 ∈ Fin)
ovnsubaddlem1.n0	⊢ (𝜑 → 𝑋 ≠ ∅)
ovnsubaddlem1.a	⊢ (𝜑 → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
ovnsubaddlem1.e	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
ovnsubaddlem1.z	⊢ 𝑍 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))})
ovnsubaddlem1.c	⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ {ℎ ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)})
ovnsubaddlem1.l	⊢ 𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑_m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))
ovnsubaddlem1.d	⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ (𝑒 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +_𝑒 𝑒)}))
ovnsubaddlem1.i	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
ovnsubaddlem1.f	⊢ (𝜑 → 𝐹:ℕ–1-1-onto→(ℕ × ℕ))
ovnsubaddlem1.g	⊢ 𝐺 = (𝑚 ∈ ℕ ↦ ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))

**Assertion**
Ref	Expression
ovnsubaddlem1	⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸))

Distinct variable groups: 𝐴,𝑎,𝑒,𝑖,𝑛 𝐴,ℎ,𝑎,𝑛 𝑧,𝐴,𝑎,𝑖,𝑛 𝐶,𝑎,𝑒,𝑖 𝐷,𝑛 𝑒,𝐸,𝑖,𝑛 𝐹,𝑎,𝑒,𝑖,𝑗,𝑚,𝑛 ℎ,𝐹,𝑘,𝑎,𝑗,𝑚,𝑛 𝑖,𝑘,𝐺,𝑗,𝑚,𝑛 ℎ,𝐼,𝑗,𝑘,𝑚,𝑛 𝑖,𝐼 𝐿,𝑎,𝑒,𝑖,𝑗,𝑚,𝑛 𝑋,𝑎,𝑒,𝑖,𝑗,𝑚,𝑛 ℎ,𝑋,𝑘 𝑧,𝑋,𝑗,𝑘 𝜑,𝑎,𝑒,𝑖,𝑗,𝑚,𝑛 𝜑,𝑘 Allowed substitution hints: 𝜑(𝑧,ℎ) 𝐴(𝑗,𝑘,𝑚) 𝐶(𝑧,ℎ,𝑗,𝑘,𝑚,𝑛) 𝐷(𝑧,𝑒,ℎ,𝑖,𝑗,𝑘,𝑚,𝑎) 𝐸(𝑧,ℎ,𝑗,𝑘,𝑚,𝑎) 𝐹(𝑧) 𝐺(𝑧,𝑒,ℎ,𝑎) 𝐼(𝑧,𝑒,𝑎) 𝐿(𝑧,ℎ,𝑘) 𝑍(𝑧,𝑒,ℎ,𝑖,𝑗,𝑘,𝑚,𝑛,𝑎)

Proof of Theorem ovnsubaddlem1 Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovnsubaddlem1.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin)
2		ovnsubaddlem1.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
3	2	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
4		simpr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
5	3, 4	ffvelcdmd 7083	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ 𝒫 (ℝ ↑_m 𝑋))
6		elpwi 4608	. . . . . . 7 ⊢ ((𝐴‘𝑛) ∈ 𝒫 (ℝ ↑_m 𝑋) → (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋))
7	5, 6	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋))
8	7	ralrimiva 3147	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋))
9		iunss 5047	. . . . 5 ⊢ (∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋) ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋))
10	8, 9	sylibr 233	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋))
11	1, 10	ovnxrcl 45220	. . 3 ⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ∈ ℝ^)
12		nfv 1918	. . . 4 ⊢ Ⅎ𝑚𝜑
13		nnex 12214	. . . . 5 ⊢ ℕ ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝜑 → ℕ ∈ V)
15		icossicc 13409	. . . . 5 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
16		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑚 ∈ ℕ)
17		simpl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝜑)
18	17, 1	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 ∈ Fin)
19		ovnsubaddlem1.l	. . . . . 6 ⊢ 𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑_m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))
20		ovnsubaddlem1.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:ℕ–1-1-onto→(ℕ × ℕ))
21		f1of 6830	. . . . . . . . . . . 12 ⊢ (𝐹:ℕ–1-1-onto→(ℕ × ℕ) → 𝐹:ℕ⟶(ℕ × ℕ))
22	20, 21	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℕ⟶(ℕ × ℕ))
23	22	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹:ℕ⟶(ℕ × ℕ))
24		simpr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
25	23, 24	ffvelcdmd 7083	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∈ (ℕ × ℕ))
26		xp1st 8002	. . . . . . . . 9 ⊢ ((𝐹‘𝑚) ∈ (ℕ × ℕ) → (1^st ‘(𝐹‘𝑚)) ∈ ℕ)
27	25, 26	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1^st ‘(𝐹‘𝑚)) ∈ ℕ)
28		xp2nd 8003	. . . . . . . . 9 ⊢ ((𝐹‘𝑚) ∈ (ℕ × ℕ) → (2^nd ‘(𝐹‘𝑚)) ∈ ℕ)
29	25, 28	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2^nd ‘(𝐹‘𝑚)) ∈ ℕ)
30		fvex 6901	. . . . . . . . 9 ⊢ (2^nd ‘(𝐹‘𝑚)) ∈ V
31		eleq1 2822	. . . . . . . . . . 11 ⊢ (𝑗 = (2^nd ‘(𝐹‘𝑚)) → (𝑗 ∈ ℕ ↔ (2^nd ‘(𝐹‘𝑚)) ∈ ℕ))
32	31	3anbi3d 1443	. . . . . . . . . 10 ⊢ (𝑗 = (2^nd ‘(𝐹‘𝑚)) → ((𝜑 ∧ (1^st ‘(𝐹‘𝑚)) ∈ ℕ ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ (1^st ‘(𝐹‘𝑚)) ∈ ℕ ∧ (2^nd ‘(𝐹‘𝑚)) ∈ ℕ)))
33		fveq2 6888	. . . . . . . . . . 11 ⊢ (𝑗 = (2^nd ‘(𝐹‘𝑚)) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗) = ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))
34	33	feq1d 6699	. . . . . . . . . 10 ⊢ (𝑗 = (2^nd ‘(𝐹‘𝑚)) → (((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗):𝑋⟶(ℝ × ℝ) ↔ ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))):𝑋⟶(ℝ × ℝ)))
35	32, 34	imbi12d 345	. . . . . . . . 9 ⊢ (𝑗 = (2^nd ‘(𝐹‘𝑚)) → (((𝜑 ∧ (1^st ‘(𝐹‘𝑚)) ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗):𝑋⟶(ℝ × ℝ)) ↔ ((𝜑 ∧ (1^st ‘(𝐹‘𝑚)) ∈ ℕ ∧ (2^nd ‘(𝐹‘𝑚)) ∈ ℕ) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))):𝑋⟶(ℝ × ℝ))))
36		fvex 6901	. . . . . . . . . 10 ⊢ (1^st ‘(𝐹‘𝑚)) ∈ V
37		eleq1 2822	. . . . . . . . . . . 12 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → (𝑛 ∈ ℕ ↔ (1^st ‘(𝐹‘𝑚)) ∈ ℕ))
38	37	3anbi2d 1442	. . . . . . . . . . 11 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ (1^st ‘(𝐹‘𝑚)) ∈ ℕ ∧ 𝑗 ∈ ℕ)))
39		fveq2 6888	. . . . . . . . . . . . 13 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → (𝐼‘𝑛) = (𝐼‘(1^st ‘(𝐹‘𝑚))))
40	39	fveq1d 6890	. . . . . . . . . . . 12 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → ((𝐼‘𝑛)‘𝑗) = ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗))
41	40	feq1d 6699	. . . . . . . . . . 11 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → (((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ) ↔ ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗):𝑋⟶(ℝ × ℝ)))
42	38, 41	imbi12d 345	. . . . . . . . . 10 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ)) ↔ ((𝜑 ∧ (1^st ‘(𝐹‘𝑚)) ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗):𝑋⟶(ℝ × ℝ))))
43		ovnsubaddlem1.c	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ {ℎ ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)})
44		sseq1 4006	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (𝐴‘𝑛) → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ↔ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)))
45	44	rabbidv 3441	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (𝐴‘𝑛) → {ℎ ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)} = {ℎ ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)})
46		ovex 7437	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∈ V
47	46	rabex 5331	. . . . . . . . . . . . . . . . . . . 20 ⊢ {ℎ ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)} ∈ V
48	47	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {ℎ ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)} ∈ V)
49	43, 45, 5, 48	fvmptd3 7017	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘(𝐴‘𝑛)) = {ℎ ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)})
50		ssrab2 4076	. . . . . . . . . . . . . . . . . . 19 ⊢ {ℎ ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)} ⊆ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)
51	50	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {ℎ ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)} ⊆ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ))
52	49, 51	eqsstrd 4019	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘(𝐴‘𝑛)) ⊆ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ))
53		ovnsubaddlem1.d	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ (𝑒 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +_𝑒 𝑒)}))
54		fveq2 6888	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (𝐴‘𝑛) → (𝐶‘𝑎) = (𝐶‘(𝐴‘𝑛)))
55	54	eleq2d 2820	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (𝐴‘𝑛) → (𝑖 ∈ (𝐶‘𝑎) ↔ 𝑖 ∈ (𝐶‘(𝐴‘𝑛))))
56		fveq2 6888	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = (𝐴‘𝑛) → ((voln‘𝑋)‘𝑎) = ((voln‘𝑋)‘(𝐴‘𝑛)))
57	56	oveq1d 7419	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (𝐴‘𝑛) → (((voln‘𝑋)‘𝑎) +_𝑒 𝑒) = (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 𝑒))
58	57	breq2d 5159	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (𝐴‘𝑛) → ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘𝑎) +_𝑒 𝑒) ↔ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 𝑒)))
59	55, 58	anbi12d 632	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = (𝐴‘𝑛) → ((𝑖 ∈ (𝐶‘𝑎) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘𝑎) +_𝑒 𝑒)) ↔ (𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 𝑒))))
60	59	rabbidva2 3435	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = (𝐴‘𝑛) → {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘𝑎) +_𝑒 𝑒)} = {𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 𝑒)})
61	60	mpteq2dv 5249	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (𝐴‘𝑛) → (𝑒 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘𝑎) +_𝑒 𝑒)}) = (𝑒 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 𝑒)}))
62		rpex 43991	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℝ⁺ ∈ V
63	62	mptex 7220	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑒 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 𝑒)}) ∈ V
64	63	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑒 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 𝑒)}) ∈ V)
65	53, 61, 5, 64	fvmptd3 7017	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘(𝐴‘𝑛)) = (𝑒 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 𝑒)}))
66		oveq2 7412	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 = (𝐸 / (2↑𝑛)) → (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 𝑒) = (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))))
67	66	breq2d 5159	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑒 = (𝐸 / (2↑𝑛)) → ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 𝑒) ↔ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛)))))
68	67	rabbidv 3441	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑒 = (𝐸 / (2↑𝑛)) → {𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 𝑒)} = {𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛)))})
69	68	adantl 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑒 = (𝐸 / (2↑𝑛))) → {𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 𝑒)} = {𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛)))})
70		ovnsubaddlem1.e	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
71	70	adantr 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ⁺)
72		2nn 12281	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 ∈ ℕ
73	72	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℕ)
74		nnnn0 12475	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
75	73, 74	nnexpcld 14204	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
76	75	nnrpd 13010	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ⁺)
77	76	adantl 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ⁺)
78	71, 77	rpdivcld 13029	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ ℝ⁺)
79		fvex 6901	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐶‘(𝐴‘𝑛)) ∈ V
80	79	rabex 5331	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛)))} ∈ V
81	80	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛)))} ∈ V)
82	65, 69, 78, 81	fvmptd 7001	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) = {𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛)))})
83		ssrab2 4076	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛)))} ⊆ (𝐶‘(𝐴‘𝑛))
84	83	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛)))} ⊆ (𝐶‘(𝐴‘𝑛)))
85	82, 84	eqsstrd 4019	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ⊆ (𝐶‘(𝐴‘𝑛)))
86		ovnsubaddlem1.i	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
87	85, 86	sseldd 3982	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛)))
88	52, 87	sseldd 3982	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ))
89		elmapfn 8855	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼‘𝑛) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) → (𝐼‘𝑛) Fn ℕ)
90	88, 89	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) Fn ℕ)
91		elmapi 8839	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼‘𝑛) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) → (𝐼‘𝑛):ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
92	88, 91	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛):ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
93	92	ffvelcdmda 7082	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑_m 𝑋))
94	93	ralrimiva 3147	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑗 ∈ ℕ ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑_m 𝑋))
95	90, 94	jca 513	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐼‘𝑛) Fn ℕ ∧ ∀𝑗 ∈ ℕ ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑_m 𝑋)))
96	95	3adant3 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛) Fn ℕ ∧ ∀𝑗 ∈ ℕ ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑_m 𝑋)))
97		ffnfv 7113	. . . . . . . . . . . . 13 ⊢ ((𝐼‘𝑛):ℕ⟶((ℝ × ℝ) ↑_m 𝑋) ↔ ((𝐼‘𝑛) Fn ℕ ∧ ∀𝑗 ∈ ℕ ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑_m 𝑋)))
98	96, 97	sylibr 233	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑛):ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
99		simp3 1139	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
100	98, 99	ffvelcdmd 7083	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑_m 𝑋))
101		elmapi 8839	. . . . . . . . . . 11 ⊢ (((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑_m 𝑋) → ((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ))
102	100, 101	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ))
103	36, 42, 102	vtocl 3549	. . . . . . . . 9 ⊢ ((𝜑 ∧ (1^st ‘(𝐹‘𝑚)) ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗):𝑋⟶(ℝ × ℝ))
104	30, 35, 103	vtocl 3549	. . . . . . . 8 ⊢ ((𝜑 ∧ (1^st ‘(𝐹‘𝑚)) ∈ ℕ ∧ (2^nd ‘(𝐹‘𝑚)) ∈ ℕ) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))):𝑋⟶(ℝ × ℝ))
105	17, 27, 29, 104	syl3anc 1372	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))):𝑋⟶(ℝ × ℝ))
106		id 22	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ)
107		fvex 6901	. . . . . . . . . . 11 ⊢ ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))) ∈ V
108	107	a1i 11	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))) ∈ V)
109		ovnsubaddlem1.g	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))
110	109	fvmpt2 7005	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))) ∈ V) → (𝐺‘𝑚) = ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))
111	106, 108, 110	syl2anc 585	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (𝐺‘𝑚) = ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))
112	111	adantl 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) = ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))
113	112	feq1d 6699	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐺‘𝑚):𝑋⟶(ℝ × ℝ) ↔ ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))):𝑋⟶(ℝ × ℝ)))
114	105, 113	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚):𝑋⟶(ℝ × ℝ))
115	16, 18, 19, 114	hoiprodcl2 45206	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐿‘(𝐺‘𝑚)) ∈ (0[,)+∞))
116	15, 115	sselid 3979	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐿‘(𝐺‘𝑚)) ∈ (0[,]+∞))
117	12, 14, 116	sge0xrclmpt 45079	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))) ∈ ℝ^*)
118		nfv 1918	. . . 4 ⊢ Ⅎ𝑛𝜑
119		0xr 11257	. . . . . 6 ⊢ 0 ∈ ℝ^*
120	119	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ^*)
121		pnfxr 11264	. . . . . 6 ⊢ +∞ ∈ ℝ^*
122	121	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ^*)
123	1	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
124		ovnsubaddlem1.z	. . . . . . . . 9 ⊢ 𝑍 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))})
125	123, 7, 124	ovnval2b 45203	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln‘𝑋)‘(𝐴‘𝑛)) = if(𝑋 = ∅, 0, inf((𝑍‘(𝐴‘𝑛)), ℝ^, < )))
126		ovnsubaddlem1.n0	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ≠ ∅)
127	126	neneqd 2946	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 = ∅)
128	127	iffalsed 4538	. . . . . . . . 9 ⊢ (𝜑 → if(𝑋 = ∅, 0, inf((𝑍‘(𝐴‘𝑛)), ℝ^, < )) = inf((𝑍‘(𝐴‘𝑛)), ℝ^, < ))
129	128	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(𝑋 = ∅, 0, inf((𝑍‘(𝐴‘𝑛)), ℝ^, < )) = inf((𝑍‘(𝐴‘𝑛)), ℝ^, < ))
130	125, 129	eqtrd 2773	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln‘𝑋)‘(𝐴‘𝑛)) = inf((𝑍‘(𝐴‘𝑛)), ℝ^, < ))
131		sseq1 4006	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝐴‘𝑛) → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ↔ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)))
132	131	anbi1d 631	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐴‘𝑛) → ((𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) ↔ ((𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
133	132	rexbidv 3179	. . . . . . . . . . 11 ⊢ (𝑎 = (𝐴‘𝑛) → (∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) ↔ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)((𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
134	133	rabbidv 3441	. . . . . . . . . 10 ⊢ (𝑎 = (𝐴‘𝑛) → {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} = {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)((𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))})
135		xrex 12967	. . . . . . . . . . . 12 ⊢ ℝ^* ∈ V
136	135	rabex 5331	. . . . . . . . . . 11 ⊢ {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)((𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} ∈ V
137	136	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)((𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} ∈ V)
138	124, 134, 5, 137	fvmptd3 7017	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑍‘(𝐴‘𝑛)) = {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)((𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))})
139		ssrab2 4076	. . . . . . . . . 10 ⊢ {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)((𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} ⊆ ℝ^*
140	139	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)((𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} ⊆ ℝ^*)
141	138, 140	eqsstrd 4019	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑍‘(𝐴‘𝑛)) ⊆ ℝ^*)
142		infxrcl 13308	. . . . . . . 8 ⊢ ((𝑍‘(𝐴‘𝑛)) ⊆ ℝ^* → inf((𝑍‘(𝐴‘𝑛)), ℝ^, < ) ∈ ℝ^)
143	141, 142	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → inf((𝑍‘(𝐴‘𝑛)), ℝ^, < ) ∈ ℝ^)
144	130, 143	eqeltrd 2834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln‘𝑋)‘(𝐴‘𝑛)) ∈ ℝ^)
145	70	rpred 13012	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℝ)
146	145	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ)
147		2re 12282	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
148	147	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ)
149	148, 74	reexpcld 14124	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ)
150	149	adantl 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ)
151	148	recnd 11238	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℂ)
152		2ne0 12312	. . . . . . . . . . 11 ⊢ 2 ≠ 0
153	152	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ≠ 0)
154		nnz 12575	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
155	151, 153, 154	expne0d 14113	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ≠ 0)
156	155	adantl 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ≠ 0)
157	146, 150, 156	redivcld 12038	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ ℝ)
158	157	rexrd 11260	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ ℝ^*)
159	144, 158	xaddcld 13276	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))) ∈ ℝ^)
160	123, 7	ovncl 45218	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln*‘𝑋)‘(𝐴‘𝑛)) ∈ (0[,]+∞))
161		xrge0ge0 43992	. . . . . . 7 ⊢ (((voln‘𝑋)‘(𝐴‘𝑛)) ∈ (0[,]+∞) → 0 ≤ ((voln‘𝑋)‘(𝐴‘𝑛)))
162	160, 161	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ ((voln*‘𝑋)‘(𝐴‘𝑛)))
163		0red 11213	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ)
164	78	rpgt0d 13015	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 < (𝐸 / (2↑𝑛)))
165	163, 157, 164	ltled 11358	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝐸 / (2↑𝑛)))
166	157	ltpnfd 13097	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) < +∞)
167	158, 122, 166	xrltled 13125	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ≤ +∞)
168	120, 122, 158, 165, 167	eliccxrd 44175	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ (0[,]+∞))
169	144, 168	xadd0ge 43965	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln‘𝑋)‘(𝐴‘𝑛)) ≤ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))))
170	120, 144, 159, 162, 169	xrletrd 13137	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))))
171		pnfge 13106	. . . . . 6 ⊢ ((((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))) ∈ ℝ^ → (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))) ≤ +∞)
172	159, 171	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))) ≤ +∞)
173	120, 122, 159, 170, 172	eliccxrd 44175	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))) ∈ (0[,]+∞))
174	118, 14, 173	sge0xrclmpt 45079	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))))) ∈ ℝ^)
175		sseq1 4006	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝐴‘(1^st ‘(𝐹‘𝑚))) → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ↔ (𝐴‘(1^st ‘(𝐹‘𝑚))) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)))
176	175	rabbidv 3441	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐴‘(1^st ‘(𝐹‘𝑚))) → {ℎ ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)} = {ℎ ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ (𝐴‘(1^st ‘(𝐹‘𝑚))) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)})
177	2	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
178	177, 27	ffvelcdmd 7083	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐴‘(1^st ‘(𝐹‘𝑚))) ∈ 𝒫 (ℝ ↑_m 𝑋))
179	46	rabex 5331	. . . . . . . . . . . . 13 ⊢ {ℎ ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ (𝐴‘(1^st ‘(𝐹‘𝑚))) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)} ∈ V
180	179	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → {ℎ ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ (𝐴‘(1^st ‘(𝐹‘𝑚))) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)} ∈ V)
181	43, 176, 178, 180	fvmptd3 7017	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) = {ℎ ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ (𝐴‘(1^st ‘(𝐹‘𝑚))) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)})
182		ssrab2 4076	. . . . . . . . . . . 12 ⊢ {ℎ ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ (𝐴‘(1^st ‘(𝐹‘𝑚))) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)} ⊆ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)
183	182	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → {ℎ ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ (𝐴‘(1^st ‘(𝐹‘𝑚))) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)} ⊆ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ))
184	181, 183	eqsstrd 4019	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ⊆ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ))
185		fveq2 6888	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (𝐴‘(1^st ‘(𝐹‘𝑚))) → (𝐶‘𝑎) = (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))))
186	185	eleq2d 2820	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝐴‘(1^st ‘(𝐹‘𝑚))) → (𝑖 ∈ (𝐶‘𝑎) ↔ 𝑖 ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚))))))
187		fveq2 6888	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (𝐴‘(1^st ‘(𝐹‘𝑚))) → ((voln‘𝑋)‘𝑎) = ((voln‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))))
188	187	oveq1d 7419	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (𝐴‘(1^st ‘(𝐹‘𝑚))) → (((voln‘𝑋)‘𝑎) +_𝑒 𝑒) = (((voln‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 𝑒))
189	188	breq2d 5159	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝐴‘(1^st ‘(𝐹‘𝑚))) → ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘𝑎) +_𝑒 𝑒) ↔ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 𝑒)))
190	186, 189	anbi12d 632	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = (𝐴‘(1^st ‘(𝐹‘𝑚))) → ((𝑖 ∈ (𝐶‘𝑎) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘𝑎) +_𝑒 𝑒)) ↔ (𝑖 ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 𝑒))))
191	190	rabbidva2 3435	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = (𝐴‘(1^st ‘(𝐹‘𝑚))) → {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘𝑎) +_𝑒 𝑒)} = {𝑖 ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 𝑒)})
192	191	mpteq2dv 5249	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝐴‘(1^st ‘(𝐹‘𝑚))) → (𝑒 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘𝑎) +_𝑒 𝑒)}) = (𝑒 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 𝑒)}))
193	62	mptex 7220	. . . . . . . . . . . . . . 15 ⊢ (𝑒 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 𝑒)}) ∈ V
194	193	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑒 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 𝑒)}) ∈ V)
195	53, 192, 178, 194	fvmptd3 7017	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐷‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) = (𝑒 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 𝑒)}))
196		oveq2 7412	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = (𝐸 / (2↑(1^st ‘(𝐹‘𝑚)))) → (((voln‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 𝑒) = (((voln‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 (𝐸 / (2↑(1^st ‘(𝐹‘𝑚))))))
197	196	breq2d 5159	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = (𝐸 / (2↑(1^st ‘(𝐹‘𝑚)))) → ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 𝑒) ↔ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 (𝐸 / (2↑(1^st ‘(𝐹‘𝑚)))))))
198	197	rabbidv 3441	. . . . . . . . . . . . . 14 ⊢ (𝑒 = (𝐸 / (2↑(1^st ‘(𝐹‘𝑚)))) → {𝑖 ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 𝑒)} = {𝑖 ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 (𝐸 / (2↑(1^st ‘(𝐹‘𝑚)))))})
199	198	adantl 483	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑒 = (𝐸 / (2↑(1^st ‘(𝐹‘𝑚))))) → {𝑖 ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 𝑒)} = {𝑖 ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 (𝐸 / (2↑(1^st ‘(𝐹‘𝑚)))))})
200	17, 70	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐸 ∈ ℝ⁺)
201		2rp 12975	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ⁺
202	201	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 2 ∈ ℝ⁺)
203	27	nnzd 12581	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1^st ‘(𝐹‘𝑚)) ∈ ℤ)
204	202, 203	rpexpcld 14206	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2↑(1^st ‘(𝐹‘𝑚))) ∈ ℝ⁺)
205	200, 204	rpdivcld 13029	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐸 / (2↑(1^st ‘(𝐹‘𝑚)))) ∈ ℝ⁺)
206		fvex 6901	. . . . . . . . . . . . . . 15 ⊢ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ∈ V
207	206	rabex 5331	. . . . . . . . . . . . . 14 ⊢ {𝑖 ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 (𝐸 / (2↑(1^st ‘(𝐹‘𝑚)))))} ∈ V
208	207	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → {𝑖 ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 (𝐸 / (2↑(1^st ‘(𝐹‘𝑚)))))} ∈ V)
209	195, 199, 205, 208	fvmptd 7001	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐷‘(𝐴‘(1^st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1^st ‘(𝐹‘𝑚))))) = {𝑖 ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 (𝐸 / (2↑(1^st ‘(𝐹‘𝑚)))))})
210		ssrab2 4076	. . . . . . . . . . . . 13 ⊢ {𝑖 ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 (𝐸 / (2↑(1^st ‘(𝐹‘𝑚)))))} ⊆ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚))))
211	210	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → {𝑖 ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 (𝐸 / (2↑(1^st ‘(𝐹‘𝑚)))))} ⊆ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))))
212	209, 211	eqsstrd 4019	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐷‘(𝐴‘(1^st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1^st ‘(𝐹‘𝑚))))) ⊆ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))))
213	37	anbi2d 630	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → ((𝜑 ∧ 𝑛 ∈ ℕ) ↔ (𝜑 ∧ (1^st ‘(𝐹‘𝑚)) ∈ ℕ)))
214		2fveq3 6893	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → (𝐷‘(𝐴‘𝑛)) = (𝐷‘(𝐴‘(1^st ‘(𝐹‘𝑚)))))
215		oveq2 7412	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → (2↑𝑛) = (2↑(1^st ‘(𝐹‘𝑚))))
216	215	oveq2d 7420	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → (𝐸 / (2↑𝑛)) = (𝐸 / (2↑(1^st ‘(𝐹‘𝑚)))))
217	214, 216	fveq12d 6895	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) = ((𝐷‘(𝐴‘(1^st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1^st ‘(𝐹‘𝑚))))))
218	39, 217	eleq12d 2828	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → ((𝐼‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ↔ (𝐼‘(1^st ‘(𝐹‘𝑚))) ∈ ((𝐷‘(𝐴‘(1^st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1^st ‘(𝐹‘𝑚)))))))
219	213, 218	imbi12d 345	. . . . . . . . . . . . 13 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → (((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) ↔ ((𝜑 ∧ (1^st ‘(𝐹‘𝑚)) ∈ ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))) ∈ ((𝐷‘(𝐴‘(1^st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1^st ‘(𝐹‘𝑚))))))))
220	36, 219, 86	vtocl 3549	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (1^st ‘(𝐹‘𝑚)) ∈ ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))) ∈ ((𝐷‘(𝐴‘(1^st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1^st ‘(𝐹‘𝑚))))))
221	17, 27, 220	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))) ∈ ((𝐷‘(𝐴‘(1^st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1^st ‘(𝐹‘𝑚))))))
222	212, 221	sseldd 3982	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))) ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))))
223	184, 222	sseldd 3982	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ))
224		elmapfn 8855	. . . . . . . . 9 ⊢ ((𝐼‘(1^st ‘(𝐹‘𝑚))) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))) Fn ℕ)
225	223, 224	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))) Fn ℕ)
226		elmapi 8839	. . . . . . . . . . 11 ⊢ ((𝐼‘(1^st ‘(𝐹‘𝑚))) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))):ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
227	223, 226	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))):ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
228	227	ffvelcdmda 7082	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗) ∈ ((ℝ × ℝ) ↑_m 𝑋))
229	228	ralrimiva 3147	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑗 ∈ ℕ ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗) ∈ ((ℝ × ℝ) ↑_m 𝑋))
230	225, 229	jca 513	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐼‘(1^st ‘(𝐹‘𝑚))) Fn ℕ ∧ ∀𝑗 ∈ ℕ ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗) ∈ ((ℝ × ℝ) ↑_m 𝑋)))
231		ffnfv 7113	. . . . . . 7 ⊢ ((𝐼‘(1^st ‘(𝐹‘𝑚))):ℕ⟶((ℝ × ℝ) ↑_m 𝑋) ↔ ((𝐼‘(1^st ‘(𝐹‘𝑚))) Fn ℕ ∧ ∀𝑗 ∈ ℕ ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗) ∈ ((ℝ × ℝ) ↑_m 𝑋)))
232	230, 231	sylibr 233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))):ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
233	232, 29	ffvelcdmd 7083	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))) ∈ ((ℝ × ℝ) ↑_m 𝑋))
234	233, 109	fmptd 7109	. . . 4 ⊢ (𝜑 → 𝐺:ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
235		simpl 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝜑)
236	86, 82	eleqtrd 2836	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ {𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛)))})
237	83, 236	sselid 3979	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛)))
238		simp3 1139	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛))) → (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛)))
239	49	3adant3 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛))) → (𝐶‘(𝐴‘𝑛)) = {ℎ ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)})
240	238, 239	eleqtrd 2836	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛))) → (𝐼‘𝑛) ∈ {ℎ ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)})
241		fveq1 6887	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = (𝐼‘𝑛) → (ℎ‘𝑗) = ((𝐼‘𝑛)‘𝑗))
242	241	coeq2d 5860	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝐼‘𝑛) → ([,) ∘ (ℎ‘𝑗)) = ([,) ∘ ((𝐼‘𝑛)‘𝑗)))
243	242	fveq1d 6890	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝐼‘𝑛) → (([,) ∘ (ℎ‘𝑗))‘𝑘) = (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
244	243	ixpeq2dv 8903	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝐼‘𝑛) → X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
245	244	iuneq2d 5025	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝐼‘𝑛) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
246	245	sseq2d 4013	. . . . . . . . . . . 12 ⊢ (ℎ = (𝐼‘𝑛) → ((𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ↔ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘)))
247	246	elrab 3682	. . . . . . . . . . 11 ⊢ ((𝐼‘𝑛) ∈ {ℎ ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)} ↔ ((𝐼‘𝑛) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘)))
248	240, 247	sylib 217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛))) → ((𝐼‘𝑛) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘)))
249	248	simprd 497	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛))) → (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
250	235, 4, 237, 249	syl3anc 1372	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
251		f1ofo 6837	. . . . . . . . . . . . . 14 ⊢ (𝐹:ℕ–1-1-onto→(ℕ × ℕ) → 𝐹:ℕ–onto→(ℕ × ℕ))
252	20, 251	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℕ–onto→(ℕ × ℕ))
253	252	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → 𝐹:ℕ–onto→(ℕ × ℕ))
254		opelxpi 5712	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ))
255	4, 254	sylan 581	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ))
256		foelcdmi 6950	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ–onto→(ℕ × ℕ) ∧ ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ)) → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩)
257	253, 255, 256	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩)
258		nfv 1918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ)
259		nfre1 3283	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚∃𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘)
260		simpl 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → 𝑚 ∈ ℕ)
261		fveq2 6888	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (1^st ‘(𝐹‘𝑚)) = (1^st ‘⟨𝑛, 𝑗⟩))
262		op1stg 7982	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ V ∧ 𝑗 ∈ V) → (1^st ‘⟨𝑛, 𝑗⟩) = 𝑛)
263	262	el2v 3483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1^st ‘⟨𝑛, 𝑗⟩) = 𝑛
264	263	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (1^st ‘⟨𝑛, 𝑗⟩) = 𝑛)
265	261, 264	eqtrd 2773	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (1^st ‘(𝐹‘𝑚)) = 𝑛)
266	265	adantl 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → (1^st ‘(𝐹‘𝑚)) = 𝑛)
267	260, 266	jca 513	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → (𝑚 ∈ ℕ ∧ (1^st ‘(𝐹‘𝑚)) = 𝑛))
268		2fveq3 6893	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑚 → (1^st ‘(𝐹‘𝑖)) = (1^st ‘(𝐹‘𝑚)))
269	268	eqeq1d 2735	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑚 → ((1^st ‘(𝐹‘𝑖)) = 𝑛 ↔ (1^st ‘(𝐹‘𝑚)) = 𝑛))
270	269	elrab 3682	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛} ↔ (𝑚 ∈ ℕ ∧ (1^st ‘(𝐹‘𝑚)) = 𝑛))
271	267, 270	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛})
272	271	3adant1 1131	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛})
273	260, 111	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → (𝐺‘𝑚) = ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))
274	265	fveq2d 6892	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (𝐼‘(1^st ‘(𝐹‘𝑚))) = (𝐼‘𝑛))
275		vex 3479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑛 ∈ V
276		vex 3479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑗 ∈ V
277	275, 276	op2ndd 7981	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (2^nd ‘(𝐹‘𝑚)) = 𝑗)
278	274, 277	fveq12d 6895	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))) = ((𝐼‘𝑛)‘𝑗))
279	278	adantl 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))) = ((𝐼‘𝑛)‘𝑗))
280	273, 279	eqtr2d 2774	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → ((𝐼‘𝑛)‘𝑗) = (𝐺‘𝑚))
281	280	coeq2d 5860	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → ([,) ∘ ((𝐼‘𝑛)‘𝑗)) = ([,) ∘ (𝐺‘𝑚)))
282	281	fveq1d 6890	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) = (([,) ∘ (𝐺‘𝑚))‘𝑘))
283	282	ixpeq2dv 8903	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
284		eqimss 4039	. . . . . . . . . . . . . . . 16 ⊢ (X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘) → X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
285	283, 284	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
286	285	3adant1 1131	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
287		rspe 3247	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛} ∧ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘)) → ∃𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
288	272, 286, 287	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → ∃𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
289	288	3exp 1120	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (𝑚 ∈ ℕ → ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → ∃𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))))
290	258, 259, 289	rexlimd 3264	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (∃𝑚 ∈ ℕ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → ∃𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘)))
291	257, 290	mpd 15	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ∃𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
292	291	ralrimiva 3147	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑗 ∈ ℕ ∃𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
293		iunss2 5051	. . . . . . . . 9 ⊢ (∀𝑗 ∈ ℕ ∃𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ ∪ 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
294	292, 293	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ ∪ 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
295	250, 294	sstrd 3991	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
296		ssrab2 4076	. . . . . . . . 9 ⊢ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛} ⊆ ℕ
297		iunss1 5010	. . . . . . . . 9 ⊢ ({𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛} ⊆ ℕ → ∪ 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
298	296, 297	ax-mp 5	. . . . . . . 8 ⊢ ∪ 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘)
299	298	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
300	295, 299	sstrd 3991	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
301	300	ralrimiva 3147	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
302		iunss 5047	. . . . 5 ⊢ (∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘) ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
303	301, 302	sylibr 233	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
304	1, 126, 19, 234, 303	ovnlecvr 45209	. . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^{^}‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))))
305	112	fveq2d 6892	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐿‘(𝐺‘𝑚)) = (𝐿‘((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚)))))
306	305	mpteq2dva 5247	. . . . . 6 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚))) = (𝑚 ∈ ℕ ↦ (𝐿‘((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))))
307	306	fveq2d 6892	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))) = (Σ^{^}‘(𝑚 ∈ ℕ ↦ (𝐿‘((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚)))))))
308		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑝𝜑
309		2fveq3 6893	. . . . . . . 8 ⊢ (𝑝 = (𝐹‘𝑚) → (𝐼‘(1^st ‘𝑝)) = (𝐼‘(1^st ‘(𝐹‘𝑚))))
310		fveq2 6888	. . . . . . . 8 ⊢ (𝑝 = (𝐹‘𝑚) → (2^nd ‘𝑝) = (2^nd ‘(𝐹‘𝑚)))
311	309, 310	fveq12d 6895	. . . . . . 7 ⊢ (𝑝 = (𝐹‘𝑚) → ((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝)) = ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))
312	311	fveq2d 6892	. . . . . 6 ⊢ (𝑝 = (𝐹‘𝑚) → (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))) = (𝐿‘((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚)))))
313		eqidd 2734	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) = (𝐹‘𝑚))
314		nfv 1918	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑝 ∈ (ℕ × ℕ))
315	1	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → 𝑋 ∈ Fin)
316		simpl 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → 𝜑)
317		xp1st 8002	. . . . . . . . . 10 ⊢ (𝑝 ∈ (ℕ × ℕ) → (1^st ‘𝑝) ∈ ℕ)
318	317	adantl 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → (1^st ‘𝑝) ∈ ℕ)
319		xp2nd 8003	. . . . . . . . . 10 ⊢ (𝑝 ∈ (ℕ × ℕ) → (2^nd ‘𝑝) ∈ ℕ)
320	319	adantl 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → (2^nd ‘𝑝) ∈ ℕ)
321		fvex 6901	. . . . . . . . . 10 ⊢ (2^nd ‘𝑝) ∈ V
322		eleq1 2822	. . . . . . . . . . . 12 ⊢ (𝑗 = (2^nd ‘𝑝) → (𝑗 ∈ ℕ ↔ (2^nd ‘𝑝) ∈ ℕ))
323	322	3anbi3d 1443	. . . . . . . . . . 11 ⊢ (𝑗 = (2^nd ‘𝑝) → ((𝜑 ∧ (1^st ‘𝑝) ∈ ℕ ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ (1^st ‘𝑝) ∈ ℕ ∧ (2^nd ‘𝑝) ∈ ℕ)))
324		fveq2 6888	. . . . . . . . . . . 12 ⊢ (𝑗 = (2^nd ‘𝑝) → ((𝐼‘(1^st ‘𝑝))‘𝑗) = ((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝)))
325	324	feq1d 6699	. . . . . . . . . . 11 ⊢ (𝑗 = (2^nd ‘𝑝) → (((𝐼‘(1^st ‘𝑝))‘𝑗):𝑋⟶(ℝ × ℝ) ↔ ((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝)):𝑋⟶(ℝ × ℝ)))
326	323, 325	imbi12d 345	. . . . . . . . . 10 ⊢ (𝑗 = (2^nd ‘𝑝) → (((𝜑 ∧ (1^st ‘𝑝) ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1^st ‘𝑝))‘𝑗):𝑋⟶(ℝ × ℝ)) ↔ ((𝜑 ∧ (1^st ‘𝑝) ∈ ℕ ∧ (2^nd ‘𝑝) ∈ ℕ) → ((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝)):𝑋⟶(ℝ × ℝ))))
327		fvex 6901	. . . . . . . . . . 11 ⊢ (1^st ‘𝑝) ∈ V
328		eleq1 2822	. . . . . . . . . . . . 13 ⊢ (𝑛 = (1^st ‘𝑝) → (𝑛 ∈ ℕ ↔ (1^st ‘𝑝) ∈ ℕ))
329	328	3anbi2d 1442	. . . . . . . . . . . 12 ⊢ (𝑛 = (1^st ‘𝑝) → ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ (1^st ‘𝑝) ∈ ℕ ∧ 𝑗 ∈ ℕ)))
330		fveq2 6888	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (1^st ‘𝑝) → (𝐼‘𝑛) = (𝐼‘(1^st ‘𝑝)))
331	330	fveq1d 6890	. . . . . . . . . . . . 13 ⊢ (𝑛 = (1^st ‘𝑝) → ((𝐼‘𝑛)‘𝑗) = ((𝐼‘(1^st ‘𝑝))‘𝑗))
332	331	feq1d 6699	. . . . . . . . . . . 12 ⊢ (𝑛 = (1^st ‘𝑝) → (((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ) ↔ ((𝐼‘(1^st ‘𝑝))‘𝑗):𝑋⟶(ℝ × ℝ)))
333	329, 332	imbi12d 345	. . . . . . . . . . 11 ⊢ (𝑛 = (1^st ‘𝑝) → (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ)) ↔ ((𝜑 ∧ (1^st ‘𝑝) ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1^st ‘𝑝))‘𝑗):𝑋⟶(ℝ × ℝ))))
334	327, 333, 102	vtocl 3549	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (1^st ‘𝑝) ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1^st ‘𝑝))‘𝑗):𝑋⟶(ℝ × ℝ))
335	321, 326, 334	vtocl 3549	. . . . . . . . 9 ⊢ ((𝜑 ∧ (1^st ‘𝑝) ∈ ℕ ∧ (2^nd ‘𝑝) ∈ ℕ) → ((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝)):𝑋⟶(ℝ × ℝ))
336	316, 318, 320, 335	syl3anc 1372	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → ((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝)):𝑋⟶(ℝ × ℝ))
337	314, 315, 19, 336	hoiprodcl2 45206	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))) ∈ (0[,)+∞))
338	15, 337	sselid 3979	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))) ∈ (0[,]+∞))
339	308, 12, 312, 14, 20, 313, 338	sge0f1o 45033	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑝 ∈ (ℕ × ℕ) ↦ (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))))) = (Σ^{^}‘(𝑚 ∈ ℕ ↦ (𝐿‘((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚)))))))
340	307, 339	eqtr4d 2776	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))) = (Σ^{^}‘(𝑝 ∈ (ℕ × ℕ) ↦ (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))))))
341		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑗𝜑
342	275, 276	op1std 7980	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → (1^st ‘𝑝) = 𝑛)
343	342	fveq2d 6892	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → (𝐼‘(1^st ‘𝑝)) = (𝐼‘𝑛))
344	275, 276	op2ndd 7981	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → (2^nd ‘𝑝) = 𝑗)
345	343, 344	fveq12d 6895	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → ((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝)) = ((𝐼‘𝑛)‘𝑗))
346	345	fveq2d 6892	. . . . . . 7 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))) = (𝐿‘((𝐼‘𝑛)‘𝑗)))
347		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ)
348	123	adantr 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
349	93, 101	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ))
350	347, 348, 19, 349	hoiprodcl2 45206	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (𝐿‘((𝐼‘𝑛)‘𝑗)) ∈ (0[,)+∞))
351	15, 350	sselid 3979	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (𝐿‘((𝐼‘𝑛)‘𝑗)) ∈ (0[,]+∞))
352	351	3impa 1111	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝐿‘((𝐼‘𝑛)‘𝑗)) ∈ (0[,]+∞))
353	341, 346, 14, 14, 352	sge0xp 45080	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))))) = (Σ^{^}‘(𝑝 ∈ (ℕ × ℕ) ↦ (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))))))
354	353	eqcomd 2739	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑝 ∈ (ℕ × ℕ) ↦ (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))))) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))))))
355	13	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ℕ ∈ V)
356		eqid 2733	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))
357	351, 356	fmptd 7109	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗))):ℕ⟶(0[,]+∞))
358	355, 357	sge0cl 45032	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))) ∈ (0[,]+∞))
359		fveq1 6887	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝐼‘𝑛) → (𝑖‘𝑗) = ((𝐼‘𝑛)‘𝑗))
360	359	fveq2d 6892	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼‘𝑛) → (𝐿‘(𝑖‘𝑗)) = (𝐿‘((𝐼‘𝑛)‘𝑗)))
361	360	mpteq2dv 5249	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘𝑛) → (𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗))))
362	361	fveq2d 6892	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘𝑛) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))))
363	362	breq1d 5157	. . . . . . . . 9 ⊢ (𝑖 = (𝐼‘𝑛) → ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))) ↔ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛)))))
364	363	elrab 3682	. . . . . . . 8 ⊢ ((𝐼‘𝑛) ∈ {𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛)))} ↔ ((𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛)) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))) ≤ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛)))))
365	236, 364	sylib 217	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛)) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛)))))
366	365	simprd 497	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))))
367	118, 14, 358, 173, 366	sge0lempt 45061	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))))) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))))))
368	354, 367	eqbrtrd 5169	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑝 ∈ (ℕ × ℕ) ↦ (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))))) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))))))
369	340, 368	eqbrtrd 5169	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))))))
370	11, 117, 174, 304, 369	xrletrd 13137	. 2 ⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))))))
371	118, 14, 160, 168	sge0xadd 45086	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))))) = ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝐸 / (2↑𝑛))))))
372	119	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ^*)
373	121	a1i 11	. . . . . 6 ⊢ (𝜑 → +∞ ∈ ℝ^*)
374	145	rexrd 11260	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ^*)
375	70	rpge0d 13016	. . . . . 6 ⊢ (𝜑 → 0 ≤ 𝐸)
376	145	ltpnfd 13097	. . . . . 6 ⊢ (𝜑 → 𝐸 < +∞)
377	372, 373, 374, 375, 376	elicod 13370	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (0[,)+∞))
378	377	sge0ad2en 45082	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝐸 / (2↑𝑛)))) = 𝐸)
379	378	oveq2d 7420	. . 3 ⊢ (𝜑 → ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝐸 / (2↑𝑛))))) = ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸))
380	371, 379	eqtrd 2773	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))))) = ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸))
381	370, 380	breqtrd 5173	1 ⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 {crab 3433 Vcvv 3475 ⊆ wss 3947 ∅c0 4321 ifcif 4527 𝒫 cpw 4601 ⟨cop 4633 ∪ ciun 4996 class class class wbr 5147 ↦ cmpt 5230 × cxp 5673 ∘ ccom 5679 Fn wfn 6535 ⟶wf 6536 –onto→wfo 6538 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7404 1^st c1st 7968 2^nd c2nd 7969 ↑_m cmap 8816 Xcixp 8887 Fincfn 8935 infcinf 9432 ℝcr 11105 0cc0 11106 +∞cpnf 11241 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 / cdiv 11867 ℕcn 12208 2c2 12263 ℝ⁺crp 12970 +_𝑒 cxad 13086 [,)cico 13322 [,]cicc 13323 ↑cexp 14023 ∏cprod 15845 volcvol 24962 Σ^{^}csumge0 45013 voln*covoln 45187

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-inf2 9632 ax-ac2 10454 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-om 7851 df-1st 7970 df-2nd 7971 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-acn 9933 df-ac 10107 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429 df-sum 15629 df-prod 15846 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-rest 17364 df-0g 17383 df-topgen 17385 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-subg 18997 df-cmn 19643 df-abl 19644 df-mgp 19980 df-ur 19997 df-ring 20049 df-cring 20050 df-oppr 20139 df-dvdsr 20160 df-unit 20161 df-invr 20191 df-dvr 20204 df-drng 20306 df-psmet 20921 df-xmet 20922 df-met 20923 df-bl 20924 df-mopn 20925 df-cnfld 20930 df-top 22378 df-topon 22395 df-bases 22431 df-cmp 22873 df-ovol 24963 df-vol 24964 df-sumge0 45014 df-ovoln 45188

This theorem is referenced by: ovnsubaddlem2 45222

W3C validator