Theorem ovnsubaddlem1 40784

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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ ↑𝑚 𝑋))
4		simpr 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
5	3, 4	ffvelrnd 6360	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
6		elpwi 4168	. . . . . . 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 2966	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑𝑚 𝑋))
9		iunss 4561	. . . . 5 ⊢ (∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑𝑚 𝑋) ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑𝑚 𝑋))
10	8, 9	sylibr 224	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑𝑚 𝑋))
11	1, 10	ovnxrcl 40783	. . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ∈ ℝ*)
12		nfv 1843	. . . 4 ⊢ Ⅎ𝑚𝜑
13		nnex 11026	. . . . 5 ⊢ ℕ ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝜑 → ℕ ∈ V)
15		icossicc 12260	. . . . 5 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
16		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑚 ∈ ℕ)
17		simpl 473	. . . . . . 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 6137	. . . . . . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹:ℕ⟶(ℕ × ℕ))
24		simpr 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
25	23, 24	ffvelrnd 6360	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∈ (ℕ × ℕ))
26		xp1st 7198	. . . . . . . . 9 ⊢ ((𝐹‘𝑚) ∈ (ℕ × ℕ) → (1st ‘(𝐹‘𝑚)) ∈ ℕ)
27	25, 26	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐹‘𝑚)) ∈ ℕ)
28		xp2nd 7199	. . . . . . . . 9 ⊢ ((𝐹‘𝑚) ∈ (ℕ × ℕ) → (2nd ‘(𝐹‘𝑚)) ∈ ℕ)
29	25, 28	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐹‘𝑚)) ∈ ℕ)
30		fvex 6201	. . . . . . . . 9 ⊢ (2nd ‘(𝐹‘𝑚)) ∈ V
31		eleq1 2689	. . . . . . . . . . 11 ⊢ (𝑗 = (2nd ‘(𝐹‘𝑚)) → (𝑗 ∈ ℕ ↔ (2nd ‘(𝐹‘𝑚)) ∈ ℕ))
32	31	3anbi3d 1405	. . . . . . . . . 10 ⊢ (𝑗 = (2nd ‘(𝐹‘𝑚)) → ((𝜑 ∧ (1st ‘(𝐹‘𝑚)) ∈ ℕ ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ (1st ‘(𝐹‘𝑚)) ∈ ℕ ∧ (2nd ‘(𝐹‘𝑚)) ∈ ℕ)))
33		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑗 = (2nd ‘(𝐹‘𝑚)) → ((𝐼‘(1st ‘(𝐹‘𝑚)))‘𝑗) = ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))))
34	33	feq1d 6030	. . . . . . . . . 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 334	. . . . . . . . 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 6201	. . . . . . . . . 10 ⊢ (1st ‘(𝐹‘𝑚)) ∈ V
37		eleq1 2689	. . . . . . . . . . . 12 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → (𝑛 ∈ ℕ ↔ (1st ‘(𝐹‘𝑚)) ∈ ℕ))
38	37	3anbi2d 1404	. . . . . . . . . . 11 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ (1st ‘(𝐹‘𝑚)) ∈ ℕ ∧ 𝑗 ∈ ℕ)))
39		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → (𝐼‘𝑛) = (𝐼‘(1st ‘(𝐹‘𝑚))))
40	39	fveq1d 6193	. . . . . . . . . . . 12 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → ((𝐼‘𝑛)‘𝑗) = ((𝐼‘(1st ‘(𝐹‘𝑚)))‘𝑗))
41	40	feq1d 6030	. . . . . . . . . . 11 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → (((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ) ↔ ((𝐼‘(1st ‘(𝐹‘𝑚)))‘𝑗):𝑋⟶(ℝ × ℝ)))
42	38, 41	imbi12d 334	. . . . . . . . . 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	. . . . . . . . . . . . . . . . . . . 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 3626	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (𝐴‘𝑛) → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ↔ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)))
46	45	rabbidv 3189	. . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . 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 6678	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∈ V
49	48	rabex 4813	. . . . . . . . . . . . . . . . . . . 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 6288	. . . . . . . . . . . . . . . . . 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 3687	. . . . . . . . . . . . . . . . . . 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 3639	. . . . . . . . . . . . . . . . 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 6191	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = (𝐴‘𝑛) → (𝐶‘𝑎) = (𝐶‘(𝐴‘𝑛)))
58	57	eleq2d 2687	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (𝐴‘𝑛) → (𝑖 ∈ (𝐶‘𝑎) ↔ 𝑖 ∈ (𝐶‘(𝐴‘𝑛))))
59		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = (𝐴‘𝑛) → ((voln*‘𝑋)‘𝑎) = ((voln*‘𝑋)‘(𝐴‘𝑛)))
60	59	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = (𝐴‘𝑛) → (((voln*‘𝑋)‘𝑎) +𝑒 𝑒) = (((voln*‘𝑋)‘(𝐴‘𝑛)) +𝑒 𝑒))
61	60	breq2d 4665	. . . . . . . . . . . . . . . . . . . . . . . . 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 747	. . . . . . . . . . . . . . . . . . . . . . . 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 3186	. . . . . . . . . . . . . . . . . . . . . . 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 4745	. . . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . . 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 39562	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℝ+ ∈ V
67	66	mptex 6486	. . . . . . . . . . . . . . . . . . . . . 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 6288	. . . . . . . . . . . . . . . . . . . 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 6658	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 = (𝐸 / (2↑𝑛)) → (((voln*‘𝑋)‘(𝐴‘𝑛)) +𝑒 𝑒) = (((voln*‘𝑋)‘(𝐴‘𝑛)) +𝑒 (𝐸 / (2↑𝑛))))
71	70	breq2d 4665	. . . . . . . . . . . . . . . . . . . . . 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 3189	. . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ+)
76		2nn 11185	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 ∈ ℕ
77	76	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℕ)
78		nnnn0 11299	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
79	77, 78	nnexpcld 13030	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
80	79	nnrpd 11870	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ+)
81	80	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ+)
82	75, 81	rpdivcld 11889	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ ℝ+)
83		fvex 6201	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐶‘(𝐴‘𝑛)) ∈ V
84	83	rabex 4813	. . . . . . . . . . . . . . . . . . . . 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 6288	. . . . . . . . . . . . . . . . . . 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 3687	. . . . . . . . . . . . . . . . . . . 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 3639	. . . . . . . . . . . . . . . . . 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 3604	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛)))
92	54, 91	sseldd 3604	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
93		elmapfn 7880	. . . . . . . . . . . . . . . 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 7879	. . . . . . . . . . . . . . . . . 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 6359	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
98	97	ralrimiva 2966	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑗 ∈ ℕ ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
99	94, 98	jca 554	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐼‘𝑛) Fn ℕ ∧ ∀𝑗 ∈ ℕ ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋)))
100	99	3adant3 1081	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛) Fn ℕ ∧ ∀𝑗 ∈ ℕ ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋)))
101		ffnfv 6388	. . . . . . . . . . . . 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 224	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑛):ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
103		simp3 1063	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
104	102, 103	ffvelrnd 6360	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑛)‘𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
105		elmapi 7879	. . . . . . . . . . 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 3259	. . . . . . . . 9 ⊢ ((𝜑 ∧ (1st ‘(𝐹‘𝑚)) ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1st ‘(𝐹‘𝑚)))‘𝑗):𝑋⟶(ℝ × ℝ))
108	30, 35, 107	vtocl 3259	. . . . . . . 8 ⊢ ((𝜑 ∧ (1st ‘(𝐹‘𝑚)) ∈ ℕ ∧ (2nd ‘(𝐹‘𝑚)) ∈ ℕ) → ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))):𝑋⟶(ℝ × ℝ))
109	17, 27, 29, 108	syl3anc 1326	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))):𝑋⟶(ℝ × ℝ))
110		id 22	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ)
111		fvex 6201	. . . . . . . . . . 11 ⊢ ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))) ∈ V
112	111	a1i 11	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))) ∈ V)
113		ovnsubaddlem1.g	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))))
114	113	fvmpt2 6291	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))) ∈ V) → (𝐺‘𝑚) = ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))))
115	110, 112, 114	syl2anc 693	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (𝐺‘𝑚) = ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))))
116	115	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) = ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))))
117	116	feq1d 6030	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐺‘𝑚):𝑋⟶(ℝ × ℝ) ↔ ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))):𝑋⟶(ℝ × ℝ)))
118	109, 117	mpbird 247	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚):𝑋⟶(ℝ × ℝ))
119	16, 18, 19, 118	hoiprodcl2 40769	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐿‘(𝐺‘𝑚)) ∈ (0[,)+∞))
120	15, 119	sseldi 3601	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐿‘(𝐺‘𝑚)) ∈ (0[,]+∞))
121	12, 14, 120	sge0xrclmpt 40645	. . 3 ⊢ (𝜑 → (Σ^‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))) ∈ ℝ*)
122		nfv 1843	. . . 4 ⊢ Ⅎ𝑛𝜑
123		0xr 10086	. . . . . 6 ⊢ 0 ∈ ℝ*
124	123	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ*)
125		pnfxr 10092	. . . . . 6 ⊢ +∞ ∈ ℝ*
126	125	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ*)
127	1	adantr 481	. . . . . . . . 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 40766	. . . . . . . 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 2799	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 = ∅)
132	131	iffalsed 4097	. . . . . . . . 9 ⊢ (𝜑 → if(𝑋 = ∅, 0, inf((𝑍‘(𝐴‘𝑛)), ℝ*, < )) = inf((𝑍‘(𝐴‘𝑛)), ℝ*, < ))
133	132	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(𝑋 = ∅, 0, inf((𝑍‘(𝐴‘𝑛)), ℝ*, < )) = inf((𝑍‘(𝐴‘𝑛)), ℝ*, < ))
134	129, 133	eqtrd 2656	. . . . . . 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 3626	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝐴‘𝑛) → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ↔ (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)))
137	136	anbi1d 741	. . . . . . . . . . . . 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 3052	. . . . . . . . . . . 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 3189	. . . . . . . . . . 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 482	. . . . . . . . . 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 11829	. . . . . . . . . . . 12 ⊢ ℝ* ∈ V
142	141	rabex 4813	. . . . . . . . . . 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 6288	. . . . . . . . 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 3687	. . . . . . . . . 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 3639	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑍‘(𝐴‘𝑛)) ⊆ ℝ*)
148		infxrcl 12163	. . . . . . . 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 2701	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln*‘𝑋)‘(𝐴‘𝑛)) ∈ ℝ*)
151	74	rpred 11872	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℝ)
152	151	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ)
153		2re 11090	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
154	153	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ)
155	154, 78	reexpcld 13025	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ)
156	155	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ)
157	154	recnd 10068	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℂ)
158		2ne0 11113	. . . . . . . . . . 11 ⊢ 2 ≠ 0
159	158	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ≠ 0)
160		nnz 11399	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
161	157, 159, 160	expne0d 13014	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ≠ 0)
162	161	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ≠ 0)
163	152, 156, 162	redivcld 10853	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ ℝ)
164	163	rexrd 10089	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ ℝ*)
165	150, 164	xaddcld 12131	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((voln*‘𝑋)‘(𝐴‘𝑛)) +𝑒 (𝐸 / (2↑𝑛))) ∈ ℝ*)
166	127, 7	ovncl 40781	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln*‘𝑋)‘(𝐴‘𝑛)) ∈ (0[,]+∞))
167		xrge0ge0 39563	. . . . . . 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 10041	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ)
170	82	rpgt0d 11875	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 < (𝐸 / (2↑𝑛)))
171	169, 163, 170	ltled 10185	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝐸 / (2↑𝑛)))
172	163	ltpnfd 11955	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) < +∞)
173	164, 126, 172	xrltled 39486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ≤ +∞)
174	124, 126, 164, 171, 173	eliccxrd 39753	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ (0[,]+∞))
175	150, 174	xadd0ge 39536	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln*‘𝑋)‘(𝐴‘𝑛)) ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +𝑒 (𝐸 / (2↑𝑛))))
176	124, 150, 165, 168, 175	xrletrd 11993	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (((voln*‘𝑋)‘(𝐴‘𝑛)) +𝑒 (𝐸 / (2↑𝑛))))
177		pnfge 11964	. . . . . 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 39753	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((voln*‘𝑋)‘(𝐴‘𝑛)) +𝑒 (𝐸 / (2↑𝑛))) ∈ (0[,]+∞))
180	122, 14, 179	sge0xrclmpt 40645	. . 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 3626	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝐴‘(1st ‘(𝐹‘𝑚))) → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ↔ (𝐴‘(1st ‘(𝐹‘𝑚))) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)))
183	182	rabbidv 3189	. . . . . . . . . . . . 13 ⊢ (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)})
184	183	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 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)})
185	2	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ ↑𝑚 𝑋))
186	185, 27	ffvelrnd 6360	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐴‘(1st ‘(𝐹‘𝑚))) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
187	48	rabex 4813	. . . . . . . . . . . . 13 ⊢ {ℎ ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ (𝐴‘(1st ‘(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‘(1st ‘(𝐹‘𝑚))) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘)} ∈ V)
189	181, 184, 186, 188	fvmptd 6288	. . . . . . . . . . 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)})
190		ssrab2 3687	. . . . . . . . . . . 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)
191	190	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))
192	189, 191	eqsstrd 3639	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐶‘(𝐴‘(1st ‘(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 6191	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (𝐴‘(1st ‘(𝐹‘𝑚))) → (𝐶‘𝑎) = (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))))
195	194	eleq2d 2687	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (𝐴‘(1st ‘(𝐹‘𝑚))) → (𝑖 ∈ (𝐶‘𝑎) ↔ 𝑖 ∈ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚))))))
196		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (𝐴‘(1st ‘(𝐹‘𝑚))) → ((voln*‘𝑋)‘𝑎) = ((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))))
197	196	oveq1d 6665	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (𝐴‘(1st ‘(𝐹‘𝑚))) → (((voln*‘𝑋)‘𝑎) +𝑒 𝑒) = (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 𝑒))
198	197	breq2d 4665	. . . . . . . . . . . . . . . . . 18 ⊢ (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)))
199	195, 198	anbi12d 747	. . . . . . . . . . . . . . . . 17 ⊢ (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))))
200	199	rabbidva2 3186	. . . . . . . . . . . . . . . 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)})
201	200	mpteq2dv 4745	. . . . . . . . . . . . . . 15 ⊢ (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)}))
202	201	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 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)}))
203	66	mptex 6486	. . . . . . . . . . . . . . 15 ⊢ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘(𝐴‘(1st ‘(𝐹‘𝑚)))) +𝑒 𝑒)}) ∈ V
204	203	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)
205	193, 202, 186, 204	fvmptd 6288	. . . . . . . . . . . . 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)}))
206		oveq2 6658	. . . . . . . . . . . . . . . 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))))))
207	206	breq2d 4665	. . . . . . . . . . . . . . 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)))))))
208	207	rabbidv 3189	. . . . . . . . . . . . . 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)))))})
209	208	adantl 482	. . . . . . . . . . . . 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)))))})
210	17, 74	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐸 ∈ ℝ+)
211		2rp 11837	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ+
212	211	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 2 ∈ ℝ+)
213	27	nnzd 11481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐹‘𝑚)) ∈ ℤ)
214	212, 213	rpexpcld 13032	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2↑(1st ‘(𝐹‘𝑚))) ∈ ℝ+)
215	210, 214	rpdivcld 11889	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐸 / (2↑(1st ‘(𝐹‘𝑚)))) ∈ ℝ+)
216		fvex 6201	. . . . . . . . . . . . . . 15 ⊢ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))) ∈ V
217	216	rabex 4813	. . . . . . . . . . . . . 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
218	217	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)
219	205, 209, 215, 218	fvmptd 6288	. . . . . . . . . . . 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)))))})
220		ssrab2 3687	. . . . . . . . . . . . 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))))
221	220	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)))))
222	219, 221	eqsstrd 3639	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐷‘(𝐴‘(1st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1st ‘(𝐹‘𝑚))))) ⊆ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))))
223	37	anbi2d 740	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → ((𝜑 ∧ 𝑛 ∈ ℕ) ↔ (𝜑 ∧ (1st ‘(𝐹‘𝑚)) ∈ ℕ)))
224		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → (𝐴‘𝑛) = (𝐴‘(1st ‘(𝐹‘𝑚))))
225	224	fveq2d 6195	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → (𝐷‘(𝐴‘𝑛)) = (𝐷‘(𝐴‘(1st ‘(𝐹‘𝑚)))))
226		oveq2 6658	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → (2↑𝑛) = (2↑(1st ‘(𝐹‘𝑚))))
227	226	oveq2d 6666	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → (𝐸 / (2↑𝑛)) = (𝐸 / (2↑(1st ‘(𝐹‘𝑚)))))
228	225, 227	fveq12d 6197	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (1st ‘(𝐹‘𝑚)) → ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) = ((𝐷‘(𝐴‘(1st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1st ‘(𝐹‘𝑚))))))
229	39, 228	eleq12d 2695	. . . . . . . . . . . . . 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)))))))
230	223, 229	imbi12d 334	. . . . . . . . . . . . 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))))))))
231	36, 230, 90	vtocl 3259	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (1st ‘(𝐹‘𝑚)) ∈ ℕ) → (𝐼‘(1st ‘(𝐹‘𝑚))) ∈ ((𝐷‘(𝐴‘(1st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1st ‘(𝐹‘𝑚))))))
232	17, 27, 231	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1st ‘(𝐹‘𝑚))) ∈ ((𝐷‘(𝐴‘(1st ‘(𝐹‘𝑚))))‘(𝐸 / (2↑(1st ‘(𝐹‘𝑚))))))
233	222, 232	sseldd 3604	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1st ‘(𝐹‘𝑚))) ∈ (𝐶‘(𝐴‘(1st ‘(𝐹‘𝑚)))))
234	192, 233	sseldd 3604	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1st ‘(𝐹‘𝑚))) ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
235		elmapfn 7880	. . . . . . . . 9 ⊢ ((𝐼‘(1st ‘(𝐹‘𝑚))) ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝐼‘(1st ‘(𝐹‘𝑚))) Fn ℕ)
236	234, 235	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1st ‘(𝐹‘𝑚))) Fn ℕ)
237		elmapi 7879	. . . . . . . . . . 11 ⊢ ((𝐼‘(1st ‘(𝐹‘𝑚))) ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝐼‘(1st ‘(𝐹‘𝑚))):ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
238	234, 237	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1st ‘(𝐹‘𝑚))):ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
239	238	ffvelrnda 6359	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1st ‘(𝐹‘𝑚)))‘𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
240	239	ralrimiva 2966	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑗 ∈ ℕ ((𝐼‘(1st ‘(𝐹‘𝑚)))‘𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
241	236, 240	jca 554	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐼‘(1st ‘(𝐹‘𝑚))) Fn ℕ ∧ ∀𝑗 ∈ ℕ ((𝐼‘(1st ‘(𝐹‘𝑚)))‘𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋)))
242		ffnfv 6388	. . . . . . 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)))
243	241, 242	sylibr 224	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐼‘(1st ‘(𝐹‘𝑚))):ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
244	243, 29	ffvelrnd 6360	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
245	244, 113	fmptd 6385	. . . 4 ⊢ (𝜑 → 𝐺:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
246		simpl 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝜑)
247	90, 86	eleqtrd 2703	. . . . . . . . . 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 3601	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛)))
249		simp3 1063	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛))) → (𝐼‘𝑛) ∈ (𝐶‘(𝐴‘𝑛)))
250	51	3adant3 1081	. . . . . . . . . . . 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 2703	. . . . . . . . . . 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 6190	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = (𝐼‘𝑛) → (ℎ‘𝑗) = ((𝐼‘𝑛)‘𝑗))
253	252	coeq2d 5284	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝐼‘𝑛) → ([,) ∘ (ℎ‘𝑗)) = ([,) ∘ ((𝐼‘𝑛)‘𝑗)))
254	253	fveq1d 6193	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝐼‘𝑛) → (([,) ∘ (ℎ‘𝑗))‘𝑘) = (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
255	254	ixpeq2dv 7924	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝐼‘𝑛) → X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
256	255	iuneq2d 4547	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝐼‘𝑛) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
257	256	sseq2d 3633	. . . . . . . . . . . 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 3363	. . . . . . . . . . 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 208	. . . . . . . . . 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 479	. . . . . . . . 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 1326	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘))
262		f1ofo 6144	. . . . . . . . . . . . . 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 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → 𝐹:ℕ–onto→(ℕ × ℕ))
265		opelxpi 5148	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ))
266	4, 265	sylan 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ))
267		foelrni 6244	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ–onto→(ℕ × ℕ) ∧ ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ)) → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩)
268	264, 266, 267	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ∃𝑚 ∈ ℕ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩)
269		nfv 1843	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ)
270		nfre1 3005	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚∃𝑚 ∈ {𝑖 ∈ ℕ ∣ (1st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘)
271		simpl 473	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → 𝑚 ∈ ℕ)
272		fveq2 6191	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (1st ‘(𝐹‘𝑚)) = (1st ‘⟨𝑛, 𝑗⟩))
273		vex 3203	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑛 ∈ V
274		vex 3203	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑗 ∈ V
275		op1stg 7180	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ V ∧ 𝑗 ∈ V) → (1st ‘⟨𝑛, 𝑗⟩) = 𝑛)
276	273, 274, 275	mp2an 708	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1st ‘⟨𝑛, 𝑗⟩) = 𝑛
277	276	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (1st ‘⟨𝑛, 𝑗⟩) = 𝑛)
278	272, 277	eqtrd 2656	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (1st ‘(𝐹‘𝑚)) = 𝑛)
279	278	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → (1st ‘(𝐹‘𝑚)) = 𝑛)
280	271, 279	jca 554	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → (𝑚 ∈ ℕ ∧ (1st ‘(𝐹‘𝑚)) = 𝑛))
281		fveq2 6191	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑚 → (𝐹‘𝑖) = (𝐹‘𝑚))
282	281	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑚 → (1st ‘(𝐹‘𝑖)) = (1st ‘(𝐹‘𝑚)))
283	282	eqeq1d 2624	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑚 → ((1st ‘(𝐹‘𝑖)) = 𝑛 ↔ (1st ‘(𝐹‘𝑚)) = 𝑛))
284	283	elrab 3363	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ {𝑖 ∈ ℕ ∣ (1st ‘(𝐹‘𝑖)) = 𝑛} ↔ (𝑚 ∈ ℕ ∧ (1st ‘(𝐹‘𝑚)) = 𝑛))
285	280, 284	sylibr 224	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1st ‘(𝐹‘𝑖)) = 𝑛})
286	285	3adant1 1079	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1st ‘(𝐹‘𝑖)) = 𝑛})
287	271, 115	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → (𝐺‘𝑚) = ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))))
288	278	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (𝐼‘(1st ‘(𝐹‘𝑚))) = (𝐼‘𝑛))
289	273, 274	op2ndd 7179	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → (2nd ‘(𝐹‘𝑚)) = 𝑗)
290	288, 289	fveq12d 6197	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘𝑚) = ⟨𝑛, 𝑗⟩ → ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))) = ((𝐼‘𝑛)‘𝑗))
291	290	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))) = ((𝐼‘𝑛)‘𝑗))
292	287, 291	eqtr2d 2657	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → ((𝐼‘𝑛)‘𝑗) = (𝐺‘𝑚))
293	292	coeq2d 5284	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → ([,) ∘ ((𝐼‘𝑛)‘𝑗)) = ([,) ∘ (𝐺‘𝑚)))
294	293	fveq1d 6193	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) = (([,) ∘ (𝐺‘𝑚))‘𝑘))
295	294	ixpeq2dv 7924	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) = ⟨𝑛, 𝑗⟩) → X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
296		eqimss 3657	. . . . . . . . . . . . . . . 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 1079	. . . . . . . . . . . . . 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 3003	. . . . . . . . . . . . . 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))
300	286, 298, 299	syl2anc 693	. . . . . . . . . . . . 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))
301	300	3exp 1264	. . . . . . . . . . . 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))))
302	269, 270, 301	rexlimd 3026	. . . . . . . . . . 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)))
303	268, 302	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))
304	303	ralrimiva 2966	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑗 ∈ ℕ ∃𝑚 ∈ {𝑖 ∈ ℕ ∣ (1st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ ((𝐼‘𝑛)‘𝑗))‘𝑘) ⊆ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
305		iunss2 4565	. . . . . . . . 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))
306	304, 305	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))
307	261, 306	sstrd 3613	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
308		ssrab2 3687	. . . . . . . . 9 ⊢ {𝑖 ∈ ℕ ∣ (1st ‘(𝐹‘𝑖)) = 𝑛} ⊆ ℕ
309		iunss1 4532	. . . . . . . . 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))
310	308, 309	ax-mp 5	. . . . . . . 8 ⊢ ∪ 𝑚 ∈ {𝑖 ∈ ℕ ∣ (1st ‘(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 ∣ (1st ‘(𝐹‘𝑖)) = 𝑛}X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
312	307, 311	sstrd 3613	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
313	312	ralrimiva 2966	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
314		iunss 4561	. . . . 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 224	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ ∪ 𝑚 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐺‘𝑚))‘𝑘))
316	1, 130, 19, 245, 315	ovnlecvr 40772	. . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))))
317	116	fveq2d 6195	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐿‘(𝐺‘𝑚)) = (𝐿‘((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚)))))
318	317	mpteq2dva 4744	. . . . . 6 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚))) = (𝑚 ∈ ℕ ↦ (𝐿‘((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))))))
319	318	fveq2d 6195	. . . . 5 ⊢ (𝜑 → (Σ^‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))) = (Σ^‘(𝑚 ∈ ℕ ↦ (𝐿‘((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚)))))))
320		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑝𝜑
321		fveq2 6191	. . . . . . . . 9 ⊢ (𝑝 = (𝐹‘𝑚) → (1st ‘𝑝) = (1st ‘(𝐹‘𝑚)))
322	321	fveq2d 6195	. . . . . . . 8 ⊢ (𝑝 = (𝐹‘𝑚) → (𝐼‘(1st ‘𝑝)) = (𝐼‘(1st ‘(𝐹‘𝑚))))
323		fveq2 6191	. . . . . . . 8 ⊢ (𝑝 = (𝐹‘𝑚) → (2nd ‘𝑝) = (2nd ‘(𝐹‘𝑚)))
324	322, 323	fveq12d 6197	. . . . . . 7 ⊢ (𝑝 = (𝐹‘𝑚) → ((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝)) = ((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚))))
325	324	fveq2d 6195	. . . . . 6 ⊢ (𝑝 = (𝐹‘𝑚) → (𝐿‘((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝))) = (𝐿‘((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚)))))
326		eqidd 2623	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) = (𝐹‘𝑚))
327		nfv 1843	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑝 ∈ (ℕ × ℕ))
328	1	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → 𝑋 ∈ Fin)
329		simpl 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → 𝜑)
330		xp1st 7198	. . . . . . . . . 10 ⊢ (𝑝 ∈ (ℕ × ℕ) → (1st ‘𝑝) ∈ ℕ)
331	330	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → (1st ‘𝑝) ∈ ℕ)
332		xp2nd 7199	. . . . . . . . . 10 ⊢ (𝑝 ∈ (ℕ × ℕ) → (2nd ‘𝑝) ∈ ℕ)
333	332	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → (2nd ‘𝑝) ∈ ℕ)
334		fvex 6201	. . . . . . . . . 10 ⊢ (2nd ‘𝑝) ∈ V
335		eleq1 2689	. . . . . . . . . . . 12 ⊢ (𝑗 = (2nd ‘𝑝) → (𝑗 ∈ ℕ ↔ (2nd ‘𝑝) ∈ ℕ))
336	335	3anbi3d 1405	. . . . . . . . . . 11 ⊢ (𝑗 = (2nd ‘𝑝) → ((𝜑 ∧ (1st ‘𝑝) ∈ ℕ ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ (1st ‘𝑝) ∈ ℕ ∧ (2nd ‘𝑝) ∈ ℕ)))
337		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑗 = (2nd ‘𝑝) → ((𝐼‘(1st ‘𝑝))‘𝑗) = ((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝)))
338	337	feq1d 6030	. . . . . . . . . . 11 ⊢ (𝑗 = (2nd ‘𝑝) → (((𝐼‘(1st ‘𝑝))‘𝑗):𝑋⟶(ℝ × ℝ) ↔ ((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝)):𝑋⟶(ℝ × ℝ)))
339	336, 338	imbi12d 334	. . . . . . . . . 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))))
340		fvex 6201	. . . . . . . . . . 11 ⊢ (1st ‘𝑝) ∈ V
341		eleq1 2689	. . . . . . . . . . . . 13 ⊢ (𝑛 = (1st ‘𝑝) → (𝑛 ∈ ℕ ↔ (1st ‘𝑝) ∈ ℕ))
342	341	3anbi2d 1404	. . . . . . . . . . . 12 ⊢ (𝑛 = (1st ‘𝑝) → ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ (1st ‘𝑝) ∈ ℕ ∧ 𝑗 ∈ ℕ)))
343		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (1st ‘𝑝) → (𝐼‘𝑛) = (𝐼‘(1st ‘𝑝)))
344	343	fveq1d 6193	. . . . . . . . . . . . 13 ⊢ (𝑛 = (1st ‘𝑝) → ((𝐼‘𝑛)‘𝑗) = ((𝐼‘(1st ‘𝑝))‘𝑗))
345	344	feq1d 6030	. . . . . . . . . . . 12 ⊢ (𝑛 = (1st ‘𝑝) → (((𝐼‘𝑛)‘𝑗):𝑋⟶(ℝ × ℝ) ↔ ((𝐼‘(1st ‘𝑝))‘𝑗):𝑋⟶(ℝ × ℝ)))
346	342, 345	imbi12d 334	. . . . . . . . . . 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))))
347	340, 346, 106	vtocl 3259	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (1st ‘𝑝) ∈ ℕ ∧ 𝑗 ∈ ℕ) → ((𝐼‘(1st ‘𝑝))‘𝑗):𝑋⟶(ℝ × ℝ))
348	334, 339, 347	vtocl 3259	. . . . . . . . 9 ⊢ ((𝜑 ∧ (1st ‘𝑝) ∈ ℕ ∧ (2nd ‘𝑝) ∈ ℕ) → ((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝)):𝑋⟶(ℝ × ℝ))
349	329, 331, 333, 348	syl3anc 1326	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → ((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝)):𝑋⟶(ℝ × ℝ))
350	327, 328, 19, 349	hoiprodcl2 40769	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → (𝐿‘((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝))) ∈ (0[,)+∞))
351	15, 350	sseldi 3601	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ × ℕ)) → (𝐿‘((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝))) ∈ (0[,]+∞))
352	320, 12, 325, 14, 20, 326, 351	sge0f1o 40599	. . . . 5 ⊢ (𝜑 → (Σ^‘(𝑝 ∈ (ℕ × ℕ) ↦ (𝐿‘((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝))))) = (Σ^‘(𝑚 ∈ ℕ ↦ (𝐿‘((𝐼‘(1st ‘(𝐹‘𝑚)))‘(2nd ‘(𝐹‘𝑚)))))))
353	319, 352	eqtr4d 2659	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑚 ∈ ℕ ↦ (𝐿‘(𝐺‘𝑚)))) = (Σ^‘(𝑝 ∈ (ℕ × ℕ) ↦ (𝐿‘((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝))))))
354		nfv 1843	. . . . . . 7 ⊢ Ⅎ𝑗𝜑
355	273, 274	op1std 7178	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → (1st ‘𝑝) = 𝑛)
356	355	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → (𝐼‘(1st ‘𝑝)) = (𝐼‘𝑛))
357	273, 274	op2ndd 7179	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → (2nd ‘𝑝) = 𝑗)
358	356, 357	fveq12d 6197	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → ((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝)) = ((𝐼‘𝑛)‘𝑗))
359	358	fveq2d 6195	. . . . . . 7 ⊢ (𝑝 = ⟨𝑛, 𝑗⟩ → (𝐿‘((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝))) = (𝐿‘((𝐼‘𝑛)‘𝑗)))
360		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ)
361	127	adantr 481	. . . . . . . . . 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 40769	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (𝐿‘((𝐼‘𝑛)‘𝑗)) ∈ (0[,)+∞))
364	15, 363	sseldi 3601	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → (𝐿‘((𝐼‘𝑛)‘𝑗)) ∈ (0[,]+∞))
365	364	3impa 1259	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝐿‘((𝐼‘𝑛)‘𝑗)) ∈ (0[,]+∞))
366	354, 359, 14, 14, 365	sge0xp 40646	. . . . . 6 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))))) = (Σ^‘(𝑝 ∈ (ℕ × ℕ) ↦ (𝐿‘((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝))))))
367	366	eqcomd 2628	. . . . 5 ⊢ (𝜑 → (Σ^‘(𝑝 ∈ (ℕ × ℕ) ↦ (𝐿‘((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝))))) = (Σ^‘(𝑛 ∈ ℕ ↦ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))))))
368	13	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ℕ ∈ V)
369		eqid 2622	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))
370	364, 369	fmptd 6385	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗))):ℕ⟶(0[,]+∞))
371	368, 370	sge0cl 40598	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))) ∈ (0[,]+∞))
372		fveq1 6190	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝐼‘𝑛) → (𝑖‘𝑗) = ((𝐼‘𝑛)‘𝑗))
373	372	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼‘𝑛) → (𝐿‘(𝑖‘𝑗)) = (𝐿‘((𝐼‘𝑛)‘𝑗)))
374	373	mpteq2dv 4745	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘𝑛) → (𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗))))
375	374	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘𝑛) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘((𝐼‘𝑛)‘𝑗)))))
376	375	breq1d 4663	. . . . . . . . 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 3363	. . . . . . . 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 208	. . . . . . 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 479	. . . . . 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 40627	. . . . 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 4675	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑝 ∈ (ℕ × ℕ) ↦ (𝐿‘((𝐼‘(1st ‘𝑝))‘(2nd ‘𝑝))))) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (((voln*‘𝑋)‘(𝐴‘𝑛)) +𝑒 (𝐸 / (2↑𝑛))))))
382	353, 381	eqbrtrd 4675	. . 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 11993	. 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 40652	. . 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 10089	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ*)
388	74	rpge0d 11876	. . . . . 6 ⊢ (𝜑 → 0 ≤ 𝐸)
389	151	ltpnfd 11955	. . . . . 6 ⊢ (𝜑 → 𝐸 < +∞)
390	385, 386, 387, 388, 389	elicod 12224	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (0[,)+∞))
391	390	sge0ad2en 40648	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝐸 / (2↑𝑛)))) = 𝐸)
392	391	oveq2d 6666	. . 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 2656	. 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 4679	1 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) +𝑒 𝐸))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ifcif 4086  𝒫 cpw 4158  ⟨cop 4183  ∪ ciun 4520   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112   ∘ ccom 5118   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167   ↑_𝑚 cmap 7857  Xcixp 7908  Fincfn 7955  infcinf 8347  ℝcr 9935  0cc0 9936  +∞cpnf 10071  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075   / cdiv 10684  ℕcn 11020  2c2 11070  ℝ⁺crp 11832   +_𝑒 cxad 11944  [,)cico 12177  [,]cicc 12178  ↑cexp 12860  ∏cprod 14635  volcvol 23232  Σ^{^}csumge0 40579  voln*covoln 40750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-prod 14636  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-0g 16102  df-topgen 16104  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-subg 17591  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234  df-sumge0 40580  df-ovoln 40751

This theorem is referenced by:  ovnsubaddlem2  40785