ovnlecvr2 - Mathbox for Glauco Siliprandi

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Theorem ovnlecvr2 47251

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. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

**Hypotheses**
Ref	Expression
ovnlecvr2.x	⊢ (𝜑 → 𝑋 ∈ Fin)
ovnlecvr2.c	⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑_m 𝑋))
ovnlecvr2.d	⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑_m 𝑋))
ovnlecvr2.s	⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
ovnlecvr2.l	⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑_m 𝑥), 𝑏 ∈ (ℝ ↑_m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))

**Assertion**
Ref	Expression
ovnlecvr2	⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))

Distinct variable groups: 𝐶,𝑎,𝑏,𝑘 𝐷,𝑎,𝑏,𝑘 𝑋,𝑎,𝑏,𝑗,𝑘,𝑥 𝜑,𝑎,𝑏,𝑗,𝑘,𝑥 Allowed substitution hints: 𝐴(𝑥,𝑗,𝑘,𝑎,𝑏) 𝐶(𝑥,𝑗) 𝐷(𝑥,𝑗) 𝐿(𝑥,𝑗,𝑘,𝑎,𝑏)

Proof of Theorem ovnlecvr2 Dummy variables 𝑖 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6882	. . . . . 6 ⊢ (𝑋 = ∅ → (voln‘𝑋) = (voln‘∅))
2	1	fveq1d 6884	. . . . 5 ⊢ (𝑋 = ∅ → ((voln‘𝑋)‘𝐴) = ((voln‘∅)‘𝐴))
3	2	adantl 486	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘𝐴) = ((voln‘∅)‘𝐴))
4		ovnlecvr2.s	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
5	4	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
6		1nn 12244	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
7		ne0i 4300	. . . . . . . . . . 11 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
8	6, 7	ax-mp 5	. . . . . . . . . 10 ⊢ ℕ ≠ ∅
9	8	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℕ ≠ ∅)
10		iunconst 4968	. . . . . . . . 9 ⊢ (ℕ ≠ ∅ → ∪ 𝑗 ∈ ℕ {∅} = {∅})
11	9, 10	syl 18	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ {∅} = {∅})
12	11	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑗 ∈ ℕ {∅} = {∅})
13		ixpeq1 8906	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = X𝑘 ∈ ∅ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
14		ixp0x 8924	. . . . . . . . . . . 12 ⊢ X𝑘 ∈ ∅ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = {∅}
15	14	a1i 11	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → X𝑘 ∈ ∅ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = {∅})
16	13, 15	eqtrd 2804	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = {∅})
17	16	adantr 485	. . . . . . . . 9 ⊢ ((𝑋 = ∅ ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = {∅})
18	17	iuneq2dv 4983	. . . . . . . 8 ⊢ (𝑋 = ∅ → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ {∅})
19	18	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ {∅})
20		reex 11191	. . . . . . . . 9 ⊢ ℝ ∈ V
21		mapdm0 8839	. . . . . . . . 9 ⊢ (ℝ ∈ V → (ℝ ↑_m ∅) = {∅})
22	20, 21	ax-mp 5	. . . . . . . 8 ⊢ (ℝ ↑_m ∅) = {∅}
23	22	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑_m ∅) = {∅})
24	12, 19, 23	3eqtr4d 2814	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = (ℝ ↑_m ∅))
25	5, 24	sseqtrd 3979	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ⊆ (ℝ ↑_m ∅))
26	25	ovn0val 47191	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘𝐴) = 0)
27	3, 26	eqtrd 2804	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) = 0)
28		nfv 1941	. . . . 5 ⊢ Ⅎ𝑗𝜑
29		nnex 12239	. . . . . 6 ⊢ ℕ ∈ V
30	29	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ∈ V)
31		icossicc 13463	. . . . . 6 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
32		ovnlecvr2.l	. . . . . . 7 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑_m 𝑥), 𝑏 ∈ (ℝ ↑_m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
33		ovnlecvr2.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ Fin)
34	33	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
35		ovnlecvr2.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑_m 𝑋))
36	35	ffvelcdmda 7080	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑_m 𝑋))
37		elmapi 8846	. . . . . . . 8 ⊢ ((𝐶‘𝑗) ∈ (ℝ ↑_m 𝑋) → (𝐶‘𝑗):𝑋⟶ℝ)
38	36, 37	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑋⟶ℝ)
39		ovnlecvr2.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑_m 𝑋))
40	39	ffvelcdmda 7080	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑_m 𝑋))
41		elmapi 8846	. . . . . . . 8 ⊢ ((𝐷‘𝑗) ∈ (ℝ ↑_m 𝑋) → (𝐷‘𝑗):𝑋⟶ℝ)
42	40, 41	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑋⟶ℝ)
43	32, 34, 38, 42	hoidmvcl 47223	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)) ∈ (0[,)+∞))
44	31, 43	sselid 3941	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)) ∈ (0[,]+∞))
45	28, 30, 44	sge0ge0mpt 47079	. . . 4 ⊢ (𝜑 → 0 ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
46	45	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
47	27, 46	eqbrtrd 5135	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
48		simpl 487	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝜑)
49		neqne 2972	. . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
50	49	adantl 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
51	33	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ∈ Fin)
52		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ≠ ∅)
53	38	ffvelcdmda 7080	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑗)‘𝑘) ∈ ℝ)
54	42	ffvelcdmda 7080	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐷‘𝑗)‘𝑘) ∈ ℝ)
55	54	rexrd 11259	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐷‘𝑗)‘𝑘) ∈ ℝ^*)
56		icossre 13455	. . . . . . . . . . . . 13 ⊢ ((((𝐶‘𝑗)‘𝑘) ∈ ℝ ∧ ((𝐷‘𝑗)‘𝑘) ∈ ℝ^*) → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ ℝ)
57	53, 55, 56	syl2anc 595	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ ℝ)
58	57	ralrimiva 3163	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∀𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ ℝ)
59		ss2ixp 8908	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ ℝ → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ℝ)
60	58, 59	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ℝ)
61	20	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℝ ∈ V)
62		ixpconstg 8904	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ Fin ∧ ℝ ∈ V) → X𝑘 ∈ 𝑋 ℝ = (ℝ ↑_m 𝑋))
63	33, 61, 62	syl2anc 595	. . . . . . . . . . 11 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ℝ = (ℝ ↑_m 𝑋))
64	63	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 ℝ = (ℝ ↑_m 𝑋))
65	60, 64	sseqtrd 3979	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ (ℝ ↑_m 𝑋))
66	65	ralrimiva 3163	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ (ℝ ↑_m 𝑋))
67		iunss 5011	. . . . . . . 8 ⊢ (∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ (ℝ ↑_m 𝑋) ↔ ∀𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ (ℝ ↑_m 𝑋))
68	66, 67	sylibr 237	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ (ℝ ↑_m 𝑋))
69	4, 68	sstrd 3953	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑_m 𝑋))
70	69	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐴 ⊆ (ℝ ↑_m 𝑋))
71		eqid 2769	. . . . 5 ⊢ {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)))))}
72	51, 52, 70, 71	ovnn0val 47192	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((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^*, < ))
73		ssrab2 4040	. . . . . 6 ⊢ {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^*
74	73	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 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)))))} ⊆ ℝ^*)
75	28, 30, 44	sge0xrclmpt 47069	. . . . . . . 8 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) ∈ ℝ^*)
76	75	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) ∈ ℝ^*)
77		opelxpi 5699	. . . . . . . . . . . . . 14 ⊢ ((((𝐶‘𝑗)‘𝑘) ∈ ℝ ∧ ((𝐷‘𝑗)‘𝑘) ∈ ℝ) → ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩ ∈ (ℝ × ℝ))
78	53, 54, 77	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩ ∈ (ℝ × ℝ))
79	78	fmpttd 7111	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩):𝑋⟶(ℝ × ℝ))
80	20, 20	xpex 7752	. . . . . . . . . . . . . 14 ⊢ (ℝ × ℝ) ∈ V
81	80	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (ℝ × ℝ) ∈ V)
82		elmapg 8836	. . . . . . . . . . . . 13 ⊢ (((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) ∈ ((ℝ × ℝ) ↑_m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩):𝑋⟶(ℝ × ℝ)))
83	81, 34, 82	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) ∈ ((ℝ × ℝ) ↑_m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩):𝑋⟶(ℝ × ℝ)))
84	79, 83	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) ∈ ((ℝ × ℝ) ↑_m 𝑋))
85	84	fmpttd 7111	. . . . . . . . . 10 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)):ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
86		ovexd 7446	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℝ × ℝ) ↑_m 𝑋) ∈ V)
87		elmapg 8836	. . . . . . . . . . 11 ⊢ ((((ℝ × ℝ) ↑_m 𝑋) ∈ V ∧ ℕ ∈ V) → ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ↔ (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)):ℕ⟶((ℝ × ℝ) ↑_m 𝑋)))
88	86, 30, 87	syl2anc 595	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ↔ (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)):ℕ⟶((ℝ × ℝ) ↑_m 𝑋)))
89	85, 88	mpbird 260	. . . . . . . . 9 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ))
90	89	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ))
91		simpr 489	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
92		mptexg 7220	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑋 ∈ Fin → (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) ∈ V)
93	33, 92	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) ∈ V)
94	93	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) ∈ V)
95		eqid 2769	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
96	95	fvmpt2 7002	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ℕ ∧ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) ∈ V) → ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗) = (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
97	91, 94, 96	syl2anc 595	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗) = (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
98	97	coeq2d 5849	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗)) = ([,) ∘ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)))
99	98	fveq1d 6884	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘) = (([,) ∘ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑘))
100	99	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘) = (([,) ∘ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑘))
101	79	adantr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩):𝑋⟶(ℝ × ℝ))
102		simpr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
103	101, 102	fvovco 45838	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑘) = ((1^st ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘))[,)(2^nd ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘))))
104		simpr 489	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
105		opex 5446	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩ ∈ V
106	105	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩ ∈ V)
107		eqid 2769	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) = (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)
108	107	fvmpt2 7002	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ 𝑋 ∧ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩ ∈ V) → ((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘) = ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)
109	104, 106, 108	syl2anc 595	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘) = ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)
110	109	fveq2d 6886	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1^st ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘)) = (1^st ‘⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
111		fvex 6895	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶‘𝑗)‘𝑘) ∈ V
112		fvex 6895	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐷‘𝑗)‘𝑘) ∈ V
113		op1stg 7998	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐶‘𝑗)‘𝑘) ∈ V ∧ ((𝐷‘𝑗)‘𝑘) ∈ V) → (1^st ‘⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) = ((𝐶‘𝑗)‘𝑘))
114	111, 112, 113	mp2an 704	. . . . . . . . . . . . . . . . . 18 ⊢ (1^st ‘⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) = ((𝐶‘𝑗)‘𝑘)
115	114	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1^st ‘⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) = ((𝐶‘𝑗)‘𝑘))
116	110, 115	eqtrd 2804	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1^st ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘)) = ((𝐶‘𝑗)‘𝑘))
117	109	fveq2d 6886	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2^nd ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘)) = (2^nd ‘⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
118	111, 112	op2nd 7995	. . . . . . . . . . . . . . . . . 18 ⊢ (2^nd ‘⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) = ((𝐷‘𝑗)‘𝑘)
119	118	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2^nd ‘⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) = ((𝐷‘𝑗)‘𝑘))
120	117, 119	eqtrd 2804	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2^nd ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘)) = ((𝐷‘𝑗)‘𝑘))
121	116, 120	oveq12d 7429	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((1^st ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘))[,)(2^nd ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘))) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
122	121	adantlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((1^st ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘))[,)(2^nd ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘))) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
123	100, 103, 122	3eqtrrd 2809	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
124	123	ixpeq2dva 8910	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
125	124	iuneq2dv 4983	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
126	4, 125	sseqtrd 3979	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
127	126	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
128		eqidd 2770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))))
129	51	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
130	52	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → 𝑋 ≠ ∅)
131	38	adantlr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑋⟶ℝ)
132	42	adantlr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑋⟶ℝ)
133	32, 129, 130, 131, 132	hoidmvn0val 47225	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)) = ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
134	133	mpteq2dva 5206	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))))
135	134	fveq2d 6886	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))))
136	123	eqcomd 2775	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
137	136	fveq2d 6886	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)) = (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
138	137	prodeq2dv 15976	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
139	138	mpteq2dva 5206	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))))
140	139	fveq2d 6886	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))))
141	140	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))))
142	128, 135, 141	3eqtr4d 2814	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))))
143	127, 142	jca 520	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))))))
144		nfcv 2931	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑖
145		nfmpt1 5212	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
146	144, 145	nfeq 2944	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
147		nfcv 2931	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝑖
148		nfcv 2931	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘ℕ
149		nfmpt1 5212	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)
150	148, 149	nfmpt 5211	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
151	147, 150	nfeq 2944	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
152		fveq1 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) → (𝑖‘𝑗) = ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))
153	152	coeq2d 5849	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗)))
154	153	fveq1d 6884	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
155	154	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
156	151, 155	ixpeq2d 45715	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) → X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
157	156	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
158	146, 157	iuneq2df 45694	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
159	158	sseq2d 3975	. . . . . . . . . 10 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) → (𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ↔ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))
160		nfv 1941	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 𝑗 ∈ ℕ
161	151, 160	nfan 1926	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) ∧ 𝑗 ∈ ℕ)
162	154	fveq2d 6886	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) → (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))
163	162	a1d 26	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) → (𝑘 ∈ 𝑋 → (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))))
164	163	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 → (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))))
165	161, 164	ralrimi 3269	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) ∧ 𝑗 ∈ ℕ) → ∀𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))
166	165	prodeq2d 15975	. . . . . . . . . . . . 13 ⊢ ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))
167	146, 166	mpteq2da 5205	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))))
168	167	fveq2d 6886	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))))
169	168	eqeq2d 2780	. . . . . . . . . 10 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) → ((Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) ↔ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))))))
170	159, 169	anbi12d 643	. . . . . . . . 9 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) → ((𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) ↔ (𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))))))
171	170	rspcev 3588	. . . . . . . 8 ⊢ (((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))))) → ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
172	90, 143, 171	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
173	76, 172	jca 520	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) ∈ ℝ^* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
174		eqeq1 2773	. . . . . . . . 9 ⊢ (𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) → (𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) ↔ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
175	174	anbi2d 641	. . . . . . . 8 ⊢ (𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘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 ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
176	175	rexbidv 3195	. . . . . . 7 ⊢ (𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘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 ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
177	176	elrab 3657	. . . . . 6 ⊢ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘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 ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) ∈ ℝ^* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
178	173, 177	sylibr 237	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘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)))))})
179		infxrlb 13361	. . . . 5 ⊢ (({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 ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘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 ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
180	74, 178, 179	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 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 ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
181	72, 180	eqbrtrd 5135	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln*‘𝑋)‘𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
182	48, 50, 181	syl2anc 595	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
183	47, 182	pm2.61dan 824	1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 ∃wrex 3095 {crab 3422 Vcvv 3461 ⊆ wss 3911 ∅c0 4292 ifcif 4490 {csn 4592 ⟨cop 4598 ∪ ciun 4958 class class class wbr 5111 ↦ cmpt 5194 × cxp 5660 ∘ ccom 5666 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ∈ cmpo 7413 1^st c1st 7984 2^nd c2nd 7985 ↑_m cmap 8824 Xcixp 8895 Fincfn 8943 infcinf 9401 ℝcr 11099 0cc0 11100 1c1 11101 +∞cpnf 11240 ℝ^*cxr 11242 < clt 11243 ≤ cle 11244 ℕcn 12233 [,)cico 13374 [,]cicc 13375 ∏cprod 15957 volcvol 25591 Σ^{^}csumge0 47003 voln*covoln 47177

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3375 df-reu 3376 df-rab 3423 df-v 3463 df-sbc 3752 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8694 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9472 df-dju 9887 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-n0 12505 df-z 12592 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-ioo 13376 df-ico 13378 df-icc 13379 df-fz 13536 df-fzo 13683 df-fl 13825 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15539 df-rlim 15540 df-sum 15738 df-prod 15958 df-rest 17475 df-topgen 17496 df-psmet 21483 df-xmet 21484 df-met 21485 df-bl 21486 df-mopn 21487 df-top 23020 df-topon 23037 df-bases 23072 df-cmp 23513 df-ovol 25592 df-vol 25593 df-sumge0 47004 df-ovoln 47178

This theorem is referenced by: hspmbllem2 47268

W3C validator