ovnlecvr2 - Mathbox for Glauco Siliprandi

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Theorem ovnlecvr2 46997

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 6844	. . . . . 6 ⊢ (𝑋 = ∅ → (voln‘𝑋) = (voln‘∅))
2	1	fveq1d 6846	. . . . 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 12170	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
7		ne0i 4295	. . . . . . . . . . 11 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
8	6, 7	ax-mp 5	. . . . . . . . . 10 ⊢ ℕ ≠ ∅
9	8	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℕ ≠ ∅)
10		iunconst 4958	. . . . . . . . 9 ⊢ (ℕ ≠ ∅ → ∪ 𝑗 ∈ ℕ {∅} = {∅})
11	9, 10	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ {∅} = {∅})
12	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑗 ∈ ℕ {∅} = {∅})
13		ixpeq1 8860	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = X𝑘 ∈ ∅ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
14		ixp0x 8878	. . . . . . . . . . . 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 2772	. . . . . . . . . 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 4973	. . . . . . . 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 11131	. . . . . . . . 9 ⊢ ℝ ∈ V
21		mapdm0 8793	. . . . . . . . 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 2782	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = (ℝ ↑_m ∅))
25	5, 24	sseqtrd 3972	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ⊆ (ℝ ↑_m ∅))
26	25	ovn0val 46937	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘𝐴) = 0)
27	3, 26	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) = 0)
28		nfv 1916	. . . . 5 ⊢ Ⅎ𝑗𝜑
29		nnex 12165	. . . . . 6 ⊢ ℕ ∈ V
30	29	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ∈ V)
31		icossicc 13366	. . . . . 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 7040	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑_m 𝑋))
37		elmapi 8800	. . . . . . . 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 7040	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑_m 𝑋))
41		elmapi 8800	. . . . . . . 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 46969	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)) ∈ (0[,)+∞))
44	31, 43	sselid 3933	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)) ∈ (0[,]+∞))
45	28, 30, 44	sge0ge0mpt 46825	. . . 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 5122	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
48		simpl 482	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝜑)
49		neqne 2941	. . . 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 7040	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑗)‘𝑘) ∈ ℝ)
54	42	ffvelcdmda 7040	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐷‘𝑗)‘𝑘) ∈ ℝ)
55	54	rexrd 11196	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐷‘𝑗)‘𝑘) ∈ ℝ^*)
56		icossre 13358	. . . . . . . . . . . . 13 ⊢ ((((𝐶‘𝑗)‘𝑘) ∈ ℝ ∧ ((𝐷‘𝑗)‘𝑘) ∈ ℝ^*) → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ ℝ)
57	53, 55, 56	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ ℝ)
58	57	ralrimiva 3130	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∀𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ ℝ)
59		ss2ixp 8862	. . . . . . . . . . 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 8858	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ Fin ∧ ℝ ∈ V) → X𝑘 ∈ 𝑋 ℝ = (ℝ ↑_m 𝑋))
63	33, 61, 62	syl2anc 585	. . . . . . . . . . 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 3972	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ (ℝ ↑_m 𝑋))
66	65	ralrimiva 3130	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ (ℝ ↑_m 𝑋))
67		iunss 5002	. . . . . . . 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 3946	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑_m 𝑋))
70	69	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐴 ⊆ (ℝ ↑_m 𝑋))
71		eqid 2737	. . . . 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 46938	. . . 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 4034	. . . . . 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 46815	. . . . . . . 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 5671	. . . . . . . . . . . . . 14 ⊢ ((((𝐶‘𝑗)‘𝑘) ∈ ℝ ∧ ((𝐷‘𝑗)‘𝑘) ∈ ℝ) → ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩ ∈ (ℝ × ℝ))
78	53, 54, 77	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩ ∈ (ℝ × ℝ))
79	78	fmpttd 7071	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩):𝑋⟶(ℝ × ℝ))
80	20, 20	xpex 7710	. . . . . . . . . . . . . 14 ⊢ (ℝ × ℝ) ∈ V
81	80	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (ℝ × ℝ) ∈ V)
82		elmapg 8790	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 7071	. . . . . . . . . 10 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)):ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
86		ovexd 7405	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℝ × ℝ) ↑_m 𝑋) ∈ V)
87		elmapg 8790	. . . . . . . . . . 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 585	. . . . . . . . . 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 7179	. . . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
96	95	fvmpt2 6963	. . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . 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 5821	. . . . . . . . . . . . . . . 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 6846	. . . . . . . . . . . . . . 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 45581	. . . . . . . . . . . . . 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 5421	. . . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) = (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)
108	107	fvmpt2 6963	. . . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘) = ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)
110	109	fveq2d 6848	. . . . . . . . . . . . . . . . 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 6857	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶‘𝑗)‘𝑘) ∈ V
112		fvex 6857	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐷‘𝑗)‘𝑘) ∈ V
113		op1stg 7957	. . . . . . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1^st ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘)) = ((𝐶‘𝑗)‘𝑘))
117	109	fveq2d 6848	. . . . . . . . . . . . . . . . 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 7954	. . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2^nd ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘)) = ((𝐷‘𝑗)‘𝑘))
121	116, 120	oveq12d 7388	. . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . 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 8864	. . . . . . . . . . . 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 4973	. . . . . . . . . . 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 3972	. . . . . . . . . 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 2738	. . . . . . . . . 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 716	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑋⟶ℝ)
132	42	adantlr 716	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑋⟶ℝ)
133	32, 129, 130, 131, 132	hoidmvn0val 46971	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)) = ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
134	133	mpteq2dva 5193	. . . . . . . . . . 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 6848	. . . . . . . . . 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 2743	. . . . . . . . . . . . . . 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 6848	. . . . . . . . . . . . . 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 15859	. . . . . . . . . . . . 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 5193	. . . . . . . . . . . 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 6848	. . . . . . . . . . 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 2782	. . . . . . . . 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 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑖
145		nfmpt1 5199	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
146	144, 145	nfeq 2913	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
147		nfcv 2899	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝑖
148		nfcv 2899	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘ℕ
149		nfmpt1 5199	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)
150	148, 149	nfmpt 5198	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
151	147, 150	nfeq 2913	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
152		fveq1 6843	. . . . . . . . . . . . . . . . 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 5821	. . . . . . . . . . . . . . . 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 6846	. . . . . . . . . . . . . . 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 45457	. . . . . . . . . . . . 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 45436	. . . . . . . . . . 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 3968	. . . . . . . . . 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 1916	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 𝑗 ∈ ℕ
161	151, 160	nfan 1901	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) ∧ 𝑗 ∈ ℕ)
162	154	fveq2d 6848	. . . . . . . . . . . . . . . . 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 3236	. . . . . . . . . . . . . 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 15858	. . . . . . . . . . . . 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 5192	. . . . . . . . . . . 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 6848	. . . . . . . . . . 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 2748	. . . . . . . . . 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 633	. . . . . . . . 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 3578	. . . . . . . 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 585	. . . . . . 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 2741	. . . . . . . . 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 631	. . . . . . . 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 3162	. . . . . . 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 3648	. . . . . 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 13264	. . . . 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 585	. . . 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 5122	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln*‘𝑋)‘𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
182	48, 50, 181	syl2anc 585	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
183	47, 182	pm2.61dan 813	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 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 {crab 3401 Vcvv 3442 ⊆ wss 3903 ∅c0 4287 ifcif 4481 {csn 4582 ⟨cop 4588 ∪ ciun 4948 class class class wbr 5100 ↦ cmpt 5181 × cxp 5632 ∘ ccom 5638 ⟶wf 6498 ‘cfv 6502 (class class class)co 7370 ∈ cmpo 7372 1^st c1st 7943 2^nd c2nd 7944 ↑_m cmap 8777 Xcixp 8849 Fincfn 8897 infcinf 9358 ℝcr 11039 0cc0 11040 1c1 11041 +∞cpnf 11177 ℝ^*cxr 11179 < clt 11180 ≤ cle 11181 ℕcn 12159 [,)cico 13277 [,]cicc 13278 ∏cprod 15840 volcvol 25437 Σ^{^}csumge0 46749 voln*covoln 46923

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-inf2 9564 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117 ax-pre-sup 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-se 5588 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-isom 6511 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-of 7634 df-om 7821 df-1st 7945 df-2nd 7946 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-2o 8410 df-er 8647 df-map 8779 df-pm 8780 df-ixp 8850 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-fi 9328 df-sup 9359 df-inf 9360 df-oi 9429 df-dju 9827 df-card 9865 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-div 11809 df-nn 12160 df-2 12222 df-3 12223 df-n0 12416 df-z 12503 df-uz 12766 df-q 12876 df-rp 12920 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-ioo 13279 df-ico 13281 df-icc 13282 df-fz 13438 df-fzo 13585 df-fl 13726 df-seq 13939 df-exp 13999 df-hash 14268 df-cj 15036 df-re 15037 df-im 15038 df-sqrt 15172 df-abs 15173 df-clim 15425 df-rlim 15426 df-sum 15624 df-prod 15841 df-rest 17356 df-topgen 17377 df-psmet 21318 df-xmet 21319 df-met 21320 df-bl 21321 df-mopn 21322 df-top 22855 df-topon 22872 df-bases 22907 df-cmp 23348 df-ovol 25438 df-vol 25439 df-sumge0 46750 df-ovoln 46924

This theorem is referenced by: hspmbllem2 47014

W3C validator