Theorem ovnsubadd 47344

Description: (voln*‘𝑋) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypotheses

Ref	Expression
ovnsubadd.1	⊢ (𝜑 → 𝑋 ∈ Fin)
ovnsubadd.2	⊢ (𝜑 → 𝐴:ℕ⟶𝒫 (ℝ ↑m 𝑋))

Assertion

Ref	Expression
ovnsubadd	⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))))

Distinct variable groups:   𝐴,𝑛   𝑛,𝑋   𝜑,𝑛

Proof of Theorem ovnsubadd

Dummy variables 𝑎 𝑒 𝑖 𝑗 𝑘 𝑙 𝑦 𝑧 𝑏 𝑑 𝑓 𝑚 ℎ 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6885	. . . . . 6 ⊢ (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅))
2	1	fveq1d 6887	. . . . 5 ⊢ (𝑋 = ∅ → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = ((voln*‘∅)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)))
3	2	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = ((voln*‘∅)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)))
4		ovnsubadd.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴:ℕ⟶𝒫 (ℝ ↑m 𝑋))
5	4	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ ↑m 𝑋))
6		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
7	5, 6	ffvelcdmd 7084	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ 𝒫 (ℝ ↑m 𝑋))
8		elpwi 4571	. . . . . . . . . 10 ⊢ ((𝐴‘𝑛) ∈ 𝒫 (ℝ ↑m 𝑋) → (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋))
9	7, 8	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋))
10	9	ralrimiva 3159	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋))
11		iunss 5011	. . . . . . . 8 ⊢ (∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋) ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋))
12	10, 11	sylibr 237	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋))
13	12	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋))
14		oveq2 7427	. . . . . . 7 ⊢ (𝑋 = ∅ → (ℝ ↑m 𝑋) = (ℝ ↑m ∅))
15	14	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑m 𝑋) = (ℝ ↑m ∅))
16	13, 15	sseqtrd 3974	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m ∅))
17	16	ovn0val 47322	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = 0)
18	3, 17	eqtrd 2800	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = 0)
19		nnex 12254	. . . . . 6 ⊢ ℕ ∈ V
20	19	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ∈ V)
21		ovnsubadd.1	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ Fin)
22	21	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
23	22, 9	ovncl 47339	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln*‘𝑋)‘(𝐴‘𝑛)) ∈ (0[,]+∞))
24		eqid 2765	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛))) = (𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))
25	23, 24	fmptd 7113	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛))):ℕ⟶(0[,]+∞))
26	20, 25	sge0ge0 47156	. . . 4 ⊢ (𝜑 → 0 ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))))
27	26	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))))
28	18, 27	eqbrtrd 5135	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))))
29	21, 12	ovnxrcl 47341	. . . 4 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ∈ ℝ*)
30	29	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ∈ ℝ*)
31	20, 25	sge0xrcl 47157	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) ∈ ℝ*)
32	31	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) ∈ ℝ*)
33	21	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝑋 ∈ Fin)
34		neqne 2968	. . . . 5 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
35	34	ad2antlr 740	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝑋 ≠ ∅)
36	4	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝐴:ℕ⟶𝒫 (ℝ ↑m 𝑋))
37		simpr 490	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
38		eqid 2765	. . . 4 ⊢ (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)))))}) = (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)))))})
39		sseq1 3963	. . . . . 6 ⊢ (𝑏 = 𝑎 → (𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) ↔ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)))
40	39	rabbidv 3425	. . . . 5 ⊢ (𝑏 = 𝑎 → {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} = {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
41	40	cbvmptv 5217	. . . 4 ⊢ (𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)}) = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
42		eqid 2765	. . . 4 ⊢ (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘))) = (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))
43		fveq2 6885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑜 = 𝑗 → (𝑙‘𝑜) = (𝑙‘𝑗))
44	43	coeq2d 5850	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑜 = 𝑗 → ([,) ∘ (𝑙‘𝑜)) = ([,) ∘ (𝑙‘𝑗)))
45	44	fveq1d 6887	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑜 = 𝑗 → (([,) ∘ (𝑙‘𝑜))‘𝑑) = (([,) ∘ (𝑙‘𝑗))‘𝑑))
46	45	ixpeq2dv 8917	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑜 = 𝑗 → X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑) = X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑑))
47		fveq2 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 = 𝑘 → (([,) ∘ (𝑙‘𝑗))‘𝑑) = (([,) ∘ (𝑙‘𝑗))‘𝑘))
48	47	cbvixpv 8919	. . . . . . . . . . . . . . . . . 18 ⊢ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑑) = X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)
49	46, 48	eqtrdi 2816	. . . . . . . . . . . . . . . . 17 ⊢ (𝑜 = 𝑗 → X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑) = X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘))
50	49	cbviunv 5005	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)
51	50	sseq2i 3967	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑) ↔ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘))
52	51	rabbii 3423	. . . . . . . . . . . . . 14 ⊢ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)} = {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)}
53	52	mpteq2i 5209	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)}) = (𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
54	53	fveq1i 6886	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)})‘𝑑) = ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑑)
55		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑎 → ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑑) = ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎))
56	54, 55	eqtrid 2812	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑎 → ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)})‘𝑑) = ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎))
57	56	eleq2d 2851	. . . . . . . . . 10 ⊢ (𝑑 = 𝑎 → (𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)})‘𝑑) ↔ 𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎)))
58		2fveq3 6890	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑘 → (vol‘(([,) ∘ ℎ)‘𝑑)) = (vol‘(([,) ∘ ℎ)‘𝑘)))
59	58	cbvprodv 15991	. . . . . . . . . . . . . . . . 17 ⊢ ∏𝑑 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑑)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘))
60	59	mpteq2i 5209	. . . . . . . . . . . . . . . 16 ⊢ (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑑 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑑))) = (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))
61	60	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑜 = 𝑗 → (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑑 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑑))) = (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘))))
62		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑜 = 𝑗 → (𝑚‘𝑜) = (𝑚‘𝑗))
63	61, 62	fveq12d 6892	. . . . . . . . . . . . . 14 ⊢ (𝑜 = 𝑗 → ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑑 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑑)))‘(𝑚‘𝑜)) = ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑚‘𝑗)))
64	63	cbvmptv 5217	. . . . . . . . . . . . 13 ⊢ (𝑜 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑑 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑑)))‘(𝑚‘𝑜))) = (𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑚‘𝑗)))
65	64	fveq2i 6888	. . . . . . . . . . . 12 ⊢ (Σ^‘(𝑜 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑑 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑑)))‘(𝑚‘𝑜)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑚‘𝑗))))
66	65	a1i 11	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑎 → (Σ^‘(𝑜 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑑 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑑)))‘(𝑚‘𝑜)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑚‘𝑗)))))
67		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑎 → ((voln*‘𝑋)‘𝑑) = ((voln*‘𝑋)‘𝑎))
68	67	oveq1d 7434	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑎 → (((voln*‘𝑋)‘𝑑) +𝑒 𝑓) = (((voln*‘𝑋)‘𝑎) +𝑒 𝑓))
69	66, 68	breq12d 5124	. . . . . . . . . 10 ⊢ (𝑑 = 𝑎 → ((Σ^‘(𝑜 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑑 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑑)))‘(𝑚‘𝑜)))) ≤ (((voln*‘𝑋)‘𝑑) +𝑒 𝑓) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑚‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)))
70	57, 69	anbi12d 644	. . . . . . . . 9 ⊢ (𝑑 = 𝑎 → ((𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)})‘𝑑) ∧ (Σ^‘(𝑜 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑑 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑑)))‘(𝑚‘𝑜)))) ≤ (((voln*‘𝑋)‘𝑑) +𝑒 𝑓)) ↔ (𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑚‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓))))
71	70	rabbidva2 3420	. . . . . . . 8 ⊢ (𝑑 = 𝑎 → {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)})‘𝑑) ∣ (Σ^‘(𝑜 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑑 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑑)))‘(𝑚‘𝑜)))) ≤ (((voln*‘𝑋)‘𝑑) +𝑒 𝑓)} = {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑚‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)})
72		fveq1 6884	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑖 → (𝑚‘𝑗) = (𝑖‘𝑗))
73	72	fveq2d 6889	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑖 → ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑚‘𝑗)) = ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑖‘𝑗)))
74	73	mpteq2dv 5207	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑖 → (𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑚‘𝑗))) = (𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑖‘𝑗))))
75	74	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝑚 = 𝑖 → (Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑚‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑖‘𝑗)))))
76	75	breq1d 5121	. . . . . . . . 9 ⊢ (𝑚 = 𝑖 → ((Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑚‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)))
77	76	cbvrabv 3428	. . . . . . . 8 ⊢ {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑚‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)} = {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)}
78	71, 77	eqtrdi 2816	. . . . . . 7 ⊢ (𝑑 = 𝑎 → {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)})‘𝑑) ∣ (Σ^‘(𝑜 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑑 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑑)))‘(𝑚‘𝑜)))) ≤ (((voln*‘𝑋)‘𝑑) +𝑒 𝑓)} = {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)})
79	78	mpteq2dv 5207	. . . . . 6 ⊢ (𝑑 = 𝑎 → (𝑓 ∈ ℝ+ ↦ {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)})‘𝑑) ∣ (Σ^‘(𝑜 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑑 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑑)))‘(𝑚‘𝑜)))) ≤ (((voln*‘𝑋)‘𝑑) +𝑒 𝑓)}) = (𝑓 ∈ ℝ+ ↦ {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)}))
80		oveq2 7427	. . . . . . . . 9 ⊢ (𝑓 = 𝑒 → (((voln*‘𝑋)‘𝑎) +𝑒 𝑓) = (((voln*‘𝑋)‘𝑎) +𝑒 𝑒))
81	80	breq2d 5123	. . . . . . . 8 ⊢ (𝑓 = 𝑒 → ((Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)))
82	81	rabbidv 3425	. . . . . . 7 ⊢ (𝑓 = 𝑒 → {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)} = {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)})
83	82	cbvmptv 5217	. . . . . 6 ⊢ (𝑓 ∈ ℝ+ ↦ {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)}) = (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)})
84	79, 83	eqtrdi 2816	. . . . 5 ⊢ (𝑑 = 𝑎 → (𝑓 ∈ ℝ+ ↦ {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)})‘𝑑) ∣ (Σ^‘(𝑜 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑑 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑑)))‘(𝑚‘𝑜)))) ≤ (((voln*‘𝑋)‘𝑑) +𝑒 𝑓)}) = (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)}))
85	84	cbvmptv 5217	. . . 4 ⊢ (𝑑 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑓 ∈ ℝ+ ↦ {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)})‘𝑑) ∣ (Σ^‘(𝑜 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑑 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑑)))‘(𝑚‘𝑜)))) ≤ (((voln*‘𝑋)‘𝑑) +𝑒 𝑓)})) = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)}))
86	33, 35, 36, 37, 38, 41, 42, 85	ovnsubaddlem2 47343	. . 3 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) +𝑒 𝑦))
87	30, 32, 86	xrlexaddrp 46126	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))))
88	28, 87	pm2.61dan 825	1 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  {crab 3418  Vcvv 3457   ⊆ wss 3906  ∅c0 4286  𝒫 cpw 4564  ∪ ciun 4958   class class class wbr 5111   ↦ cmpt 5194   × cxp 5661   ∘ ccom 5667  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419   ↑_m cmap 8830  Xcixp 8901  Fincfn 8949  ℝcr 11114  0cc0 11115  +∞cpnf 11255  ℝ^*cxr 11257   ≤ cle 11259  ℕcn 12248  ℝ⁺crp 13032   +_𝑒 cxad 13151  [,)cico 13390  [,]cicc 13391  ∏cprod 15980  volcvol 25673  Σ^{^}csumge0 47134  voln*covoln 47308

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-inf2 9617  ax-cc 10434  ax-ac2 10462  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192  ax-pre-sup 11193

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-disj 5079  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7684  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-map 8832  df-pm 8833  df-ixp 8902  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-fi 9378  df-sup 9409  df-inf 9410  df-oi 9479  df-dju 9903  df-card 9941  df-acn 9944  df-ac 10116  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-div 11887  df-nn 12249  df-2 12318  df-3 12319  df-n0 12520  df-z 12607  df-uz 12879  df-q 12989  df-rp 13033  df-xneg 13153  df-xadd 13154  df-xmul 13155  df-ioo 13392  df-ico 13394  df-icc 13395  df-fz 13552  df-fzo 13700  df-fl 13843  df-seq 14056  df-exp 14116  df-hash 14385  df-cj 15174  df-re 15175  df-im 15176  df-sqrt 15310  df-abs 15311  df-clim 15563  df-rlim 15564  df-sum 15762  df-prod 15981  df-rest 17497  df-topgen 17518  df-psmet 21564  df-xmet 21565  df-met 21566  df-bl 21567  df-mopn 21568  df-top 23101  df-topon 23118  df-bases 23153  df-cmp 23594  df-ovol 25674  df-vol 25675  df-sumge0 47135  df-ovoln 47309

This theorem is used by:  ovnome  47345  ovnsubadd2lem  47417