ovnsubadd - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ovnsubadd		Structured version Visualization version GIF version

Theorem ovnsubadd 45367

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 6891	. . . . . 6 ⊢ (𝑋 = ∅ → (voln‘𝑋) = (voln‘∅))
2	1	fveq1d 6893	. . . . 5 ⊢ (𝑋 = ∅ → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = ((voln‘∅)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)))
3	2	adantl 482	. . . 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 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
6		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
7	5, 6	ffvelcdmd 7087	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ 𝒫 (ℝ ↑_m 𝑋))
8		elpwi 4609	. . . . . . . . . 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 3146	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋))
11		iunss 5048	. . . . . . . 8 ⊢ (∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋) ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋))
12	10, 11	sylibr 233	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋))
13	12	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋))
14		oveq2 7419	. . . . . . 7 ⊢ (𝑋 = ∅ → (ℝ ↑_m 𝑋) = (ℝ ↑_m ∅))
15	14	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑_m 𝑋) = (ℝ ↑_m ∅))
16	13, 15	sseqtrd 4022	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑_m ∅))
17	16	ovn0val 45345	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = 0)
18	3, 17	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = 0)
19		nnex 12220	. . . . . 6 ⊢ ℕ ∈ V
20	19	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ∈ V)
21		ovnsubadd.1	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ Fin)
22	21	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
23	22, 9	ovncl 45362	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln*‘𝑋)‘(𝐴‘𝑛)) ∈ (0[,]+∞))
24		eqid 2732	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))
25	23, 24	fmptd 7115	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛))):ℕ⟶(0[,]+∞))
26	20, 25	sge0ge0 45179	. . . 4 ⊢ (𝜑 → 0 ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))))
27	26	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))))
28	18, 27	eqbrtrd 5170	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))))
29	21, 12	ovnxrcl 45364	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ∈ ℝ^)
30	29	adantr 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ∈ ℝ^)
31	20, 25	sge0xrcl 45180	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) ∈ ℝ^)
32	31	adantr 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) ∈ ℝ^)
33	21	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ⁺) → 𝑋 ∈ Fin)
34		neqne 2948	. . . . 5 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
35	34	ad2antlr 725	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ⁺) → 𝑋 ≠ ∅)
36	4	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ⁺) → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
37		simpr 485	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ⁺)
38		eqid 2732	. . . 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 4007	. . . . . 6 ⊢ (𝑏 = 𝑎 → (𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) ↔ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)))
40	39	rabbidv 3440	. . . . 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 5261	. . . 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 2732	. . . 4 ⊢ (ℎ ∈ ((ℝ × ℝ) ↑_m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘))) = (ℎ ∈ ((ℝ × ℝ) ↑_m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))
43		fveq2 6891	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑜 = 𝑗 → (𝑙‘𝑜) = (𝑙‘𝑗))
44	43	coeq2d 5862	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑜 = 𝑗 → ([,) ∘ (𝑙‘𝑜)) = ([,) ∘ (𝑙‘𝑗)))
45	44	fveq1d 6893	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑜 = 𝑗 → (([,) ∘ (𝑙‘𝑜))‘𝑑) = (([,) ∘ (𝑙‘𝑗))‘𝑑))
46	45	ixpeq2dv 8909	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑜 = 𝑗 → X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑) = X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑑))
47		fveq2 6891	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 = 𝑘 → (([,) ∘ (𝑙‘𝑗))‘𝑑) = (([,) ∘ (𝑙‘𝑗))‘𝑘))
48	47	cbvixpv 8911	. . . . . . . . . . . . . . . . . 18 ⊢ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑑) = X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)
49	46, 48	eqtrdi 2788	. . . . . . . . . . . . . . . . 17 ⊢ (𝑜 = 𝑗 → X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑) = X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘))
50	49	cbviunv 5043	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)
51	50	sseq2i 4011	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑) ↔ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘))
52	51	rabbii 3438	. . . . . . . . . . . . . 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 5253	. . . . . . . . . . . . 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 6892	. . . . . . . . . . . 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 6891	. . . . . . . . . . . 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 2784	. . . . . . . . . . 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 2819	. . . . . . . . . 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 6896	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑘 → (vol‘(([,) ∘ ℎ)‘𝑑)) = (vol‘(([,) ∘ ℎ)‘𝑘)))
59	58	cbvprodv 15862	. . . . . . . . . . . . . . . . 17 ⊢ ∏𝑑 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑑)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘))
60	59	mpteq2i 5253	. . . . . . . . . . . . . . . 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 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑜 = 𝑗 → (𝑚‘𝑜) = (𝑚‘𝑗))
63	61, 62	fveq12d 6898	. . . . . . . . . . . . . 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 5261	. . . . . . . . . . . . 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 6894	. . . . . . . . . . . 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 6891	. . . . . . . . . . . 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 5161	. . . . . . . . . 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 631	. . . . . . . . 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 3434	. . . . . . . 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 6890	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑖 → (𝑚‘𝑗) = (𝑖‘𝑗))
73	72	fveq2d 6895	. . . . . . . . . . . 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 5250	. . . . . . . . . . 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 6895	. . . . . . . . . 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 5158	. . . . . . . . 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 3442	. . . . . . . 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 2788	. . . . . . 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 5250	. . . . . 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 5160	. . . . . . . 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 3440	. . . . . . 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 5261	. . . . . 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 2788	. . . . 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 5261	. . . 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 45366	. . 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 44141	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))))
88	28, 87	pm2.61dan 811	1 ⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3432 Vcvv 3474 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 ∪ ciun 4997 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 ∘ ccom 5680 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ↑_m cmap 8822 Xcixp 8893 Fincfn 8941 ℝcr 11111 0cc0 11112 +∞cpnf 11247 ℝ^*cxr 11249 ≤ cle 11251 ℕcn 12214 ℝ⁺crp 12976 +_𝑒 cxad 13092 [,)cico 13328 [,]cicc 13329 ∏cprod 15851 volcvol 24987 Σ^{^}csumge0 45157 voln*covoln 45331

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cc 10432 ax-ac2 10460 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-ac 10113 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-ioo 13330 df-ico 13332 df-icc 13333 df-fz 13487 df-fzo 13630 df-fl 13759 df-seq 13969 df-exp 14030 df-hash 14293 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-rlim 15435 df-sum 15635 df-prod 15852 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-rest 17370 df-0g 17389 df-topgen 17391 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-grp 18824 df-minusg 18825 df-subg 19005 df-cmn 19652 df-abl 19653 df-mgp 19990 df-ur 20007 df-ring 20060 df-cring 20061 df-oppr 20154 df-dvdsr 20175 df-unit 20176 df-invr 20206 df-dvr 20219 df-drng 20363 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-cnfld 20951 df-top 22403 df-topon 22420 df-bases 22456 df-cmp 22898 df-ovol 24988 df-vol 24989 df-sumge0 45158 df-ovoln 45332

This theorem is referenced by: ovnome 45368 ovnsubadd2lem 45440

W3C validator