ovnsubaddlem1 - Mathbox for Glauco Siliprandi

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Theorem ovnsubaddlem1 47579

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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
4		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
5	3, 4	ffvelcdmd 7085	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ 𝒫 (ℝ ↑_m 𝑋))
6		elpwi 4564	. . . . . . 7 ⊢ ((𝐴‘𝑛) ∈ 𝒫 (ℝ ↑_m 𝑋) → (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋))
7	5, 6	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋))
8	7	ralrimiva 3155	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋))
9		iunss 5003	. . . . 5 ⊢ (∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋) ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋))
10	8, 9	sylibr 237	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋))
11	1, 10	ovnxrcl 47578	. . 3 ⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ∈ ℝ^)
12		nfv 1947	. . . 4 ⊢ Ⅎ𝑚𝜑
13		nnex 12341	. . . . 5 ⊢ ℕ ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝜑 → ℕ ∈ V)
15		icossicc 13567	. . . . 5 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
16		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑚 ∈ ℕ)
17		simpl 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝜑)
18	17, 1	syl 18	. . . . . 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 6824	. . . . . . . . . . . 12 ⊢ (𝐹:ℕ–1-1-onto→(ℕ × ℕ) → 𝐹:ℕ⟶(ℕ × ℕ))
22	20, 21	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℕ⟶(ℕ × ℕ))
23	22	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹:ℕ⟶(ℕ × ℕ))
24		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
25	23, 24	ffvelcdmd 7085	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∈ (ℕ × ℕ))
26		xp1st 8033	. . . . . . . . 9 ⊢ ((𝐹‘𝑚) ∈ (ℕ × ℕ) → (1^st ‘(𝐹‘𝑚)) ∈ ℕ)
27	25, 26	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1^st ‘(𝐹‘𝑚)) ∈ ℕ)
28		xp2nd 8034	. . . . . . . . 9 ⊢ ((𝐹‘𝑚) ∈ (ℕ × ℕ) → (2^nd ‘(𝐹‘𝑚)) ∈ ℕ)
29	25, 28	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2^nd ‘(𝐹‘𝑚)) ∈ ℕ)
30		fvex 6898	. . . . . . . . 9 ⊢ (2^nd ‘(𝐹‘𝑚)) ∈ V
31		eleq1 2849	. . . . . . . . . . 11 ⊢ (𝑗 = (2^nd ‘(𝐹‘𝑚)) → (𝑗 ∈ ℕ ↔ (2^nd ‘(𝐹‘𝑚)) ∈ ℕ))
32	31	3anbi3d 1470	. . . . . . . . . 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 6885	. . . . . . . . . . 11 ⊢ (𝑗 = (2^nd ‘(𝐹‘𝑚)) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗) = ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))
34	33	feq1d 6691	. . . . . . . . . 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 347	. . . . . . . . 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 6898	. . . . . . . . . 10 ⊢ (1^st ‘(𝐹‘𝑚)) ∈ V
37		eleq1 2849	. . . . . . . . . . . 12 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → (𝑛 ∈ ℕ ↔ (1^st ‘(𝐹‘𝑚)) ∈ ℕ))
38	37	3anbi2d 1469	. . . . . . . . . . 11 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ (1^st ‘(𝐹‘𝑚)) ∈ ℕ ∧ 𝑗 ∈ ℕ)))
39		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → (𝐼‘𝑛) = (𝐼‘(1^st ‘(𝐹‘𝑚))))
40	39	fveq1d 6887	. . . . . . . . . . . 12 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → ((𝐼‘𝑛)‘𝑗) = ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗))
41	40	feq1d 6691	. . . . . . . . . . 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 347	. . . . . . . . . 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 3956	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (𝐴‘𝑛) → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ↔ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)))
45	44	rabbidv 3420	. . . . . . . . . . . . . . . . . . 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 7453	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∈ V
47	46	rabex 5300	. . . . . . . . . . . . . . . . . . . 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 4028	. . . . . . . . . . . . . . . . . . 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 3965	. . . . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (𝐴‘𝑛) → (𝐶‘𝑎) = (𝐶‘(𝐴‘𝑛)))
55	54	eleq2d 2847	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (𝐴‘𝑛) → (𝑖 ∈ (𝐶‘𝑎) ↔ 𝑖 ∈ (𝐶‘(𝐴‘𝑛))))
56		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = (𝐴‘𝑛) → ((voln‘𝑋)‘𝑎) = ((voln‘𝑋)‘(𝐴‘𝑛)))
57	56	oveq1d 7435	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (𝐴‘𝑛) → (((voln‘𝑋)‘𝑎) +_e 𝑒) = (((voln‘𝑋)‘(𝐴‘𝑛)) +_e 𝑒))
58	57	breq2d 5115	. . . . . . . . . . . . . . . . . . . . . . . 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 644	. . . . . . . . . . . . . . . . . . . . . . 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 3415	. . . . . . . . . . . . . . . . . . . . . 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 5199	. . . . . . . . . . . . . . . . . . . . 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 46357	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℝ⁺ ∈ V
63	62	mptex 7229	. . . . . . . . . . . . . . . . . . . . . 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 7428	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 = (𝐸 / (2↑𝑛)) → (((voln‘𝑋)‘(𝐴‘𝑛)) +_e 𝑒) = (((voln‘𝑋)‘(𝐴‘𝑛)) +_e (𝐸 / (2↑𝑛))))
67	66	breq2d 5115	. . . . . . . . . . . . . . . . . . . . . 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 3420	. . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ⁺)
72		2nn 12416	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 ∈ ℕ
73	72	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℕ)
74		nnnn0 12613	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
75	73, 74	nnexpcld 14389	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
76	75	nnrpd 13162	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ⁺)
77	76	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ⁺)
78	71, 77	rpdivcld 13181	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ ℝ⁺)
79		fvex 6898	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐶‘(𝐴‘𝑛)) ∈ V
80	79	rabex 5300	. . . . . . . . . . . . . . . . . . . . 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 4028	. . . . . . . . . . . . . . . . . . . 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 3965	. . . . . . . . . . . . . . . . . 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 3932	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛)))
88	52, 87	sseldd 3932	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ))
89		elmapfn 8887	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼‘𝑛) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) → (𝐼‘𝑛) Fn ℕ)
90	88, 89	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) Fn ℕ)
91		elmapi 8869	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼‘𝑛) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) → (𝐼‘𝑛):ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
92	88, 91	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛):ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
93	92	ffvelcdmda 7084	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑_m 𝑋))
94	93	ralrimiva 3155	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑗 ∈ ℕ ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑_m 𝑋))
95	90, 94	jca 521	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐼‘𝑛) Fn ℕ ∧ ∀𝑗 ∈ ℕ ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑_m 𝑋)))
96	95	3adant3 1150	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛) Fn ℕ ∧ ∀𝑗 ∈ ℕ ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑_m 𝑋)))
97		ffnfv 7119	. . . . . . . . . . . . 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 237	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑛):ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
99		simp3 1156	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
100	98, 99	ffvelcdmd 7085	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑_m 𝑋))
101		elmapi 8869	. . . . . . . . . . 11 ⊢ (((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑_m 𝑋) → ((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ))
102	100, 101	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ))
103	36, 42, 102	vtocl 3521	. . . . . . . . 9 ⊢ ((𝜑 ∧ (1^st ‘(𝐹‘𝑚)) ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗):𝑋⟶(ℝ × ℝ))
104	30, 35, 103	vtocl 3521	. . . . . . . 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 1398	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))):𝑋⟶(ℝ × ℝ))
106		id 23	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ)
107		fvex 6898	. . . . . . . . . . 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 596	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (𝐺‘𝑚) = ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))
112	111	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) = ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))
113	112	feq1d 6691	. . . . . . 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 260	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚):𝑋⟶(ℝ × ℝ))
115	16, 18, 19, 114	hoiprodcl2 47564	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐿‘(𝐺‘𝑚)) ∈ (0[,)+∞))
116	15, 115	sselid 3929	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐿‘(𝐺‘𝑚)) ∈ (0[,]+∞))
117	12, 14, 116	sge0xrclmpt 47437	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))) ∈ ℝ^*)
118		nfv 1947	. . . 4 ⊢ Ⅎ𝑛𝜑
119		0xr 11356	. . . . . 6 ⊢ 0 ∈ ℝ^*
120	119	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ^*)
121		pnfxr 11363	. . . . . 6 ⊢ +∞ ∈ ℝ^*
122	121	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ^*)
123	1	adantr 486	. . . . . . . . 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 47561	. . . . . . . 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 2961	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 = ∅)
128	127	iffalsed 4493	. . . . . . . . 9 ⊢ (𝜑 → if(𝑋 = ∅, 0, inf((𝑍‘(𝐴‘𝑛)), ℝ^, < )) = inf((𝑍‘(𝐴‘𝑛)), ℝ^, < ))
129	128	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(𝑋 = ∅, 0, inf((𝑍‘(𝐴‘𝑛)), ℝ^, < )) = inf((𝑍‘(𝐴‘𝑛)), ℝ^, < ))
130	125, 129	eqtrd 2796	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln‘𝑋)‘(𝐴‘𝑛)) = inf((𝑍‘(𝐴‘𝑛)), ℝ^, < ))
131		sseq1 3956	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝐴‘𝑛) → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ↔ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)))
132	131	anbi1d 643	. . . . . . . . . . . 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 3187	. . . . . . . . . . 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 3420	. . . . . . . . . 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 13115	. . . . . . . . . . . 12 ⊢ ℝ^* ∈ V
136	135	rabex 5300	. . . . . . . . . . 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 4028	. . . . . . . . . 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 3965	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑍‘(𝐴‘𝑛)) ⊆ ℝ^*)
142		infxrcl 13464	. . . . . . . 8 ⊢ ((𝑍‘(𝐴‘𝑛)) ⊆ ℝ^* → inf((𝑍‘(𝐴‘𝑛)), ℝ^, < ) ∈ ℝ^)
143	141, 142	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → inf((𝑍‘(𝐴‘𝑛)), ℝ^, < ) ∈ ℝ^)
144	130, 143	eqeltrd 2861	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln‘𝑋)‘(𝐴‘𝑛)) ∈ ℝ^)
145	70	rpred 13164	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℝ)
146	145	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ)
147		2re 12417	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
148	147	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ)
149	148, 74	reexpcld 14306	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ)
150	149	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ)
151	148	recnd 11337	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℂ)
152		2ne0 12449	. . . . . . . . . . 11 ⊢ 2 ≠ 0
153	152	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ≠ 0)
154		nnz 12714	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
155	151, 153, 154	expne0d 14295	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ≠ 0)
156	155	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ≠ 0)
157	146, 150, 156	redivcld 12145	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ ℝ)
158	157	rexrd 11359	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ ℝ^*)
159	144, 158	xaddcld 13431	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((voln‘𝑋)‘(𝐴‘𝑛)) +_e (𝐸 / (2↑𝑛))) ∈ ℝ^)
160	123, 7	ovncl 47576	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln*‘𝑋)‘(𝐴‘𝑛)) ∈ (0[,]+∞))
161		xrge0ge0 46358	. . . . . . 7 ⊢ (((voln‘𝑋)‘(𝐴‘𝑛)) ∈ (0[,]+∞) → 0 ≤ ((voln‘𝑋)‘(𝐴‘𝑛)))
162	160, 161	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ ((voln*‘𝑋)‘(𝐴‘𝑛)))
163		0red 11311	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ)
164	78	rpgt0d 13167	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 < (𝐸 / (2↑𝑛)))
165	163, 157, 164	ltled 11458	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝐸 / (2↑𝑛)))
166	157	ltpnfd 13250	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) < +∞)
167	158, 122, 166	xrltled 13279	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ≤ +∞)
168	120, 122, 158, 165, 167	eliccxrd 46538	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ (0[,]+∞))
169	144, 168	xadd0ge 46333	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln‘𝑋)‘(𝐴‘𝑛)) ≤ (((voln‘𝑋)‘(𝐴‘𝑛)) +_e (𝐸 / (2↑𝑛))))
170	120, 144, 159, 162, 169	xrletrd 13291	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_e (𝐸 / (2↑𝑛))))
171		pnfge 13259	. . . . . 6 ⊢ ((((voln‘𝑋)‘(𝐴‘𝑛)) +_e (𝐸 / (2↑𝑛))) ∈ ℝ^ → (((voln*‘𝑋)‘(𝐴‘𝑛)) +_e (𝐸 / (2↑𝑛))) ≤ +∞)
172	159, 171	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((voln*‘𝑋)‘(𝐴‘𝑛)) +_e (𝐸 / (2↑𝑛))) ≤ +∞)
173	120, 122, 159, 170, 172	eliccxrd 46538	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((voln*‘𝑋)‘(𝐴‘𝑛)) +_e (𝐸 / (2↑𝑛))) ∈ (0[,]+∞))
174	118, 14, 173	sge0xrclmpt 47437	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (((voln‘𝑋)‘(𝐴‘𝑛)) +_e (𝐸 / (2↑𝑛))))) ∈ ℝ^)
175		sseq1 3956	. . . . . . . . . . . . 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 3420	. . . . . . . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
178	177, 27	ffvelcdmd 7085	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐴‘(1^st ‘(𝐹‘𝑚))) ∈ 𝒫 (ℝ ↑_m 𝑋))
179	46	rabex 5300	. . . . . . . . . . . . 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 4028	. . . . . . . . . . . 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 3965	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ⊆ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ))
185		fveq2 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (𝐴‘(1^st ‘(𝐹‘𝑚))) → (𝐶‘𝑎) = (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))))
186	185	eleq2d 2847	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝐴‘(1^st ‘(𝐹‘𝑚))) → (𝑖 ∈ (𝐶‘𝑎) ↔ 𝑖 ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚))))))
187		fveq2 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (𝐴‘(1^st ‘(𝐹‘𝑚))) → ((voln‘𝑋)‘𝑎) = ((voln‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))))
188	187	oveq1d 7435	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (𝐴‘(1^st ‘(𝐹‘𝑚))) → (((voln‘𝑋)‘𝑎) +_e 𝑒) = (((voln‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_e 𝑒))
189	188	breq2d 5115	. . . . . . . . . . . . . . . . 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 644	. . . . . . . . . . . . . . . 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 3415	. . . . . . . . . . . . . . 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 5199	. . . . . . . . . . . . . 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 7229	. . . . . . . . . . . . . . 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 7428	. . . . . . . . . . . . . . . 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 5115	. . . . . . . . . . . . . . 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 3420	. . . . . . . . . . . . . 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 487	. . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐸 ∈ ℝ⁺)
201		2rp 13125	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ⁺
202	201	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 2 ∈ ℝ⁺)
203	27	nnzd 12719	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1^st ‘(𝐹‘𝑚)) ∈ ℤ)
204	202, 203	rpexpcld 14391	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2↑(1^st ‘(𝐹‘𝑚))) ∈ ℝ⁺)
205	200, 204	rpdivcld 13181	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐸 / (2↑(1^st ‘(𝐹‘𝑚)))) ∈ ℝ⁺)
206		fvex 6898	. . . . . . . . . . . . . . 15 ⊢ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ∈ V
207	206	rabex 5300	. . . . . . . . . . . . . 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 4028	. . . . . . . . . . . . 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 3965	. . . . . . . . . . 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 642	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → ((𝜑 ∧ 𝑛 ∈ ℕ) ↔ (𝜑 ∧ (1^st ‘(𝐹‘𝑚)) ∈ ℕ)))
214		2fveq3 6890	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → (𝐷‘(𝐴‘𝑛)) = (𝐷‘(𝐴‘(1^st ‘(𝐹‘𝑚)))))
215		oveq2 7428	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → (2↑𝑛) = (2↑(1^st ‘(𝐹‘𝑚))))
216	215	oveq2d 7436	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → (𝐸 / (2↑𝑛)) = (𝐸 / (2↑(1^st ‘(𝐹‘𝑚)))))
217	214, 216	fveq12d 6892	. . . . . . . . . . . . . . 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 2855	. . . . . . . . . . . . . 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 347	. . . . . . . . . . . . 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 3521	. . . . . . . . . . . 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 596	. . . . . . . . . . 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 3932	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))) ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))))
223	184, 222	sseldd 3932	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ))
224		elmapfn 8887	. . . . . . . . 9 ⊢ ((𝐼‘(1^st ‘(𝐹‘𝑚))) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))) Fn ℕ)
225	223, 224	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))) Fn ℕ)
226		elmapi 8869	. . . . . . . . . . 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 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))):ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
228	227	ffvelcdmda 7084	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗) ∈ ((ℝ × ℝ) ↑_m 𝑋))
229	228	ralrimiva 3155	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑗 ∈ ℕ ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗) ∈ ((ℝ × ℝ) ↑_m 𝑋))
230	225, 229	jca 521	. . . . . . 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 7119	. . . . . . 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 237	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))):ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
233	232, 29	ffvelcdmd 7085	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))) ∈ ((ℝ × ℝ) ↑_m 𝑋))
234	233, 109	fmptd 7114	. . . 4 ⊢ (𝜑 → 𝐺:ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
235		simpl 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝜑)
236	86, 82	eleqtrd 2863	. . . . . . . . . 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 3929	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛)))
238		simp3 1156	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛))) → (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛)))
239	49	3adant3 1150	. . . . . . . . . . . 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 2863	. . . . . . . . . . 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 6884	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = (𝐼‘𝑛) → (ℎ‘𝑗) = ((𝐼‘𝑛)‘𝑗))
242	241	coeq2d 5840	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝐼‘𝑛) → ([,) ∘ (ℎ‘𝑗)) = ([,) ∘ ((𝐼‘𝑛)‘𝑗)))
243	242	fveq1d 6887	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝐼‘𝑛) → (([,) ∘ (ℎ‘𝑗))‘𝑘) = (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
244	243	ixpeq2dv 8941	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝐼‘𝑛) → X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
245	244	iuneq2d 4981	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝐼‘𝑛) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
246	245	sseq2d 3963	. . . . . . . . . . . 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 3645	. . . . . . . . . . 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 221	. . . . . . . . . 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 501	. . . . . . . . 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 1398	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
251		f1ofo 6832	. . . . . . . . . . . . . 14 ⊢ (𝐹:ℕ–1-1-onto→(ℕ × ℕ) → 𝐹:ℕ–onto→(ℕ × ℕ))
252	20, 251	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℕ–onto→(ℕ × ℕ))
253	252	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → 𝐹:ℕ–onto→(ℕ × ℕ))
254		opelxpi 5688	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ))
255	4, 254	sylan 592	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ))
256		foelcdmi 6946	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ–onto→(ℕ × ℕ) ∧ ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ)) → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩)
257	253, 255, 256	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩)
258		nfv 1947	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ)
259		nfre1 3288	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚∃𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘)
260		simpl 488	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → 𝑚 ∈ ℕ)
261		fveq2 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (1^st ‘(𝐹‘𝑚)) = (1^st ‘⟨𝑛, 𝑗⟩))
262		op1stg 8013	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ V ∧ 𝑗 ∈ V) → (1^st ‘⟨𝑛, 𝑗⟩) = 𝑛)
263	262	el2v 3458	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1^st ‘⟨𝑛, 𝑗⟩) = 𝑛
264	263	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (1^st ‘⟨𝑛, 𝑗⟩) = 𝑛)
265	261, 264	eqtrd 2796	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (1^st ‘(𝐹‘𝑚)) = 𝑛)
266	265	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → (1^st ‘(𝐹‘𝑚)) = 𝑛)
267	260, 266	jca 521	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → (𝑚 ∈ ℕ ∧ (1^st ‘(𝐹‘𝑚)) = 𝑛))
268		2fveq3 6890	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑚 → (1^st ‘(𝐹‘𝑖)) = (1^st ‘(𝐹‘𝑚)))
269	268	eqeq1d 2763	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑚 → ((1^st ‘(𝐹‘𝑖)) = 𝑛 ↔ (1^st ‘(𝐹‘𝑚)) = 𝑛))
270	269	elrab 3645	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛} ↔ (𝑚 ∈ ℕ ∧ (1^st ‘(𝐹‘𝑚)) = 𝑛))
271	267, 270	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛})
272	271	3adant1 1148	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛})
273	260, 111	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → (𝐺‘𝑚) = ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))
274	265	fveq2d 6889	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (𝐼‘(1^st ‘(𝐹‘𝑚))) = (𝐼‘𝑛))
275		vex 3455	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑛 ∈ V
276		vex 3455	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑗 ∈ V
277	275, 276	op2ndd 8012	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (2^nd ‘(𝐹‘𝑚)) = 𝑗)
278	274, 277	fveq12d 6892	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))) = ((𝐼‘𝑛)‘𝑗))
279	278	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))) = ((𝐼‘𝑛)‘𝑗))
280	273, 279	eqtr2d 2797	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → ((𝐼‘𝑛)‘𝑗) = (𝐺‘𝑚))
281	280	coeq2d 5840	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → ([,) ∘ ((𝐼‘𝑛)‘𝑗)) = ([,) ∘ (𝐺‘𝑚)))
282	281	fveq1d 6887	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) = (([,) ∘ (𝐺‘𝑚))‘𝑘))
283	282	ixpeq2dv 8941	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
284		eqimss 3989	. . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
286	285	3adant1 1148	. . . . . . . . . . . . . 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 3253	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 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 1137	. . . . . . . . . . . 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 3270	. . . . . . . . . . 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 16	. . . . . . . . . 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 3155	. . . . . . . . 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 5008	. . . . . . . . 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 18	. . . . . . . 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 3941	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
296		ssrab2 4028	. . . . . . . . 9 ⊢ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛} ⊆ ℕ
297		iunss1 4966	. . . . . . . . 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 3941	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
301	300	ralrimiva 3155	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
302		iunss 5003	. . . . 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 237	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
304	1, 126, 19, 234, 303	ovnlecvr 47567	. . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^{^}‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))))
305	112	fveq2d 6889	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐿‘(𝐺‘𝑚)) = (𝐿‘((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚)))))
306	305	mpteq2dva 5198	. . . . . 6 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚))) = (𝑚 ∈ ℕ ↦ (𝐿‘((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))))
307	306	fveq2d 6889	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))) = (Σ^{^}‘(𝑚 ∈ ℕ ↦ (𝐿‘((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚)))))))
308		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑝𝜑
309		2fveq3 6890	. . . . . . . 8 ⊢ (𝑝 = (𝐹‘𝑚) → (𝐼‘(1^st ‘𝑝)) = (𝐼‘(1^st ‘(𝐹‘𝑚))))
310		fveq2 6885	. . . . . . . 8 ⊢ (𝑝 = (𝐹‘𝑚) → (2^nd ‘𝑝) = (2^nd ‘(𝐹‘𝑚)))
311	309, 310	fveq12d 6892	. . . . . . 7 ⊢ (𝑝 = (𝐹‘𝑚) → ((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝)) = ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))
312	311	fveq2d 6889	. . . . . 6 ⊢ (𝑝 = (𝐹‘𝑚) → (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))) = (𝐿‘((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚)))))
313		eqidd 2762	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) = (𝐹‘𝑚))
314		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑝 ∈ (ℕ × ℕ))
315	1	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → 𝑋 ∈ Fin)
316		simpl 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → 𝜑)
317		xp1st 8033	. . . . . . . . . 10 ⊢ (𝑝 ∈ (ℕ × ℕ) → (1^st ‘𝑝) ∈ ℕ)
318	317	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → (1^st ‘𝑝) ∈ ℕ)
319		xp2nd 8034	. . . . . . . . . 10 ⊢ (𝑝 ∈ (ℕ × ℕ) → (2^nd ‘𝑝) ∈ ℕ)
320	319	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → (2^nd ‘𝑝) ∈ ℕ)
321		fvex 6898	. . . . . . . . . 10 ⊢ (2^nd ‘𝑝) ∈ V
322		eleq1 2849	. . . . . . . . . . . 12 ⊢ (𝑗 = (2^nd ‘𝑝) → (𝑗 ∈ ℕ ↔ (2^nd ‘𝑝) ∈ ℕ))
323	322	3anbi3d 1470	. . . . . . . . . . 11 ⊢ (𝑗 = (2^nd ‘𝑝) → ((𝜑 ∧ (1^st ‘𝑝) ∈ ℕ ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ (1^st ‘𝑝) ∈ ℕ ∧ (2^nd ‘𝑝) ∈ ℕ)))
324		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑗 = (2^nd ‘𝑝) → ((𝐼‘(1^st ‘𝑝))‘𝑗) = ((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝)))
325	324	feq1d 6691	. . . . . . . . . . 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 347	. . . . . . . . . 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 6898	. . . . . . . . . . 11 ⊢ (1^st ‘𝑝) ∈ V
328		eleq1 2849	. . . . . . . . . . . . 13 ⊢ (𝑛 = (1^st ‘𝑝) → (𝑛 ∈ ℕ ↔ (1^st ‘𝑝) ∈ ℕ))
329	328	3anbi2d 1469	. . . . . . . . . . . 12 ⊢ (𝑛 = (1^st ‘𝑝) → ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ (1^st ‘𝑝) ∈ ℕ ∧ 𝑗 ∈ ℕ)))
330		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (1^st ‘𝑝) → (𝐼‘𝑛) = (𝐼‘(1^st ‘𝑝)))
331	330	fveq1d 6887	. . . . . . . . . . . . 13 ⊢ (𝑛 = (1^st ‘𝑝) → ((𝐼‘𝑛)‘𝑗) = ((𝐼‘(1^st ‘𝑝))‘𝑗))
332	331	feq1d 6691	. . . . . . . . . . . 12 ⊢ (𝑛 = (1^st ‘𝑝) → (((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ) ↔ ((𝐼‘(1^st ‘𝑝))‘𝑗):𝑋⟶(ℝ × ℝ)))
333	329, 332	imbi12d 347	. . . . . . . . . . 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 3521	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (1^st ‘𝑝) ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1^st ‘𝑝))‘𝑗):𝑋⟶(ℝ × ℝ))
335	321, 326, 334	vtocl 3521	. . . . . . . . 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 1398	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → ((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝)):𝑋⟶(ℝ × ℝ))
337	314, 315, 19, 336	hoiprodcl2 47564	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))) ∈ (0[,)+∞))
338	15, 337	sselid 3929	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))) ∈ (0[,]+∞))
339	308, 12, 312, 14, 20, 313, 338	sge0f1o 47391	. . . . 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 2799	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))) = (Σ^{^}‘(𝑝 ∈ (ℕ × ℕ) ↦ (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))))))
341		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑗𝜑
342	275, 276	op1std 8011	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → (1^st ‘𝑝) = 𝑛)
343	342	fveq2d 6889	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → (𝐼‘(1^st ‘𝑝)) = (𝐼‘𝑛))
344	275, 276	op2ndd 8012	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → (2^nd ‘𝑝) = 𝑗)
345	343, 344	fveq12d 6892	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → ((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝)) = ((𝐼‘𝑛)‘𝑗))
346	345	fveq2d 6889	. . . . . . 7 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))) = (𝐿‘((𝐼‘𝑛)‘𝑗)))
347		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ)
348	123	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
349	93, 101	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ))
350	347, 348, 19, 349	hoiprodcl2 47564	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (𝐿‘((𝐼‘𝑛)‘𝑗)) ∈ (0[,)+∞))
351	15, 350	sselid 3929	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (𝐿‘((𝐼‘𝑛)‘𝑗)) ∈ (0[,]+∞))
352	351	3impa 1127	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝐿‘((𝐼‘𝑛)‘𝑗)) ∈ (0[,]+∞))
353	341, 346, 14, 14, 352	sge0xp 47438	. . . . . 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 2767	. . . . 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 2761	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))
357	351, 356	fmptd 7114	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗))):ℕ⟶(0[,]+∞))
358	355, 357	sge0cl 47390	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))) ∈ (0[,]+∞))
359		fveq1 6884	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝐼‘𝑛) → (𝑖‘𝑗) = ((𝐼‘𝑛)‘𝑗))
360	359	fveq2d 6889	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼‘𝑛) → (𝐿‘(𝑖‘𝑗)) = (𝐿‘((𝐼‘𝑛)‘𝑗)))
361	360	mpteq2dv 5199	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘𝑛) → (𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗))))
362	361	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘𝑛) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))))
363	362	breq1d 5113	. . . . . . . . 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 3645	. . . . . . . 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 221	. . . . . . 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 501	. . . . . 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 47419	. . . . 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 5127	. . . 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 5127	. . 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 13291	. 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 47444	. . 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 11359	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ^*)
375	70	rpge0d 13168	. . . . . 6 ⊢ (𝜑 → 0 ≤ 𝐸)
376	145	ltpnfd 13250	. . . . . 6 ⊢ (𝜑 → 𝐸 < +∞)
377	372, 373, 374, 375, 376	elicod 13526	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (0[,)+∞))
378	377	sge0ad2en 47440	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝐸 / (2↑𝑛)))) = 𝐸)
379	378	oveq2d 7436	. . 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 2796	. 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 5131	1 ⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_e 𝐸))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∀wral 3077 ∃wrex 3087 {crab 3413 Vcvv 3451 ⊆ wss 3899 ∅c0 4279 ifcif 4482 𝒫 cpw 4557 ⟨cop 4590 ∪ ciun 4951 class class class wbr 5103 ↦ cmpt 5186 × cxp 5649 ∘ ccom 5655 Fn wfn 6533 ⟶wf 6534 –onto→wfo 6536 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7420 1^st c1st 7999 2^nd c2nd 8000 ↑_m cmap 8847 Xcixp 8925 Fincfn 8973 infcinf 9433 ℝcr 11199 0cc0 11200 +∞cpnf 11340 ℝ^*cxr 11342 < clt 11343 ≤ cle 11344 / cdiv 11973 ℕcn 12335 2c2 12397 ℝ⁺crp 13120 +_e cxad 13239 [,)cico 13478 [,]cicc 13479 ↑cexp 14204 ∏cprod 16072 volcvol 25784 Σ^{^}csumge0 47371 voln*covoln 47545

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-dju 9982 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-n0 12607 df-z 12694 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-seq 14145 df-exp 14205 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-clim 15655 df-rlim 15656 df-sum 15854 df-prod 16073 df-rest 17593 df-topgen 17614 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-top 23212 df-topon 23229 df-bases 23264 df-cmp 23705 df-ovol 25785 df-vol 25786 df-sumge0 47372 df-ovoln 47546

This theorem is used by: ovnsubaddlem2 47580

W3C validator