Theorem ovnlecvr 47350

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 7084	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
6		elmapi 8853	. . . . . . . 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 47341	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘) ⊆ (ℝ ↑m 𝑋))
9	8	ralrimiva 3159	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘) ⊆ (ℝ ↑m 𝑋))
10		iunss 5011	. . . . 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 3948	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋))
13		eqid 2765	. . 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 47343	. 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 4035	. . . 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 12259	. . . . . . 7 ⊢ ℕ ∈ V
18	17	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ ∈ V)
19		icossicc 13484	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
20		nfv 1947	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ ℕ)
21	1	adantr 486	. . . . . . . . 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 47347	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐿‘(𝐼‘𝑗)) ∈ (0[,)+∞))
24	19, 23	sselid 3936	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐿‘(𝐼‘𝑗)) ∈ (0[,]+∞))
25		eqid 2765	. . . . . . 7 ⊢ (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))
26	24, 25	fmptd 7114	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗))):ℕ⟶(0[,]+∞))
27	18, 26	sge0xrcl 47177	. . . . 5 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) ∈ ℝ*)
28		ovex 7453	. . . . . . . . 9 ⊢ ((ℝ × ℝ) ↑m 𝑋) ∈ V
29	28, 17	pm3.2i 476	. . . . . . . 8 ⊢ (((ℝ × ℝ) ↑m 𝑋) ∈ V ∧ ℕ ∈ V)
30		elmapg 8843	. . . . . . . 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 5846	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝐼‘𝑗) → ([,) ∘ 𝑖) = ([,) ∘ (𝐼‘𝑗)))
34	33	fveq1d 6888	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼‘𝑗) → (([,) ∘ 𝑖)‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
35	34	fveq2d 6890	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘𝑗) → (vol‘(([,) ∘ 𝑖)‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
36	35	prodeq2ad 46386	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘𝑗) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
37		prodex 15987	. . . . . . . . . . 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 7018	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐿‘(𝐼‘𝑗)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
40	39	mpteq2dva 5206	. . . . . . . 8 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))
41	40	fveq2d 6890	. . . . . . 7 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
42	3, 41	jca 521	. . . . . 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 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑖 = 𝐼
44		fveq1 6885	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐼 → (𝑖‘𝑗) = (𝐼‘𝑗))
45	44	coeq2d 5850	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐼 → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ (𝐼‘𝑗)))
46	45	fveq1d 6888	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
47	46	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝐼 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
48	43, 47	ixpeq2d 45866	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘))
49	48	iuneq2d 4989	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘))
50	49	sseq2d 3970	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ↔ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘)))
51	46	fveq2d 6890	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
52	51	prodeq2ad 46386	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
53	52	mpteq2dv 5207	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))
54	53	fveq2d 6890	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
55	54	eqeq2d 2776	. . . . . . . 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 644	. . . . . . 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 3583	. . . . . 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 596	. . . . 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 521	. . . 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 2769	. . . . . . 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 642	. . . . . 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 3191	. . . . 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 3652	. . . 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 13382	. . 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 596	. 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 5135	1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  {crab 3418  Vcvv 3457   ⊆ wss 3906  ∅c0 4286  ∪ ciun 4958   class class class wbr 5111   ↦ cmpt 5194   × cxp 5661   ∘ ccom 5667  ⟶wf 6537  ‘cfv 6541  (class class class)co 7420   ↑_m cmap 8831  Xcixp 8902  Fincfn 8950  infcinf 9409  ℝcr 11119  0cc0 11120  +∞cpnf 11260  ℝ^*cxr 11262   < clt 11263   ≤ cle 11264  ℕcn 12253  [,)cico 13395  [,]cicc 13396  ∏cprod 15985  volcvol 25678  Σ^{^}csumge0 47154  voln*covoln 47328

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 7743  ax-inf2 9618  ax-cnex 11176  ax-resscn 11177  ax-1cn 11178  ax-icn 11179  ax-addcl 11180  ax-addrcl 11181  ax-mulcl 11182  ax-mulrcl 11183  ax-mulcom 11184  ax-addass 11185  ax-mulass 11186  ax-distr 11187  ax-i2m1 11188  ax-1ne0 11189  ax-1rid 11190  ax-rnegex 11191  ax-rrecex 11192  ax-cnre 11193  ax-pre-lttri 11194  ax-pre-lttrn 11195  ax-pre-ltadd 11196  ax-pre-mulgt0 11197  ax-pre-sup 11198

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-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 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7685  df-om 7870  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-1o 8460  df-2o 8461  df-er 8701  df-map 8833  df-pm 8834  df-ixp 8903  df-en 8951  df-dom 8952  df-sdom 8953  df-fin 8954  df-fi 9379  df-sup 9410  df-inf 9411  df-oi 9480  df-dju 9904  df-card 9942  df-pnf 11265  df-mnf 11266  df-xr 11267  df-ltxr 11268  df-le 11269  df-sub 11463  df-neg 11464  df-div 11892  df-nn 12254  df-2 12323  df-3 12324  df-n0 12525  df-z 12612  df-uz 12884  df-q 12994  df-rp 13038  df-xneg 13158  df-xadd 13159  df-xmul 13160  df-ioo 13397  df-ico 13399  df-icc 13400  df-fz 13557  df-fzo 13705  df-fl 13848  df-seq 14061  df-exp 14121  df-hash 14390  df-cj 15179  df-re 15180  df-im 15181  df-sqrt 15315  df-abs 15316  df-clim 15568  df-rlim 15569  df-sum 15767  df-prod 15986  df-rest 17502  df-topgen 17523  df-psmet 21569  df-xmet 21570  df-met 21571  df-bl 21572  df-mopn 21573  df-top 23106  df-topon 23123  df-bases 23158  df-cmp 23599  df-ovol 25679  df-vol 25680  df-sumge0 47155  df-ovoln 47329

This theorem is used by:  ovnsubaddlem1  47362