ovnsubaddlem1 - Mathbox for Glauco Siliprandi

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Theorem ovnsubaddlem1 39460

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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ ↑_𝑚 𝑋))
4		simpr 476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
5	3, 4	ffvelrnd 6268	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ 𝒫 (ℝ ↑_𝑚 𝑋))
6		elpwi 4117	. . . . . . 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 2949	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑_𝑚 𝑋))
9		iunss 4497	. . . . 5 ⊢ (∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑_𝑚 𝑋) ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑_𝑚 𝑋))
10	8, 9	sylibr 223	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑_𝑚 𝑋))
11	1, 10	ovnxrcl 39459	. . 3 ⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ∈ ℝ^)
12		nfv 1830	. . . 4 ⊢ Ⅎ𝑚𝜑
13		nnex 10903	. . . . 5 ⊢ ℕ ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝜑 → ℕ ∈ V)
15		icossicc 12131	. . . . 5 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
16		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑚 ∈ ℕ)
17		simpl 472	. . . . . . 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 6050	. . . . . . . . . . . 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 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
25	23, 24	ffvelrnd 6268	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∈ (ℕ × ℕ))
26		xp1st 7089	. . . . . . . . 9 ⊢ ((𝐹‘𝑚) ∈ (ℕ × ℕ) → (1^st ‘(𝐹‘𝑚)) ∈ ℕ)
27	25, 26	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1^st ‘(𝐹‘𝑚)) ∈ ℕ)
28		xp2nd 7090	. . . . . . . . 9 ⊢ ((𝐹‘𝑚) ∈ (ℕ × ℕ) → (2^nd ‘(𝐹‘𝑚)) ∈ ℕ)
29	25, 28	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2^nd ‘(𝐹‘𝑚)) ∈ ℕ)
30		fvex 6113	. . . . . . . . 9 ⊢ (2^nd ‘(𝐹‘𝑚)) ∈ V
31		eleq1 2676	. . . . . . . . . . 11 ⊢ (𝑗 = (2^nd ‘(𝐹‘𝑚)) → (𝑗 ∈ ℕ ↔ (2^nd ‘(𝐹‘𝑚)) ∈ ℕ))
32	31	3anbi3d 1397	. . . . . . . . . 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 6103	. . . . . . . . . . 11 ⊢ (𝑗 = (2^nd ‘(𝐹‘𝑚)) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗) = ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))
34	33	feq1d 5943	. . . . . . . . . 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 333	. . . . . . . . 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 6113	. . . . . . . . . 10 ⊢ (1^st ‘(𝐹‘𝑚)) ∈ V
37		eleq1 2676	. . . . . . . . . . . 12 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → (𝑛 ∈ ℕ ↔ (1^st ‘(𝐹‘𝑚)) ∈ ℕ))
38	37	3anbi2d 1396	. . . . . . . . . . 11 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ (1^st ‘(𝐹‘𝑚)) ∈ ℕ ∧ 𝑗 ∈ ℕ)))
39		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → (𝐼‘𝑛) = (𝐼‘(1^st ‘(𝐹‘𝑚))))
40	39	fveq1d 6105	. . . . . . . . . . . 12 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → ((𝐼‘𝑛)‘𝑗) = ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗))
41	40	feq1d 5943	. . . . . . . . . . 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 333	. . . . . . . . . 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	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑_𝑚 𝑋) ↦ {ℎ ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)})
44	43	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑_𝑚 𝑋) ↦ {ℎ ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)}))
45		sseq1 3589	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (𝐴‘𝑛) → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ↔ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)))
46	45	rabbidv 3164	. . . . . . . . . . . . . . . . . . . 20 ⊢ (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)})
47	46	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 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)})
48		ovex 6577	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∈ V
49	48	rabex 4740	. . . . . . . . . . . . . . . . . . . 20 ⊢ {ℎ ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∣ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)} ∈ V
50	49	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {ℎ ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∣ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)} ∈ V)
51	44, 47, 5, 50	fvmptd 6197	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘(𝐴‘𝑛)) = {ℎ ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∣ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)})
52		ssrab2 3650	. . . . . . . . . . . . . . . . . . 19 ⊢ {ℎ ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∣ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)} ⊆ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)
53	52	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))
54	51, 53	eqsstrd 3602	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘(𝐴‘𝑛)) ⊆ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ))
55		ovnsubaddlem1.d	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑_𝑚 𝑋) ↦ (𝑒 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +_𝑒 𝑒)}))
56	55	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑_𝑚 𝑋) ↦ (𝑒 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +_𝑒 𝑒)})))
57		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = (𝐴‘𝑛) → (𝐶‘𝑎) = (𝐶‘(𝐴‘𝑛)))
58	57	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (𝐴‘𝑛) → (𝑖 ∈ (𝐶‘𝑎) ↔ 𝑖 ∈ (𝐶‘(𝐴‘𝑛))))
59		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = (𝐴‘𝑛) → ((voln‘𝑋)‘𝑎) = ((voln‘𝑋)‘(𝐴‘𝑛)))
60	59	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = (𝐴‘𝑛) → (((voln‘𝑋)‘𝑎) +_𝑒 𝑒) = (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 𝑒))
61	60	breq2d 4595	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (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)))
62	58, 61	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (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))))
63	62	rabbidva2 3162	. . . . . . . . . . . . . . . . . . . . . . 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)})
64	63	mpteq2dv 4673	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (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)}))
65	64	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 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)}))
66		rpex 38503	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℝ⁺ ∈ V
67	66	mptex 6390	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑒 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 𝑒)}) ∈ V
68	67	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)
69	56, 65, 5, 68	fvmptd 6197	. . . . . . . . . . . . . . . . . . . 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)}))
70		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 = (𝐸 / (2↑𝑛)) → (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 𝑒) = (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))))
71	70	breq2d 4595	. . . . . . . . . . . . . . . . . . . . . 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)))))
72	71	rabbidv 3164	. . . . . . . . . . . . . . . . . . . . 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)))})
73	72	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)))})
74		ovnsubaddlem1.e	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
75	74	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ⁺)
76		2nn 11062	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 ∈ ℕ
77	76	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℕ)
78		nnnn0 11176	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
79	77, 78	nnexpcld 12892	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
80	79	nnrpd 11746	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ⁺)
81	80	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ⁺)
82	75, 81	rpdivcld 11765	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ ℝ⁺)
83		fvex 6113	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐶‘(𝐴‘𝑛)) ∈ V
84	83	rabex 4740	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛)))} ∈ V
85	84	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)
86	69, 73, 82, 85	fvmptd 6197	. . . . . . . . . . . . . . . . . . 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)))})
87		ssrab2 3650	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛)))} ⊆ (𝐶‘(𝐴‘𝑛))
88	87	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)))
89	86, 88	eqsstrd 3602	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ⊆ (𝐶‘(𝐴‘𝑛)))
90		ovnsubaddlem1.i	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
91	89, 90	sseldd 3569	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛)))
92	54, 91	sseldd 3569	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ))
93		elmapfn 7766	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼‘𝑛) ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝐼‘𝑛) Fn ℕ)
94	92, 93	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) Fn ℕ)
95		elmapi 7765	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼‘𝑛) ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝐼‘𝑛):ℕ⟶((ℝ × ℝ) ↑_𝑚 𝑋))
96	92, 95	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛):ℕ⟶((ℝ × ℝ) ↑_𝑚 𝑋))
97	96	ffvelrnda 6267	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋))
98	97	ralrimiva 2949	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑗 ∈ ℕ ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋))
99	94, 98	jca 553	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐼‘𝑛) Fn ℕ ∧ ∀𝑗 ∈ ℕ ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋)))
100	99	3adant3 1074	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛) Fn ℕ ∧ ∀𝑗 ∈ ℕ ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋)))
101		ffnfv 6295	. . . . . . . . . . . . 13 ⊢ ((𝐼‘𝑛):ℕ⟶((ℝ × ℝ) ↑_𝑚 𝑋) ↔ ((𝐼‘𝑛) Fn ℕ ∧ ∀𝑗 ∈ ℕ ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋)))
102	100, 101	sylibr 223	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑛):ℕ⟶((ℝ × ℝ) ↑_𝑚 𝑋))
103		simp3 1056	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
104	102, 103	ffvelrnd 6268	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋))
105		elmapi 7765	. . . . . . . . . . 11 ⊢ (((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋) → ((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ))
106	104, 105	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ))
107	36, 42, 106	vtocl 3232	. . . . . . . . 9 ⊢ ((𝜑 ∧ (1^st ‘(𝐹‘𝑚)) ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗):𝑋⟶(ℝ × ℝ))
108	30, 35, 107	vtocl 3232	. . . . . . . 8 ⊢ ((𝜑 ∧ (1^st ‘(𝐹‘𝑚)) ∈ ℕ ∧ (2^nd ‘(𝐹‘𝑚)) ∈ ℕ) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))):𝑋⟶(ℝ × ℝ))
109	17, 27, 29, 108	syl3anc 1318	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))):𝑋⟶(ℝ × ℝ))
110		id 22	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ)
111		fvex 6113	. . . . . . . . . . 11 ⊢ ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))) ∈ V
112	111	a1i 11	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))) ∈ V)
113		ovnsubaddlem1.g	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))
114	113	fvmpt2 6200	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))) ∈ V) → (𝐺‘𝑚) = ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))
115	110, 112, 114	syl2anc 691	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (𝐺‘𝑚) = ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))
116	115	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) = ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))
117	116	feq1d 5943	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐺‘𝑚):𝑋⟶(ℝ × ℝ) ↔ ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))):𝑋⟶(ℝ × ℝ)))
118	109, 117	mpbird 246	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚):𝑋⟶(ℝ × ℝ))
119	16, 18, 19, 118	hoiprodcl2 39445	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐿‘(𝐺‘𝑚)) ∈ (0[,)+∞))
120	15, 119	sseldi 3566	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐿‘(𝐺‘𝑚)) ∈ (0[,]+∞))
121	12, 14, 120	sge0xrclmpt 39321	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))) ∈ ℝ^*)
122		nfv 1830	. . . 4 ⊢ Ⅎ𝑛𝜑
123		0xr 9965	. . . . . 6 ⊢ 0 ∈ ℝ^*
124	123	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ^*)
125		pnfxr 9971	. . . . . 6 ⊢ +∞ ∈ ℝ^*
126	125	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ^*)
127	1	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
128		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)))))})
129	127, 7, 128	ovnval2b 39442	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln‘𝑋)‘(𝐴‘𝑛)) = if(𝑋 = ∅, 0, inf((𝑍‘(𝐴‘𝑛)), ℝ^, < )))
130		ovnsubaddlem1.n0	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ≠ ∅)
131	130	neneqd 2787	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 = ∅)
132	131	iffalsed 4047	. . . . . . . . 9 ⊢ (𝜑 → if(𝑋 = ∅, 0, inf((𝑍‘(𝐴‘𝑛)), ℝ^, < )) = inf((𝑍‘(𝐴‘𝑛)), ℝ^, < ))
133	132	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(𝑋 = ∅, 0, inf((𝑍‘(𝐴‘𝑛)), ℝ^, < )) = inf((𝑍‘(𝐴‘𝑛)), ℝ^, < ))
134	129, 133	eqtrd 2644	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln‘𝑋)‘(𝐴‘𝑛)) = inf((𝑍‘(𝐴‘𝑛)), ℝ^, < ))
135	128	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 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)))))}))
136		sseq1 3589	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝐴‘𝑛) → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ↔ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)))
137	136	anbi1d 737	. . . . . . . . . . . . 13 ⊢ (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)))))))
138	137	rexbidv 3034	. . . . . . . . . . . 12 ⊢ (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)))))))
139	138	rabbidv 3164	. . . . . . . . . . 11 ⊢ (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)))))})
140	139	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 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)))))})
141		xrex 11705	. . . . . . . . . . . 12 ⊢ ℝ^* ∈ V
142	141	rabex 4740	. . . . . . . . . . 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
143	142	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)
144	135, 140, 5, 143	fvmptd 6197	. . . . . . . . 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)))))})
145		ssrab2 3650	. . . . . . . . . 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^*
146	145	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^*)
147	144, 146	eqsstrd 3602	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑍‘(𝐴‘𝑛)) ⊆ ℝ^*)
148		infxrcl 12035	. . . . . . . 8 ⊢ ((𝑍‘(𝐴‘𝑛)) ⊆ ℝ^* → inf((𝑍‘(𝐴‘𝑛)), ℝ^, < ) ∈ ℝ^)
149	147, 148	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → inf((𝑍‘(𝐴‘𝑛)), ℝ^, < ) ∈ ℝ^)
150	134, 149	eqeltrd 2688	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln‘𝑋)‘(𝐴‘𝑛)) ∈ ℝ^)
151	74	rpred 11748	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℝ)
152	151	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ)
153		2re 10967	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
154	153	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ)
155	154, 78	reexpcld 12887	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ)
156	155	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ)
157	154	recnd 9947	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℂ)
158		2ne0 10990	. . . . . . . . . . 11 ⊢ 2 ≠ 0
159	158	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ≠ 0)
160		nnz 11276	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
161	157, 159, 160	expne0d 12876	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ≠ 0)
162	161	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ≠ 0)
163	152, 156, 162	redivcld 10732	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ ℝ)
164	163	rexrd 9968	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ ℝ^*)
165	150, 164	xaddcld 12003	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))) ∈ ℝ^)
166	127, 7	ovncl 39457	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln*‘𝑋)‘(𝐴‘𝑛)) ∈ (0[,]+∞))
167		xrge0ge0 38504	. . . . . . 7 ⊢ (((voln‘𝑋)‘(𝐴‘𝑛)) ∈ (0[,]+∞) → 0 ≤ ((voln‘𝑋)‘(𝐴‘𝑛)))
168	166, 167	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ ((voln*‘𝑋)‘(𝐴‘𝑛)))
169		0red 9920	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ)
170	82	rpgt0d 11751	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 < (𝐸 / (2↑𝑛)))
171	169, 163, 170	ltled 10064	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝐸 / (2↑𝑛)))
172	163	ltpnfd 11831	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) < +∞)
173	164, 126, 172	xrltled 38427	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ≤ +∞)
174	124, 126, 164, 171, 173	eliccxrd 38600	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ (0[,]+∞))
175	150, 174	xadd0ge 38477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln‘𝑋)‘(𝐴‘𝑛)) ≤ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))))
176	124, 150, 165, 168, 175	xrletrd 11869	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))))
177		pnfge 11840	. . . . . 6 ⊢ ((((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))) ∈ ℝ^ → (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))) ≤ +∞)
178	165, 177	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))) ≤ +∞)
179	124, 126, 165, 176, 178	eliccxrd 38600	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))) ∈ (0[,]+∞))
180	122, 14, 179	sge0xrclmpt 39321	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))))) ∈ ℝ^)
181	43	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑_𝑚 𝑋) ↦ {ℎ ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)}))
182		sseq1 3589	. . . . . . . . . . . . . 14 ⊢ (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)))
183	182	rabbidv 3164	. . . . . . . . . . . . 13 ⊢ (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)})
184	183	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 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)})
185	2	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ ↑_𝑚 𝑋))
186	185, 27	ffvelrnd 6268	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐴‘(1^st ‘(𝐹‘𝑚))) ∈ 𝒫 (ℝ ↑_𝑚 𝑋))
187	48	rabex 4740	. . . . . . . . . . . . 13 ⊢ {ℎ ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∣ (𝐴‘(1^st ‘(𝐹‘𝑚))) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)} ∈ V
188	187	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)
189	181, 184, 186, 188	fvmptd 6197	. . . . . . . . . . 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)})
190		ssrab2 3650	. . . . . . . . . . . 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)
191	190	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))
192	189, 191	eqsstrd 3602	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ⊆ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ))
193	55	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑_𝑚 𝑋) ↦ (𝑒 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +_𝑒 𝑒)})))
194		fveq2 6103	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (𝐴‘(1^st ‘(𝐹‘𝑚))) → (𝐶‘𝑎) = (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))))
195	194	eleq2d 2673	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (𝐴‘(1^st ‘(𝐹‘𝑚))) → (𝑖 ∈ (𝐶‘𝑎) ↔ 𝑖 ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚))))))
196		fveq2 6103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (𝐴‘(1^st ‘(𝐹‘𝑚))) → ((voln‘𝑋)‘𝑎) = ((voln‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))))
197	196	oveq1d 6564	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (𝐴‘(1^st ‘(𝐹‘𝑚))) → (((voln‘𝑋)‘𝑎) +_𝑒 𝑒) = (((voln‘𝑋)‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) +_𝑒 𝑒))
198	197	breq2d 4595	. . . . . . . . . . . . . . . . . 18 ⊢ (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)))
199	195, 198	anbi12d 743	. . . . . . . . . . . . . . . . 17 ⊢ (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))))
200	199	rabbidva2 3162	. . . . . . . . . . . . . . . 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)})
201	200	mpteq2dv 4673	. . . . . . . . . . . . . . 15 ⊢ (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)}))
202	201	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 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)}))
203	66	mptex 6390	. . . . . . . . . . . . . . 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
204	203	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)
205	193, 202, 186, 204	fvmptd 6197	. . . . . . . . . . . . 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)}))
206		oveq2 6557	. . . . . . . . . . . . . . . 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))))))
207	206	breq2d 4595	. . . . . . . . . . . . . . 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)))))))
208	207	rabbidv 3164	. . . . . . . . . . . . . 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)))))})
209	208	adantl 481	. . . . . . . . . . . . 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)))))})
210	17, 74	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐸 ∈ ℝ⁺)
211		2rp 11713	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ⁺
212	211	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 2 ∈ ℝ⁺)
213	27	nnzd 11357	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1^st ‘(𝐹‘𝑚)) ∈ ℤ)
214	212, 213	rpexpcld 12894	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2↑(1^st ‘(𝐹‘𝑚))) ∈ ℝ⁺)
215	210, 214	rpdivcld 11765	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐸 / (2↑(1^st ‘(𝐹‘𝑚)))) ∈ ℝ⁺)
216		fvex 6113	. . . . . . . . . . . . . . 15 ⊢ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))) ∈ V
217	216	rabex 4740	. . . . . . . . . . . . . 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
218	217	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)
219	205, 209, 215, 218	fvmptd 6197	. . . . . . . . . . . 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)))))})
220		ssrab2 3650	. . . . . . . . . . . . 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))))
221	220	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)))))
222	219, 221	eqsstrd 3602	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐷‘(𝐴‘(1^st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1^st ‘(𝐹‘𝑚))))) ⊆ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))))
223	37	anbi2d 736	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → ((𝜑 ∧ 𝑛 ∈ ℕ) ↔ (𝜑 ∧ (1^st ‘(𝐹‘𝑚)) ∈ ℕ)))
224		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → (𝐴‘𝑛) = (𝐴‘(1^st ‘(𝐹‘𝑚))))
225	224	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → (𝐷‘(𝐴‘𝑛)) = (𝐷‘(𝐴‘(1^st ‘(𝐹‘𝑚)))))
226		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → (2↑𝑛) = (2↑(1^st ‘(𝐹‘𝑚))))
227	226	oveq2d 6565	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → (𝐸 / (2↑𝑛)) = (𝐸 / (2↑(1^st ‘(𝐹‘𝑚)))))
228	225, 227	fveq12d 6109	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (1^st ‘(𝐹‘𝑚)) → ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) = ((𝐷‘(𝐴‘(1^st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1^st ‘(𝐹‘𝑚))))))
229	39, 228	eleq12d 2682	. . . . . . . . . . . . . 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)))))))
230	223, 229	imbi12d 333	. . . . . . . . . . . . 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))))))))
231	36, 230, 90	vtocl 3232	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (1^st ‘(𝐹‘𝑚)) ∈ ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))) ∈ ((𝐷‘(𝐴‘(1^st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1^st ‘(𝐹‘𝑚))))))
232	17, 27, 231	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))) ∈ ((𝐷‘(𝐴‘(1^st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1^st ‘(𝐹‘𝑚))))))
233	222, 232	sseldd 3569	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))) ∈ (𝐶‘(𝐴‘(1^st ‘(𝐹‘𝑚)))))
234	192, 233	sseldd 3569	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))) ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ))
235		elmapfn 7766	. . . . . . . . 9 ⊢ ((𝐼‘(1^st ‘(𝐹‘𝑚))) ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))) Fn ℕ)
236	234, 235	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))) Fn ℕ)
237		elmapi 7765	. . . . . . . . . . 11 ⊢ ((𝐼‘(1^st ‘(𝐹‘𝑚))) ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))):ℕ⟶((ℝ × ℝ) ↑_𝑚 𝑋))
238	234, 237	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))):ℕ⟶((ℝ × ℝ) ↑_𝑚 𝑋))
239	238	ffvelrnda 6267	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋))
240	239	ralrimiva 2949	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑗 ∈ ℕ ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋))
241	236, 240	jca 553	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐼‘(1^st ‘(𝐹‘𝑚))) Fn ℕ ∧ ∀𝑗 ∈ ℕ ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘𝑗) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋)))
242		ffnfv 6295	. . . . . . 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)))
243	241, 242	sylibr 223	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1^st ‘(𝐹‘𝑚))):ℕ⟶((ℝ × ℝ) ↑_𝑚 𝑋))
244	243, 29	ffvelrnd 6268	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋))
245	244, 113	fmptd 6292	. . . 4 ⊢ (𝜑 → 𝐺:ℕ⟶((ℝ × ℝ) ↑_𝑚 𝑋))
246		simpl 472	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝜑)
247	90, 86	eleqtrd 2690	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ {𝑖 ∈ (𝐶‘(𝐴‘𝑛)) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛)))})
248	87, 247	sseldi 3566	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛)))
249		simp3 1056	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛))) → (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛)))
250	51	3adant3 1074	. . . . . . . . . . . 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)})
251	249, 250	eleqtrd 2690	. . . . . . . . . . 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)})
252		fveq1 6102	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = (𝐼‘𝑛) → (ℎ‘𝑗) = ((𝐼‘𝑛)‘𝑗))
253	252	coeq2d 5206	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝐼‘𝑛) → ([,) ∘ (ℎ‘𝑗)) = ([,) ∘ ((𝐼‘𝑛)‘𝑗)))
254	253	fveq1d 6105	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝐼‘𝑛) → (([,) ∘ (ℎ‘𝑗))‘𝑘) = (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
255	254	ixpeq2dv 7810	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝐼‘𝑛) → X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
256	255	iuneq2d 4483	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝐼‘𝑛) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
257	256	sseq2d 3596	. . . . . . . . . . . 12 ⊢ (ℎ = (𝐼‘𝑛) → ((𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ↔ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘)))
258	257	elrab 3331	. . . . . . . . . . 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)))
259	251, 258	sylib 207	. . . . . . . . . 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)))
260	259	simprd 478	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛))) → (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
261	246, 4, 248, 260	syl3anc 1318	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
262		f1ofo 6057	. . . . . . . . . . . . . 14 ⊢ (𝐹:ℕ–1-1-onto→(ℕ × ℕ) → 𝐹:ℕ–onto→(ℕ × ℕ))
263	20, 262	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℕ–onto→(ℕ × ℕ))
264	263	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → 𝐹:ℕ–onto→(ℕ × ℕ))
265		opelxpi 5072	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ))
266	4, 265	sylan 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ))
267		foelrni 6154	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ–onto→(ℕ × ℕ) ∧ ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ)) → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩)
268	264, 266, 267	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩)
269		nfv 1830	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ)
270		nfre1 2988	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚∃𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘)
271		simpl 472	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → 𝑚 ∈ ℕ)
272		fveq2 6103	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (1^st ‘(𝐹‘𝑚)) = (1^st ‘⟨𝑛, 𝑗⟩))
273		vex 3176	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑛 ∈ V
274		vex 3176	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑗 ∈ V
275		op1stg 7071	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ V ∧ 𝑗 ∈ V) → (1^st ‘⟨𝑛, 𝑗⟩) = 𝑛)
276	273, 274, 275	mp2an 704	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1^st ‘⟨𝑛, 𝑗⟩) = 𝑛
277	276	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (1^st ‘⟨𝑛, 𝑗⟩) = 𝑛)
278	272, 277	eqtrd 2644	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (1^st ‘(𝐹‘𝑚)) = 𝑛)
279	278	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → (1^st ‘(𝐹‘𝑚)) = 𝑛)
280	271, 279	jca 553	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → (𝑚 ∈ ℕ ∧ (1^st ‘(𝐹‘𝑚)) = 𝑛))
281		fveq2 6103	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑚 → (𝐹‘𝑖) = (𝐹‘𝑚))
282	281	fveq2d 6107	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑚 → (1^st ‘(𝐹‘𝑖)) = (1^st ‘(𝐹‘𝑚)))
283	282	eqeq1d 2612	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑚 → ((1^st ‘(𝐹‘𝑖)) = 𝑛 ↔ (1^st ‘(𝐹‘𝑚)) = 𝑛))
284	283	elrab 3331	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛} ↔ (𝑚 ∈ ℕ ∧ (1^st ‘(𝐹‘𝑚)) = 𝑛))
285	280, 284	sylibr 223	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛})
286	285	3adant1 1072	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛})
287	271, 115	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → (𝐺‘𝑚) = ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))
288	278	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (𝐼‘(1^st ‘(𝐹‘𝑚))) = (𝐼‘𝑛))
289	273, 274	op2ndd 7070	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (2^nd ‘(𝐹‘𝑚)) = 𝑗)
290	288, 289	fveq12d 6109	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))) = ((𝐼‘𝑛)‘𝑗))
291	290	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))) = ((𝐼‘𝑛)‘𝑗))
292	287, 291	eqtr2d 2645	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → ((𝐼‘𝑛)‘𝑗) = (𝐺‘𝑚))
293	292	coeq2d 5206	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → ([,) ∘ ((𝐼‘𝑛)‘𝑗)) = ([,) ∘ (𝐺‘𝑚)))
294	293	fveq1d 6105	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) = (([,) ∘ (𝐺‘𝑚))‘𝑘))
295	294	ixpeq2dv 7810	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
296		eqimss 3620	. . . . . . . . . . . . . . . 16 ⊢ (X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘) → X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
297	295, 296	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
298	297	3adant1 1072	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
299		rspe 2986	. . . . . . . . . . . . . 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))
300	286, 298, 299	syl2anc 691	. . . . . . . . . . . . 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))
301	300	3exp 1256	. . . . . . . . . . . 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))))
302	269, 270, 301	rexlimd 3008	. . . . . . . . . . 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)))
303	268, 302	mpd 15	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ∃𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
304	303	ralrimiva 2949	. . . . . . . . 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))
305		iunss2 4501	. . . . . . . . 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))
306	304, 305	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ ∪ 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
307	261, 306	sstrd 3578	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
308		ssrab2 3650	. . . . . . . . 9 ⊢ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛} ⊆ ℕ
309		iunss1 4468	. . . . . . . . 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))
310	308, 309	ax-mp 5	. . . . . . . 8 ⊢ ∪ 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘)
311	310	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))
312	307, 311	sstrd 3578	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
313	312	ralrimiva 2949	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
314		iunss 4497	. . . . 5 ⊢ (∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘) ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
315	313, 314	sylibr 223	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
316	1, 130, 19, 245, 315	ovnlecvr 39448	. . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^{^}‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))))
317	116	fveq2d 6107	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐿‘(𝐺‘𝑚)) = (𝐿‘((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚)))))
318	317	mpteq2dva 4672	. . . . . 6 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚))) = (𝑚 ∈ ℕ ↦ (𝐿‘((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))))
319	318	fveq2d 6107	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))) = (Σ^{^}‘(𝑚 ∈ ℕ ↦ (𝐿‘((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚)))))))
320		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑝𝜑
321		fveq2 6103	. . . . . . . . 9 ⊢ (𝑝 = (𝐹‘𝑚) → (1^st ‘𝑝) = (1^st ‘(𝐹‘𝑚)))
322	321	fveq2d 6107	. . . . . . . 8 ⊢ (𝑝 = (𝐹‘𝑚) → (𝐼‘(1^st ‘𝑝)) = (𝐼‘(1^st ‘(𝐹‘𝑚))))
323		fveq2 6103	. . . . . . . 8 ⊢ (𝑝 = (𝐹‘𝑚) → (2^nd ‘𝑝) = (2^nd ‘(𝐹‘𝑚)))
324	322, 323	fveq12d 6109	. . . . . . 7 ⊢ (𝑝 = (𝐹‘𝑚) → ((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝)) = ((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚))))
325	324	fveq2d 6107	. . . . . 6 ⊢ (𝑝 = (𝐹‘𝑚) → (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))) = (𝐿‘((𝐼‘(1^st ‘(𝐹‘𝑚)))‘(2^nd ‘(𝐹‘𝑚)))))
326		eqidd 2611	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) = (𝐹‘𝑚))
327		nfv 1830	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑝 ∈ (ℕ × ℕ))
328	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → 𝑋 ∈ Fin)
329		simpl 472	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → 𝜑)
330		xp1st 7089	. . . . . . . . . 10 ⊢ (𝑝 ∈ (ℕ × ℕ) → (1^st ‘𝑝) ∈ ℕ)
331	330	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → (1^st ‘𝑝) ∈ ℕ)
332		xp2nd 7090	. . . . . . . . . 10 ⊢ (𝑝 ∈ (ℕ × ℕ) → (2^nd ‘𝑝) ∈ ℕ)
333	332	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → (2^nd ‘𝑝) ∈ ℕ)
334		fvex 6113	. . . . . . . . . 10 ⊢ (2^nd ‘𝑝) ∈ V
335		eleq1 2676	. . . . . . . . . . . 12 ⊢ (𝑗 = (2^nd ‘𝑝) → (𝑗 ∈ ℕ ↔ (2^nd ‘𝑝) ∈ ℕ))
336	335	3anbi3d 1397	. . . . . . . . . . 11 ⊢ (𝑗 = (2^nd ‘𝑝) → ((𝜑 ∧ (1^st ‘𝑝) ∈ ℕ ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ (1^st ‘𝑝) ∈ ℕ ∧ (2^nd ‘𝑝) ∈ ℕ)))
337		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑗 = (2^nd ‘𝑝) → ((𝐼‘(1^st ‘𝑝))‘𝑗) = ((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝)))
338	337	feq1d 5943	. . . . . . . . . . 11 ⊢ (𝑗 = (2^nd ‘𝑝) → (((𝐼‘(1^st ‘𝑝))‘𝑗):𝑋⟶(ℝ × ℝ) ↔ ((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝)):𝑋⟶(ℝ × ℝ)))
339	336, 338	imbi12d 333	. . . . . . . . . 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))))
340		fvex 6113	. . . . . . . . . . 11 ⊢ (1^st ‘𝑝) ∈ V
341		eleq1 2676	. . . . . . . . . . . . 13 ⊢ (𝑛 = (1^st ‘𝑝) → (𝑛 ∈ ℕ ↔ (1^st ‘𝑝) ∈ ℕ))
342	341	3anbi2d 1396	. . . . . . . . . . . 12 ⊢ (𝑛 = (1^st ‘𝑝) → ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ (1^st ‘𝑝) ∈ ℕ ∧ 𝑗 ∈ ℕ)))
343		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (1^st ‘𝑝) → (𝐼‘𝑛) = (𝐼‘(1^st ‘𝑝)))
344	343	fveq1d 6105	. . . . . . . . . . . . 13 ⊢ (𝑛 = (1^st ‘𝑝) → ((𝐼‘𝑛)‘𝑗) = ((𝐼‘(1^st ‘𝑝))‘𝑗))
345	344	feq1d 5943	. . . . . . . . . . . 12 ⊢ (𝑛 = (1^st ‘𝑝) → (((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ) ↔ ((𝐼‘(1^st ‘𝑝))‘𝑗):𝑋⟶(ℝ × ℝ)))
346	342, 345	imbi12d 333	. . . . . . . . . . 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))))
347	340, 346, 106	vtocl 3232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (1^st ‘𝑝) ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1^st ‘𝑝))‘𝑗):𝑋⟶(ℝ × ℝ))
348	334, 339, 347	vtocl 3232	. . . . . . . . 9 ⊢ ((𝜑 ∧ (1^st ‘𝑝) ∈ ℕ ∧ (2^nd ‘𝑝) ∈ ℕ) → ((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝)):𝑋⟶(ℝ × ℝ))
349	329, 331, 333, 348	syl3anc 1318	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → ((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝)):𝑋⟶(ℝ × ℝ))
350	327, 328, 19, 349	hoiprodcl2 39445	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))) ∈ (0[,)+∞))
351	15, 350	sseldi 3566	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))) ∈ (0[,]+∞))
352	320, 12, 325, 14, 20, 326, 351	sge0f1o 39275	. . . . 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)))))))
353	319, 352	eqtr4d 2647	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))) = (Σ^{^}‘(𝑝 ∈ (ℕ × ℕ) ↦ (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))))))
354		nfv 1830	. . . . . . 7 ⊢ Ⅎ𝑗𝜑
355	273, 274	op1std 7069	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → (1^st ‘𝑝) = 𝑛)
356	355	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → (𝐼‘(1^st ‘𝑝)) = (𝐼‘𝑛))
357	273, 274	op2ndd 7070	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → (2^nd ‘𝑝) = 𝑗)
358	356, 357	fveq12d 6109	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → ((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝)) = ((𝐼‘𝑛)‘𝑗))
359	358	fveq2d 6107	. . . . . . 7 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))) = (𝐿‘((𝐼‘𝑛)‘𝑗)))
360		nfv 1830	. . . . . . . . . 10 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ)
361	127	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
362	97, 105	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ))
363	360, 361, 19, 362	hoiprodcl2 39445	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (𝐿‘((𝐼‘𝑛)‘𝑗)) ∈ (0[,)+∞))
364	15, 363	sseldi 3566	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (𝐿‘((𝐼‘𝑛)‘𝑗)) ∈ (0[,]+∞))
365	364	3impa 1251	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝐿‘((𝐼‘𝑛)‘𝑗)) ∈ (0[,]+∞))
366	354, 359, 14, 14, 365	sge0xp 39322	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))))) = (Σ^{^}‘(𝑝 ∈ (ℕ × ℕ) ↦ (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))))))
367	366	eqcomd 2616	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑝 ∈ (ℕ × ℕ) ↦ (𝐿‘((𝐼‘(1^st ‘𝑝))‘(2^nd ‘𝑝))))) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))))))
368	13	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ℕ ∈ V)
369		eqid 2610	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))
370	364, 369	fmptd 6292	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗))):ℕ⟶(0[,]+∞))
371	368, 370	sge0cl 39274	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))) ∈ (0[,]+∞))
372		fveq1 6102	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝐼‘𝑛) → (𝑖‘𝑗) = ((𝐼‘𝑛)‘𝑗))
373	372	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼‘𝑛) → (𝐿‘(𝑖‘𝑗)) = (𝐿‘((𝐼‘𝑛)‘𝑗)))
374	373	mpteq2dv 4673	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘𝑛) → (𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗))))
375	374	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘𝑛) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))))
376	375	breq1d 4593	. . . . . . . . 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)))))
377	376	elrab 3331	. . . . . . . 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)))))
378	247, 377	sylib 207	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛)) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛)))))
379	378	simprd 478	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))))
380	122, 14, 371, 179, 379	sge0lempt 39303	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))))) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))))))
381	367, 380	eqbrtrd 4605	. . . 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))))))
382	353, 381	eqbrtrd 4605	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (((voln*‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))))))
383	11, 121, 180, 316, 382	xrletrd 11869	. 2 ⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))))))
384	122, 14, 166, 174	sge0xadd 39328	. . 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))))))
385	123	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ^*)
386	125	a1i 11	. . . . . 6 ⊢ (𝜑 → +∞ ∈ ℝ^*)
387	151	rexrd 9968	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ^*)
388	74	rpge0d 11752	. . . . . 6 ⊢ (𝜑 → 0 ≤ 𝐸)
389	151	ltpnfd 11831	. . . . . 6 ⊢ (𝜑 → 𝐸 < +∞)
390	385, 386, 387, 388, 389	elicod 12095	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (0[,)+∞))
391	390	sge0ad2en 39324	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝐸 / (2↑𝑛)))) = 𝐸)
392	391	oveq2d 6565	. . 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))
393	384, 392	eqtrd 2644	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (((voln‘𝑋)‘(𝐴‘𝑛)) +_𝑒 (𝐸 / (2↑𝑛))))) = ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸))
394	383, 393	breqtrd 4609	1 ⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 ifcif 4036 𝒫 cpw 4108 ⟨cop 4131 ∪ ciun 4455 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 –onto→wfo 5802 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 ↑_𝑚 cmap 7744 Xcixp 7794 Fincfn 7841 infcinf 8230 ℝcr 9814 0cc0 9815 +∞cpnf 9950 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 / cdiv 10563 ℕcn 10897 2c2 10947 ℝ⁺crp 11708 +_𝑒 cxad 11820 [,)cico 12048 [,]cicc 12049 ↑cexp 12722 ∏cprod 14474 volcvol 23039 Σ^{^}csumge0 39255 voln*covoln 39426

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-ac2 9168 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-ac 8822 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-prod 14475 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-rest 15906 df-0g 15925 df-topgen 15927 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-subg 17414 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-cmp 21000 df-ovol 23040 df-vol 23041 df-sumge0 39256 df-ovoln 39427

This theorem is referenced by: ovnsubaddlem2 39461

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