Theorem ovnlecvr 47300

Description: Given a subset of multidimensional reals and a set of half-open intervals that covers it, the Lebesgue outer measure of the set is bounded by the generalized sum of the pre-measure of the half-open intervals. The statement would also be true with 𝑋 the empty set, but covers are not used for the zero-dimensional case. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypotheses

Ref	Expression
ovnlecvr.x	⊢ (𝜑 → 𝑋 ∈ Fin)
ovnlecvr.n0	⊢ (𝜑 → 𝑋 ≠ ∅)
ovnlecvr.l	⊢ 𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))
ovnlecvr.i	⊢ (𝜑 → 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
ovnlecvr.ss	⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘))

Assertion

Ref	Expression
ovnlecvr	⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))))

Distinct variable groups:   𝐴,𝑖   𝑖,𝐼,𝑗,𝑘   𝑖,𝐿   𝑖,𝑋,𝑗,𝑘   𝜑,𝑖,𝑗,𝑘

Allowed substitution hints:   𝐴(𝑗, 𝑘)   𝐿(𝑗, 𝑘)

Proof of Theorem ovnlecvr

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovnlecvr.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
2		ovnlecvr.n0	. . 3 ⊢ (𝜑 → 𝑋 ≠ ∅)
3		ovnlecvr.ss	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘))
4		ovnlecvr.i	. . . . . . . . 9 ⊢ (𝜑 → 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
5	4	ffvelcdmda 7079	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
6		elmapi 8842	. . . . . . . 8 ⊢ ((𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
7	5, 6	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
8	7	hoissrrn 47291	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘) ⊆ (ℝ ↑m 𝑋))
9	8	ralrimiva 3157	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘) ⊆ (ℝ ↑m 𝑋))
10		iunss 5009	. . . . 5 ⊢ (∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘) ⊆ (ℝ ↑m 𝑋) ↔ ∀𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘) ⊆ (ℝ ↑m 𝑋))
11	9, 10	sylibr 237	. . . 4 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘) ⊆ (ℝ ↑m 𝑋))
12	3, 11	sstrd 3947	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋))
13		eqid 2763	. . 3 ⊢ {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 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}
14	1, 2, 12, 13	ovnn0val 47293	. 2 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) = inf({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)))))}, ℝ*, < ))
15		ssrab2 4034	. . . 4 ⊢ {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)))))} ⊆ ℝ*
16	15	a1i 11	. . 3 ⊢ (𝜑 → {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)))))} ⊆ ℝ*)
17		nnex 12243	. . . . . . 7 ⊢ ℕ ∈ V
18	17	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ ∈ V)
19		icossicc 13467	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
20		nfv 1944	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ ℕ)
21	1	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
22		ovnlecvr.l	. . . . . . . . 9 ⊢ 𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))
23	20, 21, 22, 7	hoiprodcl2 47297	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐿‘(𝐼‘𝑗)) ∈ (0[,)+∞))
24	19, 23	sselid 3935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐿‘(𝐼‘𝑗)) ∈ (0[,]+∞))
25		eqid 2763	. . . . . . 7 ⊢ (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))
26	24, 25	fmptd 7109	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗))):ℕ⟶(0[,]+∞))
27	18, 26	sge0xrcl 47127	. . . . 5 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) ∈ ℝ*)
28		ovex 7443	. . . . . . . . 9 ⊢ ((ℝ × ℝ) ↑m 𝑋) ∈ V
29	28, 17	pm3.2i 475	. . . . . . . 8 ⊢ (((ℝ × ℝ) ↑m 𝑋) ∈ V ∧ ℕ ∈ V)
30		elmapg 8832	. . . . . . . 8 ⊢ ((((ℝ × ℝ) ↑m 𝑋) ∈ V ∧ ℕ ∈ V) → (𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↔ 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋)))
31	29, 30	ax-mp 5	. . . . . . 7 ⊢ (𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↔ 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
32	4, 31	sylibr 237	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
33		coeq2 5844	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝐼‘𝑗) → ([,) ∘ 𝑖) = ([,) ∘ (𝐼‘𝑗)))
34	33	fveq1d 6883	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼‘𝑗) → (([,) ∘ 𝑖)‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
35	34	fveq2d 6885	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘𝑗) → (vol‘(([,) ∘ 𝑖)‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
36	35	prodeq2ad 46336	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘𝑗) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
37		prodex 15964	. . . . . . . . . . 11 ⊢ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) ∈ V
38	37	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) ∈ V)
39	22, 36, 5, 38	fvmptd3 7013	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐿‘(𝐼‘𝑗)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
40	39	mpteq2dva 5204	. . . . . . . 8 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))
41	40	fveq2d 6885	. . . . . . 7 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
42	3, 41	jca 520	. . . . . 6 ⊢ (𝜑 → (𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))))
43		nfv 1944	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑖 = 𝐼
44		fveq1 6880	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐼 → (𝑖‘𝑗) = (𝐼‘𝑗))
45	44	coeq2d 5848	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐼 → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ (𝐼‘𝑗)))
46	45	fveq1d 6883	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
47	46	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝐼 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
48	43, 47	ixpeq2d 45816	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘))
49	48	iuneq2d 4987	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘))
50	49	sseq2d 3969	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ↔ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘)))
51	46	fveq2d 6885	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
52	51	prodeq2ad 46336	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
53	52	mpteq2dv 5205	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))
54	53	fveq2d 6885	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
55	54	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))))
56	50, 55	anbi12d 643	. . . . . . 7 ⊢ (𝑖 = 𝐼 → ((𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) ↔ (𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))))
57	56	rspcev 3581	. . . . . 6 ⊢ ((𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))) → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
58	32, 42, 57	syl2anc 595	. . . . 5 ⊢ (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
59	27, 58	jca 520	. . . 4 ⊢ (𝜑 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
60		eqeq1 2767	. . . . . . 7 ⊢ (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) → (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
61	60	anbi2d 641	. . . . . 6 ⊢ (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) → ((𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) ↔ (𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
62	61	rexbidv 3189	. . . . 5 ⊢ (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) → (∃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 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
63	62	elrab 3650	. . . 4 ⊢ ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) ∈ {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)))))} ↔ ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
64	59, 63	sylibr 237	. . 3 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) ∈ {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)))))})
65		infxrlb 13365	. . 3 ⊢ (({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)))))} ⊆ ℝ* ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) ∈ {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)))))}) → inf({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)))))}, ℝ*, < ) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))))
66	16, 64, 65	syl2anc 595	. 2 ⊢ (𝜑 → inf({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)))))}, ℝ*, < ) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))))
67	14, 66	eqbrtrd 5133	1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  {crab 3416  Vcvv 3455   ⊆ wss 3905  ∅c0 4286  ∪ ciun 4956   class class class wbr 5109   ↦ cmpt 5192   × cxp 5659   ∘ ccom 5665  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8820  Xcixp 8891  Fincfn 8939  infcinf 9397  ℝcr 11103  0cc0 11104  +∞cpnf 11244  ℝ^*cxr 11246   < clt 11247   ≤ cle 11248  ℕcn 12237  [,)cico 13378  [,]cicc 13379  ∏cprod 15962  volcvol 25631  Σ^{^}csumge0 47104  voln*covoln 47278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606  ax-cnex 11160  ax-resscn 11161  ax-1cn 11162  ax-icn 11163  ax-addcl 11164  ax-addrcl 11165  ax-mulcl 11166  ax-mulrcl 11167  ax-mulcom 11168  ax-addass 11169  ax-mulass 11170  ax-distr 11171  ax-i2m1 11172  ax-1ne0 11173  ax-1rid 11174  ax-rnegex 11175  ax-rrecex 11176  ax-cnre 11177  ax-pre-lttri 11178  ax-pre-lttrn 11179  ax-pre-ltadd 11180  ax-pre-mulgt0 11181  ax-pre-sup 11182

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fi 9367  df-sup 9398  df-inf 9399  df-oi 9468  df-dju 9892  df-card 9930  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11447  df-neg 11448  df-div 11876  df-nn 12238  df-2 12307  df-3 12308  df-n0 12509  df-z 12596  df-uz 12867  df-q 12977  df-rp 13021  df-xneg 13141  df-xadd 13142  df-xmul 13143  df-ioo 13380  df-ico 13382  df-icc 13383  df-fz 13540  df-fzo 13688  df-fl 13830  df-seq 14043  df-exp 14103  df-hash 14372  df-cj 15155  df-re 15156  df-im 15157  df-sqrt 15291  df-abs 15292  df-clim 15544  df-rlim 15545  df-sum 15743  df-prod 15963  df-rest 17479  df-topgen 17500  df-psmet 21523  df-xmet 21524  df-met 21525  df-bl 21526  df-mopn 21527  df-top 23060  df-topon 23077  df-bases 23112  df-cmp 23553  df-ovol 25632  df-vol 25633  df-sumge0 47105  df-ovoln 47279

This theorem is used by:  ovnsubaddlem1  47312