Theorem ovnlecvr 45573

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 7086	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
6		elmapi 8845	. . . . . . . 8 ⊢ ((𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
7	5, 6	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
8	7	hoissrrn 45564	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘) ⊆ (ℝ ↑m 𝑋))
9	8	ralrimiva 3146	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘) ⊆ (ℝ ↑m 𝑋))
10		iunss 5048	. . . . 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 233	. . . 4 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘) ⊆ (ℝ ↑m 𝑋))
12	3, 11	sstrd 3992	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋))
13		eqid 2732	. . 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 45566	. 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 4077	. . . 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 12222	. . . . . . 7 ⊢ ℕ ∈ V
18	17	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ ∈ V)
19		icossicc 13417	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
20		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ ℕ)
21	1	adantr 481	. . . . . . . . 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 45570	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐿‘(𝐼‘𝑗)) ∈ (0[,)+∞))
24	19, 23	sselid 3980	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐿‘(𝐼‘𝑗)) ∈ (0[,]+∞))
25		eqid 2732	. . . . . . 7 ⊢ (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))
26	24, 25	fmptd 7115	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗))):ℕ⟶(0[,]+∞))
27	18, 26	sge0xrcl 45400	. . . . 5 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) ∈ ℝ*)
28		ovex 7444	. . . . . . . . 9 ⊢ ((ℝ × ℝ) ↑m 𝑋) ∈ V
29	28, 17	pm3.2i 471	. . . . . . . 8 ⊢ (((ℝ × ℝ) ↑m 𝑋) ∈ V ∧ ℕ ∈ V)
30		elmapg 8835	. . . . . . . 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 233	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
33		coeq2 5858	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝐼‘𝑗) → ([,) ∘ 𝑖) = ([,) ∘ (𝐼‘𝑗)))
34	33	fveq1d 6893	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼‘𝑗) → (([,) ∘ 𝑖)‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
35	34	fveq2d 6895	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘𝑗) → (vol‘(([,) ∘ 𝑖)‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
36	35	prodeq2ad 44607	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘𝑗) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
37		prodex 15855	. . . . . . . . . . 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 7021	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐿‘(𝐼‘𝑗)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
40	39	mpteq2dva 5248	. . . . . . . 8 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))
41	40	fveq2d 6895	. . . . . . 7 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
42	3, 41	jca 512	. . . . . 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 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑖 = 𝐼
44		fveq1 6890	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐼 → (𝑖‘𝑗) = (𝐼‘𝑗))
45	44	coeq2d 5862	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐼 → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ (𝐼‘𝑗)))
46	45	fveq1d 6893	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
47	46	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝐼 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
48	43, 47	ixpeq2d 44057	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘))
49	48	iuneq2d 5026	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘))
50	49	sseq2d 4014	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ↔ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘)))
51	46	fveq2d 6895	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
52	51	prodeq2ad 44607	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
53	52	mpteq2dv 5250	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))
54	53	fveq2d 6895	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
55	54	eqeq2d 2743	. . . . . . . 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 631	. . . . . . 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 3612	. . . . . 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 584	. . . . 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 512	. . . 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 2736	. . . . . . 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 629	. . . . . 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 3178	. . . . 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 3683	. . . 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 233	. . 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 13317	. . 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 584	. 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 5170	1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3432  Vcvv 3474   ⊆ wss 3948  ∅c0 4322  ∪ 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  infcinf 9438  ℝcr 11111  0cc0 11112  +∞cpnf 11249  ℝ^*cxr 11251   < clt 11252   ≤ cle 11253  ℕcn 12216  [,)cico 13330  [,]cicc 13331  ∏cprod 15853  volcvol 25204  Σ^{^}csumge0 45377  voln*covoln 45551

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-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

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-op 4635  df-uni 4909  df-int 4951  df-iun 4999  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-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-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-q 12937  df-rp 12979  df-xneg 13096  df-xadd 13097  df-xmul 13098  df-ioo 13332  df-ico 13334  df-icc 13335  df-fz 13489  df-fzo 13632  df-fl 13761  df-seq 13971  df-exp 14032  df-hash 14295  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15436  df-rlim 15437  df-sum 15637  df-prod 15854  df-rest 17372  df-topgen 17393  df-psmet 21136  df-xmet 21137  df-met 21138  df-bl 21139  df-mopn 21140  df-top 22616  df-topon 22633  df-bases 22669  df-cmp 23111  df-ovol 25205  df-vol 25206  df-sumge0 45378  df-ovoln 45552

This theorem is referenced by:  ovnsubaddlem1  45585