Theorem ovnsubaddlem1 43998

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‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ ↑m 𝑋))
4		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
5	3, 4	ffvelrnd 6944	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ 𝒫 (ℝ ↑m 𝑋))
6		elpwi 4539	. . . . . . 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 3107	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋))
9		iunss 4971	. . . . 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 43997	. . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ∈ ℝ*)
12		nfv 1918	. . . 4 ⊢ Ⅎ𝑚𝜑
13		nnex 11909	. . . . 5 ⊢ ℕ ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝜑 → ℕ ∈ V)
15		icossicc 13097	. . . . 5 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
16		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑚 ∈ ℕ)
17		simpl 482	. . . . . . 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 6700	. . . . . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹:ℕ⟶(ℕ × ℕ))
24		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
25	23, 24	ffvelrnd 6944	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∈ (ℕ × ℕ))
26		xp1st 7836	. . . . . . . . 9 ⊢ ((𝐹‘𝑚) ∈ (ℕ × ℕ) → (1st ‘(𝐹‘𝑚)) ∈ ℕ)
27	25, 26	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐹‘𝑚)) ∈ ℕ)
28		xp2nd 7837	. . . . . . . . 9 ⊢ ((𝐹‘𝑚) ∈ (ℕ × ℕ) → (2nd ‘(𝐹‘𝑚)) ∈ ℕ)
29	25, 28	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐹‘𝑚)) ∈ ℕ)
30		fvex 6769	. . . . . . . . 9 ⊢ (2nd ‘(𝐹‘𝑚)) ∈ V
31		eleq1 2826	. . . . . . . . . . 11 ⊢ (𝑗 = (2nd ‘(𝐹‘𝑚)) → (𝑗 ∈ ℕ ↔ (2nd ‘(𝐹‘𝑚)) ∈ ℕ))
32	31	3anbi3d 1440	. . . . . . . . . 10 ⊢ (𝑗 = (2nd ‘(𝐹‘𝑚)) → ((𝜑 ∧ (1st ‘(𝐹‘𝑚)) ∈ ℕ ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ (1st ‘(𝐹‘𝑚)) ∈ ℕ ∧ (2nd ‘(𝐹‘𝑚)) ∈ ℕ)))
33		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑗 = (2nd ‘(𝐹‘𝑚)) → ((𝐼‘(1st ‘(𝐹‘𝑚)))‘𝑗) = ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))))
34	33	feq1d 6569	. . . . . . . . . 10 ⊢ (𝑗 = (2nd ‘(𝐹‘𝑚)) → (((𝐼‘(1st ‘(𝐹‘𝑚)))‘𝑗):𝑋⟶(ℝ × ℝ) ↔ ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))):𝑋⟶(ℝ × ℝ)))
35	32, 34	imbi12d 344	. . . . . . . . 9 ⊢ (𝑗 = (2nd ‘(𝐹‘𝑚)) → (((𝜑 ∧ (1st ‘(𝐹‘𝑚)) ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1st ‘(𝐹‘𝑚)))‘𝑗):𝑋⟶(ℝ × ℝ)) ↔ ((𝜑 ∧ (1st ‘(𝐹‘𝑚)) ∈ ℕ ∧ (2nd ‘(𝐹‘𝑚)) ∈ ℕ) → ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))):𝑋⟶(ℝ × ℝ))))
36		fvex 6769	. . . . . . . . . 10 ⊢ (1st ‘(𝐹‘𝑚)) ∈ V
37		eleq1 2826	. . . . . . . . . . . 12 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → (𝑛 ∈ ℕ ↔ (1st ‘(𝐹‘𝑚)) ∈ ℕ))
38	37	3anbi2d 1439	. . . . . . . . . . 11 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ (1st ‘(𝐹‘𝑚)) ∈ ℕ ∧ 𝑗 ∈ ℕ)))
39		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → (𝐼‘𝑛) = (𝐼‘(1st ‘(𝐹‘𝑚))))
40	39	fveq1d 6758	. . . . . . . . . . . 12 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → ((𝐼‘𝑛)‘𝑗) = ((𝐼‘(1st ‘(𝐹‘𝑚)))‘𝑗))
41	40	feq1d 6569	. . . . . . . . . . 11 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → (((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ) ↔ ((𝐼‘(1st ‘(𝐹‘𝑚)))‘𝑗):𝑋⟶(ℝ × ℝ)))
42	38, 41	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ)) ↔ ((𝜑 ∧ (1st ‘(𝐹‘𝑚)) ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1st ‘(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 3942	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (𝐴‘𝑛) → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ↔ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)))
45	44	rabbidv 3404	. . . . . . . . . . . . . . . . . . 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 7288	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∈ V
47	46	rabex 5251	. . . . . . . . . . . . . . . . . . . 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 6880	. . . . . . . . . . . . . . . . . 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 4009	. . . . . . . . . . . . . . . . . . 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 3955	. . . . . . . . . . . . . . . . 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 6756	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (𝐴‘𝑛) → (𝐶‘𝑎) = (𝐶‘(𝐴‘𝑛)))
55	54	eleq2d 2824	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (𝐴‘𝑛) → (𝑖 ∈ (𝐶‘𝑎) ↔ 𝑖 ∈ (𝐶‘(𝐴‘𝑛))))
56		fveq2 6756	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = (𝐴‘𝑛) → ((voln*‘𝑋)‘𝑎) = ((voln*‘𝑋)‘(𝐴‘𝑛)))
57	56	oveq1d 7270	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (𝐴‘𝑛) → (((voln*‘𝑋)‘𝑎) +𝑒 𝑒) = (((voln*‘𝑋)‘(𝐴‘𝑛)) +𝑒 𝑒))
58	57	breq2d 5082	. . . . . . . . . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . . . . . . . . . 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 3400	. . . . . . . . . . . . . . . . . . . . . 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 5172	. . . . . . . . . . . . . . . . . . . . 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 42775	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℝ+ ∈ V
63	62	mptex 7081	. . . . . . . . . . . . . . . . . . . . . 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 6880	. . . . . . . . . . . . . . . . . . . 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 7263	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 = (𝐸 / (2↑𝑛)) → (((voln*‘𝑋)‘(𝐴‘𝑛)) +𝑒 𝑒) = (((voln*‘𝑋)‘(𝐴‘𝑛)) +𝑒 (𝐸 / (2↑𝑛))))
67	66	breq2d 5082	. . . . . . . . . . . . . . . . . . . . . 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 3404	. . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ+)
72		2nn 11976	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 ∈ ℕ
73	72	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℕ)
74		nnnn0 12170	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
75	73, 74	nnexpcld 13888	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
76	75	nnrpd 12699	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ+)
77	76	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ+)
78	71, 77	rpdivcld 12718	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ ℝ+)
79		fvex 6769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐶‘(𝐴‘𝑛)) ∈ V
80	79	rabex 5251	. . . . . . . . . . . . . . . . . . . . 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 6864	. . . . . . . . . . . . . . . . . . 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 4009	. . . . . . . . . . . . . . . . . . . 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 3955	. . . . . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛)))
88	52, 87	sseldd 3918	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
89		elmapfn 8611	. . . . . . . . . . . . . . . 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 8595	. . . . . . . . . . . . . . . . . 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	ffvelrnda 6943	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
94	93	ralrimiva 3107	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑗 ∈ ℕ ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
95	90, 94	jca 511	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐼‘𝑛) Fn ℕ ∧ ∀𝑗 ∈ ℕ ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋)))
96	95	3adant3 1130	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛) Fn ℕ ∧ ∀𝑗 ∈ ℕ ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋)))
97		ffnfv 6974	. . . . . . . . . . . . 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 1136	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
100	98, 99	ffvelrnd 6944	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
101		elmapi 8595	. . . . . . . . . . 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 3488	. . . . . . . . 9 ⊢ ((𝜑 ∧ (1st ‘(𝐹‘𝑚)) ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1st ‘(𝐹‘𝑚)))‘𝑗):𝑋⟶(ℝ × ℝ))
104	30, 35, 103	vtocl 3488	. . . . . . . 8 ⊢ ((𝜑 ∧ (1st ‘(𝐹‘𝑚)) ∈ ℕ ∧ (2nd ‘(𝐹‘𝑚)) ∈ ℕ) → ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))):𝑋⟶(ℝ × ℝ))
105	17, 27, 29, 104	syl3anc 1369	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))):𝑋⟶(ℝ × ℝ))
106		id 22	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ)
107		fvex 6769	. . . . . . . . . . 11 ⊢ ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))) ∈ V
108	107	a1i 11	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))) ∈ V)
109		ovnsubaddlem1.g	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))))
110	109	fvmpt2 6868	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))) ∈ V) → (𝐺‘𝑚) = ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))))
111	106, 108, 110	syl2anc 583	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (𝐺‘𝑚) = ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))))
112	111	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) = ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))))
113	112	feq1d 6569	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐺‘𝑚):𝑋⟶(ℝ × ℝ) ↔ ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))):𝑋⟶(ℝ × ℝ)))
114	105, 113	mpbird 256	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚):𝑋⟶(ℝ × ℝ))
115	16, 18, 19, 114	hoiprodcl2 43983	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐿‘(𝐺‘𝑚)) ∈ (0[,)+∞))
116	15, 115	sselid 3915	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐿‘(𝐺‘𝑚)) ∈ (0[,]+∞))
117	12, 14, 116	sge0xrclmpt 43856	. . 3 ⊢ (𝜑 → (Σ^‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))) ∈ ℝ*)
118		nfv 1918	. . . 4 ⊢ Ⅎ𝑛𝜑
119		0xr 10953	. . . . . 6 ⊢ 0 ∈ ℝ*
120	119	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ*)
121		pnfxr 10960	. . . . . 6 ⊢ +∞ ∈ ℝ*
122	121	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ*)
123	1	adantr 480	. . . . . . . . 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 43980	. . . . . . . 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 2947	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 = ∅)
128	127	iffalsed 4467	. . . . . . . . 9 ⊢ (𝜑 → if(𝑋 = ∅, 0, inf((𝑍‘(𝐴‘𝑛)), ℝ*, < )) = inf((𝑍‘(𝐴‘𝑛)), ℝ*, < ))
129	128	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(𝑋 = ∅, 0, inf((𝑍‘(𝐴‘𝑛)), ℝ*, < )) = inf((𝑍‘(𝐴‘𝑛)), ℝ*, < ))
130	125, 129	eqtrd 2778	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln*‘𝑋)‘(𝐴‘𝑛)) = inf((𝑍‘(𝐴‘𝑛)), ℝ*, < ))
131		sseq1 3942	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝐴‘𝑛) → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ↔ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)))
132	131	anbi1d 629	. . . . . . . . . . . 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 3225	. . . . . . . . . . 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 3404	. . . . . . . . . 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 12656	. . . . . . . . . . . 12 ⊢ ℝ* ∈ V
136	135	rabex 5251	. . . . . . . . . . 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 6880	. . . . . . . . 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 4009	. . . . . . . . . 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 3955	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑍‘(𝐴‘𝑛)) ⊆ ℝ*)
142		infxrcl 12996	. . . . . . . 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 2839	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln*‘𝑋)‘(𝐴‘𝑛)) ∈ ℝ*)
145	70	rpred 12701	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℝ)
146	145	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ)
147		2re 11977	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
148	147	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ)
149	148, 74	reexpcld 13809	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ)
150	149	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ)
151	148	recnd 10934	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℂ)
152		2ne0 12007	. . . . . . . . . . 11 ⊢ 2 ≠ 0
153	152	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ≠ 0)
154		nnz 12272	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
155	151, 153, 154	expne0d 13798	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ≠ 0)
156	155	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ≠ 0)
157	146, 150, 156	redivcld 11733	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ ℝ)
158	157	rexrd 10956	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ ℝ*)
159	144, 158	xaddcld 12964	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((voln*‘𝑋)‘(𝐴‘𝑛)) +𝑒 (𝐸 / (2↑𝑛))) ∈ ℝ*)
160	123, 7	ovncl 43995	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln*‘𝑋)‘(𝐴‘𝑛)) ∈ (0[,]+∞))
161		xrge0ge0 42776	. . . . . . 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 10909	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ)
164	78	rpgt0d 12704	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 < (𝐸 / (2↑𝑛)))
165	163, 157, 164	ltled 11053	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝐸 / (2↑𝑛)))
166	157	ltpnfd 12786	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) < +∞)
167	158, 122, 166	xrltled 12813	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ≤ +∞)
168	120, 122, 158, 165, 167	eliccxrd 42955	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ (0[,]+∞))
169	144, 168	xadd0ge 42749	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln*‘𝑋)‘(𝐴‘𝑛)) ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +𝑒 (𝐸 / (2↑𝑛))))
170	120, 144, 159, 162, 169	xrletrd 12825	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +𝑒 (𝐸 / (2↑𝑛))))
171		pnfge 12795	. . . . . 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 42955	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((voln*‘𝑋)‘(𝐴‘𝑛)) +𝑒 (𝐸 / (2↑𝑛))) ∈ (0[,]+∞))
174	118, 14, 173	sge0xrclmpt 43856	. . 3 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (((voln*‘𝑋)‘(𝐴‘𝑛)) +𝑒 (𝐸 / (2↑𝑛))))) ∈ ℝ*)
175		sseq1 3942	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝐴‘(1st ‘(𝐹‘𝑚))) → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ↔ (𝐴‘(1st ‘(𝐹‘𝑚))) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)))
176	175	rabbidv 3404	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐴‘(1st ‘(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‘(1st ‘(𝐹‘𝑚))) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)})
177	2	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ ↑m 𝑋))
178	177, 27	ffvelrnd 6944	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐴‘(1st ‘(𝐹‘𝑚))) ∈ 𝒫 (ℝ ↑m 𝑋))
179	46	rabex 5251	. . . . . . . . . . . . 13 ⊢ {ℎ ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ (𝐴‘(1st ‘(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‘(1st ‘(𝐹‘𝑚))) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)} ∈ V)
181	43, 176, 178, 180	fvmptd3 6880	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))) = {ℎ ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ (𝐴‘(1st ‘(𝐹‘𝑚))) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)})
182		ssrab2 4009	. . . . . . . . . . . 12 ⊢ {ℎ ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ (𝐴‘(1st ‘(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‘(1st ‘(𝐹‘𝑚))) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)} ⊆ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
184	181, 183	eqsstrd 3955	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))) ⊆ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
185		fveq2 6756	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (𝐴‘(1st ‘(𝐹‘𝑚))) → (𝐶‘𝑎) = (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))))
186	185	eleq2d 2824	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝐴‘(1st ‘(𝐹‘𝑚))) → (𝑖 ∈ (𝐶‘𝑎) ↔ 𝑖 ∈ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚))))))
187		fveq2 6756	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (𝐴‘(1st ‘(𝐹‘𝑚))) → ((voln*‘𝑋)‘𝑎) = ((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))))
188	187	oveq1d 7270	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (𝐴‘(1st ‘(𝐹‘𝑚))) → (((voln*‘𝑋)‘𝑎) +𝑒 𝑒) = (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 𝑒))
189	188	breq2d 5082	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝐴‘(1st ‘(𝐹‘𝑚))) → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 𝑒)))
190	186, 189	anbi12d 630	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = (𝐴‘(1st ‘(𝐹‘𝑚))) → ((𝑖 ∈ (𝐶‘𝑎) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)) ↔ (𝑖 ∈ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 𝑒))))
191	190	rabbidva2 3400	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = (𝐴‘(1st ‘(𝐹‘𝑚))) → {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)} = {𝑖 ∈ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 𝑒)})
192	191	mpteq2dv 5172	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝐴‘(1st ‘(𝐹‘𝑚))) → (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)}) = (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 𝑒)}))
193	62	mptex 7081	. . . . . . . . . . . . . . 15 ⊢ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 𝑒)}) ∈ V
194	193	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 𝑒)}) ∈ V)
195	53, 192, 178, 194	fvmptd3 6880	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐷‘(𝐴‘(1st ‘(𝐹‘𝑚)))) = (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 𝑒)}))
196		oveq2 7263	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = (𝐸 / (2↑(1st ‘(𝐹‘𝑚)))) → (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 𝑒) = (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 (𝐸 / (2↑(1st ‘(𝐹‘𝑚))))))
197	196	breq2d 5082	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = (𝐸 / (2↑(1st ‘(𝐹‘𝑚)))) → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 𝑒) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 (𝐸 / (2↑(1st ‘(𝐹‘𝑚)))))))
198	197	rabbidv 3404	. . . . . . . . . . . . . 14 ⊢ (𝑒 = (𝐸 / (2↑(1st ‘(𝐹‘𝑚)))) → {𝑖 ∈ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 𝑒)} = {𝑖 ∈ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 (𝐸 / (2↑(1st ‘(𝐹‘𝑚)))))})
199	198	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑒 = (𝐸 / (2↑(1st ‘(𝐹‘𝑚))))) → {𝑖 ∈ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 𝑒)} = {𝑖 ∈ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 (𝐸 / (2↑(1st ‘(𝐹‘𝑚)))))})
200	17, 70	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐸 ∈ ℝ+)
201		2rp 12664	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ+
202	201	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 2 ∈ ℝ+)
203	27	nnzd 12354	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐹‘𝑚)) ∈ ℤ)
204	202, 203	rpexpcld 13890	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2↑(1st ‘(𝐹‘𝑚))) ∈ ℝ+)
205	200, 204	rpdivcld 12718	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐸 / (2↑(1st ‘(𝐹‘𝑚)))) ∈ ℝ+)
206		fvex 6769	. . . . . . . . . . . . . . 15 ⊢ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))) ∈ V
207	206	rabex 5251	. . . . . . . . . . . . . 14 ⊢ {𝑖 ∈ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 (𝐸 / (2↑(1st ‘(𝐹‘𝑚)))))} ∈ V
208	207	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → {𝑖 ∈ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 (𝐸 / (2↑(1st ‘(𝐹‘𝑚)))))} ∈ V)
209	195, 199, 205, 208	fvmptd 6864	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐷‘(𝐴‘(1st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1st ‘(𝐹‘𝑚))))) = {𝑖 ∈ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 (𝐸 / (2↑(1st ‘(𝐹‘𝑚)))))})
210		ssrab2 4009	. . . . . . . . . . . . 13 ⊢ {𝑖 ∈ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 (𝐸 / (2↑(1st ‘(𝐹‘𝑚)))))} ⊆ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚))))
211	210	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → {𝑖 ∈ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 (𝐸 / (2↑(1st ‘(𝐹‘𝑚)))))} ⊆ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))))
212	209, 211	eqsstrd 3955	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐷‘(𝐴‘(1st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1st ‘(𝐹‘𝑚))))) ⊆ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))))
213	37	anbi2d 628	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → ((𝜑 ∧ 𝑛 ∈ ℕ) ↔ (𝜑 ∧ (1st ‘(𝐹‘𝑚)) ∈ ℕ)))
214		2fveq3 6761	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → (𝐷‘(𝐴‘𝑛)) = (𝐷‘(𝐴‘(1st ‘(𝐹‘𝑚)))))
215		oveq2 7263	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → (2↑𝑛) = (2↑(1st ‘(𝐹‘𝑚))))
216	215	oveq2d 7271	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → (𝐸 / (2↑𝑛)) = (𝐸 / (2↑(1st ‘(𝐹‘𝑚)))))
217	214, 216	fveq12d 6763	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) = ((𝐷‘(𝐴‘(1st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1st ‘(𝐹‘𝑚))))))
218	39, 217	eleq12d 2833	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → ((𝐼‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ↔ (𝐼‘(1st ‘(𝐹‘𝑚))) ∈ ((𝐷‘(𝐴‘(1st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1st ‘(𝐹‘𝑚)))))))
219	213, 218	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → (((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) ↔ ((𝜑 ∧ (1st ‘(𝐹‘𝑚)) ∈ ℕ) → (𝐼‘(1st ‘(𝐹‘𝑚))) ∈ ((𝐷‘(𝐴‘(1st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1st ‘(𝐹‘𝑚))))))))
220	36, 219, 86	vtocl 3488	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (1st ‘(𝐹‘𝑚)) ∈ ℕ) → (𝐼‘(1st ‘(𝐹‘𝑚))) ∈ ((𝐷‘(𝐴‘(1st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1st ‘(𝐹‘𝑚))))))
221	17, 27, 220	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1st ‘(𝐹‘𝑚))) ∈ ((𝐷‘(𝐴‘(1st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1st ‘(𝐹‘𝑚))))))
222	212, 221	sseldd 3918	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1st ‘(𝐹‘𝑚))) ∈ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))))
223	184, 222	sseldd 3918	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1st ‘(𝐹‘𝑚))) ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
224		elmapfn 8611	. . . . . . . . 9 ⊢ ((𝐼‘(1st ‘(𝐹‘𝑚))) ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝐼‘(1st ‘(𝐹‘𝑚))) Fn ℕ)
225	223, 224	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1st ‘(𝐹‘𝑚))) Fn ℕ)
226		elmapi 8595	. . . . . . . . . . 11 ⊢ ((𝐼‘(1st ‘(𝐹‘𝑚))) ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝐼‘(1st ‘(𝐹‘𝑚))):ℕ⟶((ℝ × ℝ) ↑m 𝑋))
227	223, 226	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1st ‘(𝐹‘𝑚))):ℕ⟶((ℝ × ℝ) ↑m 𝑋))
228	227	ffvelrnda 6943	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1st ‘(𝐹‘𝑚)))‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
229	228	ralrimiva 3107	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑗 ∈ ℕ ((𝐼‘(1st ‘(𝐹‘𝑚)))‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
230	225, 229	jca 511	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐼‘(1st ‘(𝐹‘𝑚))) Fn ℕ ∧ ∀𝑗 ∈ ℕ ((𝐼‘(1st ‘(𝐹‘𝑚)))‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋)))
231		ffnfv 6974	. . . . . . 7 ⊢ ((𝐼‘(1st ‘(𝐹‘𝑚))):ℕ⟶((ℝ × ℝ) ↑m 𝑋) ↔ ((𝐼‘(1st ‘(𝐹‘𝑚))) Fn ℕ ∧ ∀𝑗 ∈ ℕ ((𝐼‘(1st ‘(𝐹‘𝑚)))‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋)))
232	230, 231	sylibr 233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1st ‘(𝐹‘𝑚))):ℕ⟶((ℝ × ℝ) ↑m 𝑋))
233	232, 29	ffvelrnd 6944	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))) ∈ ((ℝ × ℝ) ↑m 𝑋))
234	233, 109	fmptd 6970	. . . 4 ⊢ (𝜑 → 𝐺:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
235		simpl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝜑)
236	86, 82	eleqtrd 2841	. . . . . . . . . 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 3915	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛)))
238		simp3 1136	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛))) → (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛)))
239	49	3adant3 1130	. . . . . . . . . . . 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 2841	. . . . . . . . . . 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 6755	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = (𝐼‘𝑛) → (ℎ‘𝑗) = ((𝐼‘𝑛)‘𝑗))
242	241	coeq2d 5760	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝐼‘𝑛) → ([,) ∘ (ℎ‘𝑗)) = ([,) ∘ ((𝐼‘𝑛)‘𝑗)))
243	242	fveq1d 6758	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝐼‘𝑛) → (([,) ∘ (ℎ‘𝑗))‘𝑘) = (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
244	243	ixpeq2dv 8659	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝐼‘𝑛) → X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
245	244	iuneq2d 4950	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝐼‘𝑛) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
246	245	sseq2d 3949	. . . . . . . . . . . 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 3617	. . . . . . . . . . 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 495	. . . . . . . . 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 1369	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
251		f1ofo 6707	. . . . . . . . . . . . . 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 722	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → 𝐹:ℕ–onto→(ℕ × ℕ))
254		opelxpi 5617	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ))
255	4, 254	sylan 579	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ))
256		foelrni 6813	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ–onto→(ℕ × ℕ) ∧ ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ)) → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩)
257	253, 255, 256	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩)
258		nfv 1918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ)
259		nfre1 3234	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚∃𝑚 ∈ {𝑖 ∈ ℕ ∣ (1st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘)
260		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → 𝑚 ∈ ℕ)
261		fveq2 6756	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (1st ‘(𝐹‘𝑚)) = (1st ‘⟨𝑛, 𝑗⟩))
262		op1stg 7816	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ V ∧ 𝑗 ∈ V) → (1st ‘⟨𝑛, 𝑗⟩) = 𝑛)
263	262	el2v 3430	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1st ‘⟨𝑛, 𝑗⟩) = 𝑛
264	263	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (1st ‘⟨𝑛, 𝑗⟩) = 𝑛)
265	261, 264	eqtrd 2778	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (1st ‘(𝐹‘𝑚)) = 𝑛)
266	265	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → (1st ‘(𝐹‘𝑚)) = 𝑛)
267	260, 266	jca 511	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → (𝑚 ∈ ℕ ∧ (1st ‘(𝐹‘𝑚)) = 𝑛))
268		2fveq3 6761	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑚 → (1st ‘(𝐹‘𝑖)) = (1st ‘(𝐹‘𝑚)))
269	268	eqeq1d 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑚 → ((1st ‘(𝐹‘𝑖)) = 𝑛 ↔ (1st ‘(𝐹‘𝑚)) = 𝑛))
270	269	elrab 3617	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ {𝑖 ∈ ℕ ∣ (1st ‘(𝐹‘𝑖)) = 𝑛} ↔ (𝑚 ∈ ℕ ∧ (1st ‘(𝐹‘𝑚)) = 𝑛))
271	267, 270	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1st ‘(𝐹‘𝑖)) = 𝑛})
272	271	3adant1 1128	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1st ‘(𝐹‘𝑖)) = 𝑛})
273	260, 111	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → (𝐺‘𝑚) = ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))))
274	265	fveq2d 6760	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (𝐼‘(1st ‘(𝐹‘𝑚))) = (𝐼‘𝑛))
275		vex 3426	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑛 ∈ V
276		vex 3426	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑗 ∈ V
277	275, 276	op2ndd 7815	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (2nd ‘(𝐹‘𝑚)) = 𝑗)
278	274, 277	fveq12d 6763	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))) = ((𝐼‘𝑛)‘𝑗))
279	278	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))) = ((𝐼‘𝑛)‘𝑗))
280	273, 279	eqtr2d 2779	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → ((𝐼‘𝑛)‘𝑗) = (𝐺‘𝑚))
281	280	coeq2d 5760	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → ([,) ∘ ((𝐼‘𝑛)‘𝑗)) = ([,) ∘ (𝐺‘𝑚)))
282	281	fveq1d 6758	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) = (([,) ∘ (𝐺‘𝑚))‘𝑘))
283	282	ixpeq2dv 8659	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
284		eqimss 3973	. . . . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . . 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 3232	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ {𝑖 ∈ ℕ ∣ (1st ‘(𝐹‘𝑖)) = 𝑛} ∧ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘)) → ∃𝑚 ∈ {𝑖 ∈ ℕ ∣ (1st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
288	272, 286, 287	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → ∃𝑚 ∈ {𝑖 ∈ ℕ ∣ (1st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
289	288	3exp 1117	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (𝑚 ∈ ℕ → ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → ∃𝑚 ∈ {𝑖 ∈ ℕ ∣ (1st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))))
290	258, 259, 289	rexlimd 3245	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (∃𝑚 ∈ ℕ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → ∃𝑚 ∈ {𝑖 ∈ ℕ ∣ (1st ‘(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 ∣ (1st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
292	291	ralrimiva 3107	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑗 ∈ ℕ ∃𝑚 ∈ {𝑖 ∈ ℕ ∣ (1st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
293		iunss2 4975	. . . . . . . . 9 ⊢ (∀𝑗 ∈ ℕ ∃𝑚 ∈ {𝑖 ∈ ℕ ∣ (1st ‘(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 ∣ (1st ‘(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 ∣ (1st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
295	250, 294	sstrd 3927	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
296		ssrab2 4009	. . . . . . . . 9 ⊢ {𝑖 ∈ ℕ ∣ (1st ‘(𝐹‘𝑖)) = 𝑛} ⊆ ℕ
297		iunss1 4935	. . . . . . . . 9 ⊢ ({𝑖 ∈ ℕ ∣ (1st ‘(𝐹‘𝑖)) = 𝑛} ⊆ ℕ → ∪ 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1st ‘(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 ∣ (1st ‘(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 ∣ (1st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
300	295, 299	sstrd 3927	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
301	300	ralrimiva 3107	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
302		iunss 4971	. . . . 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 43986	. . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))))
305	112	fveq2d 6760	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐿‘(𝐺‘𝑚)) = (𝐿‘((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚)))))
306	305	mpteq2dva 5170	. . . . . 6 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚))) = (𝑚 ∈ ℕ ↦ (𝐿‘((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))))))
307	306	fveq2d 6760	. . . . 5 ⊢ (𝜑 → (Σ^‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))) = (Σ^‘(𝑚 ∈ ℕ ↦ (𝐿‘((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚)))))))
308		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑝𝜑
309		2fveq3 6761	. . . . . . . 8 ⊢ (𝑝 = (𝐹‘𝑚) → (𝐼‘(1st ‘𝑝)) = (𝐼‘(1st ‘(𝐹‘𝑚))))
310		fveq2 6756	. . . . . . . 8 ⊢ (𝑝 = (𝐹‘𝑚) → (2nd ‘𝑝) = (2nd ‘(𝐹‘𝑚)))
311	309, 310	fveq12d 6763	. . . . . . 7 ⊢ (𝑝 = (𝐹‘𝑚) → ((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝)) = ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))))
312	311	fveq2d 6760	. . . . . 6 ⊢ (𝑝 = (𝐹‘𝑚) → (𝐿‘((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝))) = (𝐿‘((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚)))))
313		eqidd 2739	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) = (𝐹‘𝑚))
314		nfv 1918	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑝 ∈ (ℕ × ℕ))
315	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → 𝑋 ∈ Fin)
316		simpl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → 𝜑)
317		xp1st 7836	. . . . . . . . . 10 ⊢ (𝑝 ∈ (ℕ × ℕ) → (1st ‘𝑝) ∈ ℕ)
318	317	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → (1st ‘𝑝) ∈ ℕ)
319		xp2nd 7837	. . . . . . . . . 10 ⊢ (𝑝 ∈ (ℕ × ℕ) → (2nd ‘𝑝) ∈ ℕ)
320	319	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → (2nd ‘𝑝) ∈ ℕ)
321		fvex 6769	. . . . . . . . . 10 ⊢ (2nd ‘𝑝) ∈ V
322		eleq1 2826	. . . . . . . . . . . 12 ⊢ (𝑗 = (2nd ‘𝑝) → (𝑗 ∈ ℕ ↔ (2nd ‘𝑝) ∈ ℕ))
323	322	3anbi3d 1440	. . . . . . . . . . 11 ⊢ (𝑗 = (2nd ‘𝑝) → ((𝜑 ∧ (1st ‘𝑝) ∈ ℕ ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ (1st ‘𝑝) ∈ ℕ ∧ (2nd ‘𝑝) ∈ ℕ)))
324		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑗 = (2nd ‘𝑝) → ((𝐼‘(1st ‘𝑝))‘𝑗) = ((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝)))
325	324	feq1d 6569	. . . . . . . . . . 11 ⊢ (𝑗 = (2nd ‘𝑝) → (((𝐼‘(1st ‘𝑝))‘𝑗):𝑋⟶(ℝ × ℝ) ↔ ((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝)):𝑋⟶(ℝ × ℝ)))
326	323, 325	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑗 = (2nd ‘𝑝) → (((𝜑 ∧ (1st ‘𝑝) ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1st ‘𝑝))‘𝑗):𝑋⟶(ℝ × ℝ)) ↔ ((𝜑 ∧ (1st ‘𝑝) ∈ ℕ ∧ (2nd ‘𝑝) ∈ ℕ) → ((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝)):𝑋⟶(ℝ × ℝ))))
327		fvex 6769	. . . . . . . . . . 11 ⊢ (1st ‘𝑝) ∈ V
328		eleq1 2826	. . . . . . . . . . . . 13 ⊢ (𝑛 = (1st ‘𝑝) → (𝑛 ∈ ℕ ↔ (1st ‘𝑝) ∈ ℕ))
329	328	3anbi2d 1439	. . . . . . . . . . . 12 ⊢ (𝑛 = (1st ‘𝑝) → ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ (1st ‘𝑝) ∈ ℕ ∧ 𝑗 ∈ ℕ)))
330		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (1st ‘𝑝) → (𝐼‘𝑛) = (𝐼‘(1st ‘𝑝)))
331	330	fveq1d 6758	. . . . . . . . . . . . 13 ⊢ (𝑛 = (1st ‘𝑝) → ((𝐼‘𝑛)‘𝑗) = ((𝐼‘(1st ‘𝑝))‘𝑗))
332	331	feq1d 6569	. . . . . . . . . . . 12 ⊢ (𝑛 = (1st ‘𝑝) → (((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ) ↔ ((𝐼‘(1st ‘𝑝))‘𝑗):𝑋⟶(ℝ × ℝ)))
333	329, 332	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑛 = (1st ‘𝑝) → (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ)) ↔ ((𝜑 ∧ (1st ‘𝑝) ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1st ‘𝑝))‘𝑗):𝑋⟶(ℝ × ℝ))))
334	327, 333, 102	vtocl 3488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (1st ‘𝑝) ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1st ‘𝑝))‘𝑗):𝑋⟶(ℝ × ℝ))
335	321, 326, 334	vtocl 3488	. . . . . . . . 9 ⊢ ((𝜑 ∧ (1st ‘𝑝) ∈ ℕ ∧ (2nd ‘𝑝) ∈ ℕ) → ((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝)):𝑋⟶(ℝ × ℝ))
336	316, 318, 320, 335	syl3anc 1369	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → ((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝)):𝑋⟶(ℝ × ℝ))
337	314, 315, 19, 336	hoiprodcl2 43983	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → (𝐿‘((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝))) ∈ (0[,)+∞))
338	15, 337	sselid 3915	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → (𝐿‘((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝))) ∈ (0[,]+∞))
339	308, 12, 312, 14, 20, 313, 338	sge0f1o 43810	. . . . 5 ⊢ (𝜑 → (Σ^‘(𝑝 ∈ (ℕ × ℕ) ↦ (𝐿‘((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝))))) = (Σ^‘(𝑚 ∈ ℕ ↦ (𝐿‘((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚)))))))
340	307, 339	eqtr4d 2781	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))) = (Σ^‘(𝑝 ∈ (ℕ × ℕ) ↦ (𝐿‘((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝))))))
341		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑗𝜑
342	275, 276	op1std 7814	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → (1st ‘𝑝) = 𝑛)
343	342	fveq2d 6760	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → (𝐼‘(1st ‘𝑝)) = (𝐼‘𝑛))
344	275, 276	op2ndd 7815	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → (2nd ‘𝑝) = 𝑗)
345	343, 344	fveq12d 6763	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → ((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝)) = ((𝐼‘𝑛)‘𝑗))
346	345	fveq2d 6760	. . . . . . 7 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → (𝐿‘((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝))) = (𝐿‘((𝐼‘𝑛)‘𝑗)))
347		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ)
348	123	adantr 480	. . . . . . . . . 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 43983	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (𝐿‘((𝐼‘𝑛)‘𝑗)) ∈ (0[,)+∞))
351	15, 350	sselid 3915	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (𝐿‘((𝐼‘𝑛)‘𝑗)) ∈ (0[,]+∞))
352	351	3impa 1108	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝐿‘((𝐼‘𝑛)‘𝑗)) ∈ (0[,]+∞))
353	341, 346, 14, 14, 352	sge0xp 43857	. . . . . 6 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))))) = (Σ^‘(𝑝 ∈ (ℕ × ℕ) ↦ (𝐿‘((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝))))))
354	353	eqcomd 2744	. . . . 5 ⊢ (𝜑 → (Σ^‘(𝑝 ∈ (ℕ × ℕ) ↦ (𝐿‘((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝))))) = (Σ^‘(𝑛 ∈ ℕ ↦ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))))))
355	13	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ℕ ∈ V)
356		eqid 2738	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))
357	351, 356	fmptd 6970	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗))):ℕ⟶(0[,]+∞))
358	355, 357	sge0cl 43809	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))) ∈ (0[,]+∞))
359		fveq1 6755	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝐼‘𝑛) → (𝑖‘𝑗) = ((𝐼‘𝑛)‘𝑗))
360	359	fveq2d 6760	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼‘𝑛) → (𝐿‘(𝑖‘𝑗)) = (𝐿‘((𝐼‘𝑛)‘𝑗)))
361	360	mpteq2dv 5172	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘𝑛) → (𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗))))
362	361	fveq2d 6760	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘𝑛) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))))
363	362	breq1d 5080	. . . . . . . . 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 3617	. . . . . . . 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 495	. . . . . 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 43838	. . . . 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 5092	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑝 ∈ (ℕ × ℕ) ↦ (𝐿‘((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝))))) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (((voln*‘𝑋)‘(𝐴‘𝑛)) +𝑒 (𝐸 / (2↑𝑛))))))
369	340, 368	eqbrtrd 5092	. . 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 12825	. 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 43863	. . 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 10956	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ*)
375	70	rpge0d 12705	. . . . . 6 ⊢ (𝜑 → 0 ≤ 𝐸)
376	145	ltpnfd 12786	. . . . . 6 ⊢ (𝜑 → 𝐸 < +∞)
377	372, 373, 374, 375, 376	elicod 13058	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (0[,)+∞))
378	377	sge0ad2en 43859	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝐸 / (2↑𝑛)))) = 𝐸)
379	378	oveq2d 7271	. . 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 2778	. 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 5096	1 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) +𝑒 𝐸))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  ifcif 4456  𝒫 cpw 4530  ⟨cop 4564  ∪ ciun 4921   class class class wbr 5070   ↦ cmpt 5153   × cxp 5578   ∘ ccom 5584   Fn wfn 6413  ⟶wf 6414  –onto→wfo 6416  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803   ↑_m cmap 8573  Xcixp 8643  Fincfn 8691  infcinf 9130  ℝcr 10801  0cc0 10802  +∞cpnf 10937  ℝ^*cxr 10939   < clt 10940   ≤ cle 10941   / cdiv 11562  ℕcn 11903  2c2 11958  ℝ⁺crp 12659   +_𝑒 cxad 12775  [,)cico 13010  [,]cicc 13011  ↑cexp 13710  ∏cprod 15543  volcvol 24532  Σ^{^}csumge0 43790  voln*covoln 43964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-ac2 10150  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880  ax-addf 10881  ax-mulf 10882

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-disj 5036  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-tpos 8013  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-map 8575  df-pm 8576  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fi 9100  df-sup 9131  df-inf 9132  df-oi 9199  df-dju 9590  df-card 9628  df-acn 9631  df-ac 9803  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-q 12618  df-rp 12660  df-xneg 12777  df-xadd 12778  df-xmul 12779  df-ioo 13012  df-ico 13014  df-icc 13015  df-fz 13169  df-fzo 13312  df-fl 13440  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-rlim 15126  df-sum 15326  df-prod 15544  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-starv 16903  df-tset 16907  df-ple 16908  df-ds 16910  df-unif 16911  df-rest 17050  df-0g 17069  df-topgen 17071  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-minusg 18496  df-subg 18667  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-ring 19700  df-cring 19701  df-oppr 19777  df-dvdsr 19798  df-unit 19799  df-invr 19829  df-dvr 19840  df-drng 19908  df-psmet 20502  df-xmet 20503  df-met 20504  df-bl 20505  df-mopn 20506  df-cnfld 20511  df-top 21951  df-topon 21968  df-bases 22004  df-cmp 22446  df-ovol 24533  df-vol 24534  df-sumge0 43791  df-ovoln 43965

This theorem is referenced by:  ovnsubaddlem2  43999