ovnlecvr2 - Mathbox for Glauco Siliprandi

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Theorem ovnlecvr2 46707

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 6822	. . . . . 6 ⊢ (𝑋 = ∅ → (voln‘𝑋) = (voln‘∅))
2	1	fveq1d 6824	. . . . 5 ⊢ (𝑋 = ∅ → ((voln‘𝑋)‘𝐴) = ((voln‘∅)‘𝐴))
3	2	adantl 481	. . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
6		1nn 12136	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
7		ne0i 4288	. . . . . . . . . . 11 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
8	6, 7	ax-mp 5	. . . . . . . . . 10 ⊢ ℕ ≠ ∅
9	8	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℕ ≠ ∅)
10		iunconst 4949	. . . . . . . . 9 ⊢ (ℕ ≠ ∅ → ∪ 𝑗 ∈ ℕ {∅} = {∅})
11	9, 10	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ {∅} = {∅})
12	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑗 ∈ ℕ {∅} = {∅})
13		ixpeq1 8832	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = X𝑘 ∈ ∅ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
14		ixp0x 8850	. . . . . . . . . . . 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 2766	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = {∅})
17	16	adantr 480	. . . . . . . . 9 ⊢ ((𝑋 = ∅ ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = {∅})
18	17	iuneq2dv 4964	. . . . . . . 8 ⊢ (𝑋 = ∅ → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ {∅})
19	18	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ {∅})
20		reex 11097	. . . . . . . . 9 ⊢ ℝ ∈ V
21		mapdm0 8766	. . . . . . . . 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 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = (ℝ ↑_m ∅))
25	5, 24	sseqtrd 3966	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ⊆ (ℝ ↑_m ∅))
26	25	ovn0val 46647	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘𝐴) = 0)
27	3, 26	eqtrd 2766	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) = 0)
28		nfv 1915	. . . . 5 ⊢ Ⅎ𝑗𝜑
29		nnex 12131	. . . . . 6 ⊢ ℕ ∈ V
30	29	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ∈ V)
31		icossicc 13336	. . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
35		ovnlecvr2.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑_m 𝑋))
36	35	ffvelcdmda 7017	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑_m 𝑋))
37		elmapi 8773	. . . . . . . 8 ⊢ ((𝐶‘𝑗) ∈ (ℝ ↑_m 𝑋) → (𝐶‘𝑗):𝑋⟶ℝ)
38	36, 37	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑋⟶ℝ)
39		ovnlecvr2.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑_m 𝑋))
40	39	ffvelcdmda 7017	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑_m 𝑋))
41		elmapi 8773	. . . . . . . 8 ⊢ ((𝐷‘𝑗) ∈ (ℝ ↑_m 𝑋) → (𝐷‘𝑗):𝑋⟶ℝ)
42	40, 41	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑋⟶ℝ)
43	32, 34, 38, 42	hoidmvcl 46679	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)) ∈ (0[,)+∞))
44	31, 43	sselid 3927	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)) ∈ (0[,]+∞))
45	28, 30, 44	sge0ge0mpt 46535	. . . 4 ⊢ (𝜑 → 0 ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
46	45	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
47	27, 46	eqbrtrd 5111	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
48		simpl 482	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝜑)
49		neqne 2936	. . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
50	49	adantl 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
51	33	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ∈ Fin)
52		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ≠ ∅)
53	38	ffvelcdmda 7017	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑗)‘𝑘) ∈ ℝ)
54	42	ffvelcdmda 7017	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐷‘𝑗)‘𝑘) ∈ ℝ)
55	54	rexrd 11162	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐷‘𝑗)‘𝑘) ∈ ℝ^*)
56		icossre 13328	. . . . . . . . . . . . 13 ⊢ ((((𝐶‘𝑗)‘𝑘) ∈ ℝ ∧ ((𝐷‘𝑗)‘𝑘) ∈ ℝ^*) → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ ℝ)
57	53, 55, 56	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ ℝ)
58	57	ralrimiva 3124	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∀𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ ℝ)
59		ss2ixp 8834	. . . . . . . . . . 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 17	. . . . . . . . . 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 8830	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ Fin ∧ ℝ ∈ V) → X𝑘 ∈ 𝑋 ℝ = (ℝ ↑_m 𝑋))
63	33, 61, 62	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ℝ = (ℝ ↑_m 𝑋))
64	63	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 ℝ = (ℝ ↑_m 𝑋))
65	60, 64	sseqtrd 3966	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ (ℝ ↑_m 𝑋))
66	65	ralrimiva 3124	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ (ℝ ↑_m 𝑋))
67		iunss 4992	. . . . . . . 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 234	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ (ℝ ↑_m 𝑋))
69	4, 68	sstrd 3940	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑_m 𝑋))
70	69	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐴 ⊆ (ℝ ↑_m 𝑋))
71		eqid 2731	. . . . 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 46648	. . . 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 4027	. . . . . 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 46525	. . . . . . . 8 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) ∈ ℝ^*)
76	75	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) ∈ ℝ^*)
77		opelxpi 5651	. . . . . . . . . . . . . 14 ⊢ ((((𝐶‘𝑗)‘𝑘) ∈ ℝ ∧ ((𝐷‘𝑗)‘𝑘) ∈ ℝ) → ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩ ∈ (ℝ × ℝ))
78	53, 54, 77	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩ ∈ (ℝ × ℝ))
79	78	fmpttd 7048	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩):𝑋⟶(ℝ × ℝ))
80	20, 20	xpex 7686	. . . . . . . . . . . . . 14 ⊢ (ℝ × ℝ) ∈ V
81	80	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (ℝ × ℝ) ∈ V)
82		elmapg 8763	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 257	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) ∈ ((ℝ × ℝ) ↑_m 𝑋))
85	84	fmpttd 7048	. . . . . . . . . 10 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)):ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
86		ovexd 7381	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℝ × ℝ) ↑_m 𝑋) ∈ V)
87		elmapg 8763	. . . . . . . . . . 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 584	. . . . . . . . . 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 257	. . . . . . . . 9 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ))
90	89	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ))
91		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
92		mptexg 7155	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑋 ∈ Fin → (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) ∈ V)
93	33, 92	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) ∈ V)
94	93	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) ∈ V)
95		eqid 2731	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
96	95	fvmpt2 6940	. . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . 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 5801	. . . . . . . . . . . . . . . 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 6824	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩):𝑋⟶(ℝ × ℝ))
102		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
103	101, 102	fvovco 45289	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
105		opex 5402	. . . . . . . . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) = (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)
108	107	fvmpt2 6940	. . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘) = ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)
110	109	fveq2d 6826	. . . . . . . . . . . . . . . . 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 6835	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶‘𝑗)‘𝑘) ∈ V
112		fvex 6835	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐷‘𝑗)‘𝑘) ∈ V
113		op1stg 7933	. . . . . . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1^st ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘)) = ((𝐶‘𝑗)‘𝑘))
117	109	fveq2d 6826	. . . . . . . . . . . . . . . . 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 7930	. . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2^nd ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘)) = ((𝐷‘𝑗)‘𝑘))
121	116, 120	oveq12d 7364	. . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . 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 8836	. . . . . . . . . . . 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 4964	. . . . . . . . . . 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 3966	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
127	126	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
128		eqidd 2732	. . . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
130	52	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → 𝑋 ≠ ∅)
131	38	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑋⟶ℝ)
132	42	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑋⟶ℝ)
133	32, 129, 130, 131, 132	hoidmvn0val 46681	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)) = ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
134	133	mpteq2dva 5182	. . . . . . . . . . 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 6826	. . . . . . . . . 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 2737	. . . . . . . . . . . . . . 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 6826	. . . . . . . . . . . . . 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 15829	. . . . . . . . . . . . 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 5182	. . . . . . . . . . . 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 6826	. . . . . . . . . . 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 480	. . . . . . . . . 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 2776	. . . . . . . . 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 511	. . . . . . . 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 2894	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑖
145		nfmpt1 5188	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
146	144, 145	nfeq 2908	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
147		nfcv 2894	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝑖
148		nfcv 2894	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘ℕ
149		nfmpt1 5188	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)
150	148, 149	nfmpt 5187	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
151	147, 150	nfeq 2908	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
152		fveq1 6821	. . . . . . . . . . . . . . . . 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 5801	. . . . . . . . . . . . . . . 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 6824	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 45164	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 45143	. . . . . . . . . . 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 3962	. . . . . . . . . 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 1915	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 𝑗 ∈ ℕ
161	151, 160	nfan 1900	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) ∧ 𝑗 ∈ ℕ)
162	154	fveq2d 6826	. . . . . . . . . . . . . . . . 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 25	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 3230	. . . . . . . . . . . . . 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 15828	. . . . . . . . . . . . 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 5181	. . . . . . . . . . . 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 6826	. . . . . . . . . . 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 2742	. . . . . . . . . 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 632	. . . . . . . . 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 3572	. . . . . . . 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 584	. . . . . . 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 511	. . . . . 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 2735	. . . . . . . . 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 630	. . . . . . . 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 3156	. . . . . . 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 3642	. . . . . 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 234	. . . . 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 13234	. . . . 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 584	. . . 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 5111	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln*‘𝑋)‘𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
182	48, 50, 181	syl2anc 584	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
183	47, 182	pm2.61dan 812	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 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 {crab 3395 Vcvv 3436 ⊆ wss 3897 ∅c0 4280 ifcif 4472 {csn 4573 ⟨cop 4579 ∪ ciun 4939 class class class wbr 5089 ↦ cmpt 5170 × cxp 5612 ∘ ccom 5618 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 1^st c1st 7919 2^nd c2nd 7920 ↑_m cmap 8750 Xcixp 8821 Fincfn 8869 infcinf 9325 ℝcr 11005 0cc0 11006 1c1 11007 +∞cpnf 11143 ℝ^*cxr 11145 < clt 11146 ≤ cle 11147 ℕcn 12125 [,)cico 13247 [,]cicc 13248 ∏cprod 15810 volcvol 25391 Σ^{^}csumge0 46459 voln*covoln 46633

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-dju 9794 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-n0 12382 df-z 12469 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-rlim 15396 df-sum 15594 df-prod 15811 df-rest 17326 df-topgen 17347 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-top 22809 df-topon 22826 df-bases 22861 df-cmp 23302 df-ovol 25392 df-vol 25393 df-sumge0 46460 df-ovoln 46634

This theorem is referenced by: hspmbllem2 46724

W3C validator