Theorem ovnsubadd 47143

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 6867	. . . . . 6 ⊢ (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅))
2	1	fveq1d 6869	. . . . 5 ⊢ (𝑋 = ∅ → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = ((voln*‘∅)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)))
3	2	adantl 485	. . . 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 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ ↑m 𝑋))
6		simpr 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
7	5, 6	ffvelcdmd 7066	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ 𝒫 (ℝ ↑m 𝑋))
8		elpwi 4562	. . . . . . . . . 10 ⊢ ((𝐴‘𝑛) ∈ 𝒫 (ℝ ↑m 𝑋) → (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋))
9	7, 8	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋))
10	9	ralrimiva 3154	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋))
11		iunss 5002	. . . . . . . 8 ⊢ (∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋) ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋))
12	10, 11	sylibr 236	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋))
13	12	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋))
14		oveq2 7404	. . . . . . 7 ⊢ (𝑋 = ∅ → (ℝ ↑m 𝑋) = (ℝ ↑m ∅))
15	14	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑m 𝑋) = (ℝ ↑m ∅))
16	13, 15	sseqtrd 3972	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m ∅))
17	16	ovn0val 47121	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = 0)
18	3, 17	eqtrd 2797	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = 0)
19		nnex 12216	. . . . . 6 ⊢ ℕ ∈ V
20	19	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ∈ V)
21		ovnsubadd.1	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ Fin)
22	21	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
23	22, 9	ovncl 47138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln*‘𝑋)‘(𝐴‘𝑛)) ∈ (0[,]+∞))
24		eqid 2762	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛))) = (𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))
25	23, 24	fmptd 7095	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛))):ℕ⟶(0[,]+∞))
26	20, 25	sge0ge0 46955	. . . 4 ⊢ (𝜑 → 0 ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))))
27	26	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))))
28	18, 27	eqbrtrd 5122	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))))
29	21, 12	ovnxrcl 47140	. . . 4 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ∈ ℝ*)
30	29	adantr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ∈ ℝ*)
31	20, 25	sge0xrcl 46956	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) ∈ ℝ*)
32	31	adantr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) ∈ ℝ*)
33	21	ad2antrr 736	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝑋 ∈ Fin)
34		neqne 2965	. . . . 5 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
35	34	ad2antlr 737	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝑋 ≠ ∅)
36	4	ad2antrr 736	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝐴:ℕ⟶𝒫 (ℝ ↑m 𝑋))
37		simpr 488	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
38		eqid 2762	. . . 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 3961	. . . . . 6 ⊢ (𝑏 = 𝑎 → (𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) ↔ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)))
40	39	rabbidv 3421	. . . . 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 5204	. . . 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 2762	. . . 4 ⊢ (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘))) = (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))
43		fveq2 6867	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑜 = 𝑗 → (𝑙‘𝑜) = (𝑙‘𝑗))
44	43	coeq2d 5834	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑜 = 𝑗 → ([,) ∘ (𝑙‘𝑜)) = ([,) ∘ (𝑙‘𝑗)))
45	44	fveq1d 6869	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑜 = 𝑗 → (([,) ∘ (𝑙‘𝑜))‘𝑑) = (([,) ∘ (𝑙‘𝑗))‘𝑑))
46	45	ixpeq2dv 8895	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑜 = 𝑗 → X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑) = X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑑))
47		fveq2 6867	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 = 𝑘 → (([,) ∘ (𝑙‘𝑗))‘𝑑) = (([,) ∘ (𝑙‘𝑗))‘𝑘))
48	47	cbvixpv 8897	. . . . . . . . . . . . . . . . . 18 ⊢ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑑) = X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)
49	46, 48	eqtrdi 2813	. . . . . . . . . . . . . . . . 17 ⊢ (𝑜 = 𝑗 → X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑) = X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘))
50	49	cbviunv 4996	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)
51	50	sseq2i 3965	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑) ↔ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘))
52	51	rabbii 3419	. . . . . . . . . . . . . 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 5196	. . . . . . . . . . . . 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 6868	. . . . . . . . . . . 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 6867	. . . . . . . . . . . 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 2809	. . . . . . . . . . 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 2848	. . . . . . . . . 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 6872	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑘 → (vol‘(([,) ∘ ℎ)‘𝑑)) = (vol‘(([,) ∘ ℎ)‘𝑘)))
59	58	cbvprodv 15944	. . . . . . . . . . . . . . . . 17 ⊢ ∏𝑑 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑑)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘))
60	59	mpteq2i 5196	. . . . . . . . . . . . . . . 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 6867	. . . . . . . . . . . . . . 15 ⊢ (𝑜 = 𝑗 → (𝑚‘𝑜) = (𝑚‘𝑗))
63	61, 62	fveq12d 6874	. . . . . . . . . . . . . 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 5204	. . . . . . . . . . . . 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 6870	. . . . . . . . . . . 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 6867	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑎 → ((voln*‘𝑋)‘𝑑) = ((voln*‘𝑋)‘𝑎))
68	67	oveq1d 7411	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑎 → (((voln*‘𝑋)‘𝑑) +𝑒 𝑓) = (((voln*‘𝑋)‘𝑎) +𝑒 𝑓))
69	66, 68	breq12d 5113	. . . . . . . . . 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 641	. . . . . . . . 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 3416	. . . . . . . 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 6866	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑖 → (𝑚‘𝑗) = (𝑖‘𝑗))
73	72	fveq2d 6871	. . . . . . . . . . . 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 5194	. . . . . . . . . . 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 6871	. . . . . . . . . 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 5110	. . . . . . . . 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 3424	. . . . . . . 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 2813	. . . . . . 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 5194	. . . . . 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 7404	. . . . . . . . 9 ⊢ (𝑓 = 𝑒 → (((voln*‘𝑋)‘𝑎) +𝑒 𝑓) = (((voln*‘𝑋)‘𝑎) +𝑒 𝑒))
81	80	breq2d 5112	. . . . . . . 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 3421	. . . . . . 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 5204	. . . . . 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 2813	. . . . 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 5204	. . . 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 47142	. . 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 45925	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))))
88	28, 87	pm2.61dan 822	1 ⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ∀wral 3076  ∃wrex 3086  {crab 3414  Vcvv 3454   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4555  ∪ ciun 4949   class class class wbr 5100   ↦ cmpt 5181   × cxp 5645   ∘ ccom 5651  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396   ↑_m cmap 8808  Xcixp 8879  Fincfn 8927  ℝcr 11072  0cc0 11073  +∞cpnf 11213  ℝ^*cxr 11215   ≤ cle 11217  ℕcn 12210  ℝ⁺crp 12993   +_𝑒 cxad 13112  [,)cico 13351  [,]cicc 13352  ∏cprod 15933  volcvol 25522  Σ^{^}csumge0 46933  voln*covoln 47107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-inf2 9596  ax-cc 10392  ax-ac2 10420  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150  ax-pre-sup 11151

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-disj 5068  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7660  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8678  df-map 8810  df-pm 8811  df-ixp 8880  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fi 9357  df-sup 9388  df-inf 9389  df-oi 9458  df-dju 9859  df-card 9897  df-acn 9900  df-ac 10072  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-div 11845  df-nn 12211  df-2 12280  df-3 12281  df-n0 12482  df-z 12569  df-uz 12840  df-q 12950  df-rp 12994  df-xneg 13114  df-xadd 13115  df-xmul 13116  df-ioo 13353  df-ico 13355  df-icc 13356  df-fz 13513  df-fzo 13660  df-fl 13802  df-seq 14015  df-exp 14075  df-hash 14344  df-cj 15126  df-re 15127  df-im 15128  df-sqrt 15262  df-abs 15263  df-clim 15515  df-rlim 15516  df-sum 15714  df-prod 15934  df-rest 17451  df-topgen 17472  df-psmet 21413  df-xmet 21414  df-met 21415  df-bl 21416  df-mopn 21417  df-top 22951  df-topon 22968  df-bases 23003  df-cmp 23444  df-ovol 25523  df-vol 25524  df-sumge0 46934  df-ovoln 47108

This theorem is referenced by:  ovnome  47144  ovnsubadd2lem  47216