Theorem ovnsubadd 47177

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 6882	. . . . . 6 ⊢ (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅))
2	1	fveq1d 6884	. . . . 5 ⊢ (𝑋 = ∅ → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = ((voln*‘∅)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)))
3	2	adantl 486	. . . 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 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ ↑m 𝑋))
6		simpr 489	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
7	5, 6	ffvelcdmd 7081	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ 𝒫 (ℝ ↑m 𝑋))
8		elpwi 4574	. . . . . . . . . 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 3163	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋))
11		iunss 5013	. . . . . . . 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 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋))
14		oveq2 7419	. . . . . . 7 ⊢ (𝑋 = ∅ → (ℝ ↑m 𝑋) = (ℝ ↑m ∅))
15	14	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑m 𝑋) = (ℝ ↑m ∅))
16	13, 15	sseqtrd 3981	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m ∅))
17	16	ovn0val 47155	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = 0)
18	3, 17	eqtrd 2804	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = 0)
19		nnex 12238	. . . . . 6 ⊢ ℕ ∈ V
20	19	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ∈ V)
21		ovnsubadd.1	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ Fin)
22	21	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
23	22, 9	ovncl 47172	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln*‘𝑋)‘(𝐴‘𝑛)) ∈ (0[,]+∞))
24		eqid 2769	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛))) = (𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))
25	23, 24	fmptd 7110	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛))):ℕ⟶(0[,]+∞))
26	20, 25	sge0ge0 46989	. . . 4 ⊢ (𝜑 → 0 ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))))
27	26	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))))
28	18, 27	eqbrtrd 5137	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))))
29	21, 12	ovnxrcl 47174	. . . 4 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ∈ ℝ*)
30	29	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ∈ ℝ*)
31	20, 25	sge0xrcl 46990	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) ∈ ℝ*)
32	31	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) ∈ ℝ*)
33	21	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝑋 ∈ Fin)
34		neqne 2972	. . . . 5 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
35	34	ad2antlr 739	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝑋 ≠ ∅)
36	4	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝐴:ℕ⟶𝒫 (ℝ ↑m 𝑋))
37		simpr 489	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
38		eqid 2769	. . . 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 3970	. . . . . 6 ⊢ (𝑏 = 𝑎 → (𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) ↔ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)))
40	39	rabbidv 3430	. . . . 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 5219	. . . 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 2769	. . . 4 ⊢ (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘))) = (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))
43		fveq2 6882	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑜 = 𝑗 → (𝑙‘𝑜) = (𝑙‘𝑗))
44	43	coeq2d 5849	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑜 = 𝑗 → ([,) ∘ (𝑙‘𝑜)) = ([,) ∘ (𝑙‘𝑗)))
45	44	fveq1d 6884	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑜 = 𝑗 → (([,) ∘ (𝑙‘𝑜))‘𝑑) = (([,) ∘ (𝑙‘𝑗))‘𝑑))
46	45	ixpeq2dv 8910	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑜 = 𝑗 → X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑) = X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑑))
47		fveq2 6882	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 = 𝑘 → (([,) ∘ (𝑙‘𝑗))‘𝑑) = (([,) ∘ (𝑙‘𝑗))‘𝑘))
48	47	cbvixpv 8912	. . . . . . . . . . . . . . . . . 18 ⊢ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑑) = X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)
49	46, 48	eqtrdi 2820	. . . . . . . . . . . . . . . . 17 ⊢ (𝑜 = 𝑗 → X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑) = X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘))
50	49	cbviunv 5007	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)
51	50	sseq2i 3974	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑) ↔ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘))
52	51	rabbii 3428	. . . . . . . . . . . . . 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 5211	. . . . . . . . . . . . 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 6883	. . . . . . . . . . . 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 6882	. . . . . . . . . . . 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 2816	. . . . . . . . . . 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 2855	. . . . . . . . . 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 6887	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑘 → (vol‘(([,) ∘ ℎ)‘𝑑)) = (vol‘(([,) ∘ ℎ)‘𝑘)))
59	58	cbvprodv 15967	. . . . . . . . . . . . . . . . 17 ⊢ ∏𝑑 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑑)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘))
60	59	mpteq2i 5211	. . . . . . . . . . . . . . . 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 6882	. . . . . . . . . . . . . . 15 ⊢ (𝑜 = 𝑗 → (𝑚‘𝑜) = (𝑚‘𝑗))
63	61, 62	fveq12d 6889	. . . . . . . . . . . . . 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 5219	. . . . . . . . . . . . 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 6885	. . . . . . . . . . . 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 6882	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑎 → ((voln*‘𝑋)‘𝑑) = ((voln*‘𝑋)‘𝑎))
68	67	oveq1d 7426	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑎 → (((voln*‘𝑋)‘𝑑) +𝑒 𝑓) = (((voln*‘𝑋)‘𝑎) +𝑒 𝑓))
69	66, 68	breq12d 5126	. . . . . . . . . 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 643	. . . . . . . . 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 3425	. . . . . . . 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 6881	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑖 → (𝑚‘𝑗) = (𝑖‘𝑗))
73	72	fveq2d 6886	. . . . . . . . . . . 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 5209	. . . . . . . . . . 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 6886	. . . . . . . . . 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 5123	. . . . . . . . 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 3433	. . . . . . . 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 2820	. . . . . . 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 5209	. . . . . 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 7419	. . . . . . . . 9 ⊢ (𝑓 = 𝑒 → (((voln*‘𝑋)‘𝑎) +𝑒 𝑓) = (((voln*‘𝑋)‘𝑎) +𝑒 𝑒))
81	80	breq2d 5125	. . . . . . . 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 3430	. . . . . . 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 5219	. . . . . 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 2820	. . . . 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 5219	. . . 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 47176	. . 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 45959	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))))
88	28, 87	pm2.61dan 824	1 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  {crab 3423  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  ∪ ciun 4960   class class class wbr 5113   ↦ cmpt 5196   × cxp 5660   ∘ ccom 5666  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ↑_m cmap 8823  Xcixp 8894  Fincfn 8942  ℝcr 11098  0cc0 11099  +∞cpnf 11239  ℝ^*cxr 11241   ≤ cle 11243  ℕcn 12232  ℝ⁺crp 13015   +_𝑒 cxad 13134  [,)cico 13373  [,]cicc 13374  ∏cprod 15956  volcvol 25590  Σ^{^}csumge0 46967  voln*covoln 47141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9609  ax-cc 10418  ax-ac2 10446  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-disj 5081  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7675  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-er 8693  df-map 8825  df-pm 8826  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fi 9370  df-sup 9401  df-inf 9402  df-oi 9471  df-dju 9886  df-card 9924  df-acn 9927  df-ac 10099  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-3 12303  df-n0 12504  df-z 12591  df-uz 12862  df-q 12972  df-rp 13016  df-xneg 13136  df-xadd 13137  df-xmul 13138  df-ioo 13375  df-ico 13377  df-icc 13378  df-fz 13535  df-fzo 13682  df-fl 13824  df-seq 14037  df-exp 14097  df-hash 14366  df-cj 15149  df-re 15150  df-im 15151  df-sqrt 15285  df-abs 15286  df-clim 15538  df-rlim 15539  df-sum 15737  df-prod 15957  df-rest 17474  df-topgen 17495  df-psmet 21482  df-xmet 21483  df-met 21484  df-bl 21485  df-mopn 21486  df-top 23019  df-topon 23036  df-bases 23071  df-cmp 23512  df-ovol 25591  df-vol 25592  df-sumge0 46968  df-ovoln 47142

This theorem is referenced by:  ovnome  47178  ovnsubadd2lem  47250