ovnlecvr2 - Mathbox for Glauco Siliprandi

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Theorem ovnlecvr2 46625

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 6906	. . . . . 6 ⊢ (𝑋 = ∅ → (voln‘𝑋) = (voln‘∅))
2	1	fveq1d 6908	. . . . 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 12277	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
7		ne0i 4341	. . . . . . . . . . 11 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
8	6, 7	ax-mp 5	. . . . . . . . . 10 ⊢ ℕ ≠ ∅
9	8	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℕ ≠ ∅)
10		iunconst 5001	. . . . . . . . 9 ⊢ (ℕ ≠ ∅ → ∪ 𝑗 ∈ ℕ {∅} = {∅})
11	9, 10	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ {∅} = {∅})
12	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑗 ∈ ℕ {∅} = {∅})
13		ixpeq1 8948	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = X𝑘 ∈ ∅ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
14		ixp0x 8966	. . . . . . . . . . . 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 2777	. . . . . . . . . 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 5016	. . . . . . . 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 11246	. . . . . . . . 9 ⊢ ℝ ∈ V
21		mapdm0 8882	. . . . . . . . 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 2787	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = (ℝ ↑_m ∅))
25	5, 24	sseqtrd 4020	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ⊆ (ℝ ↑_m ∅))
26	25	ovn0val 46565	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘𝐴) = 0)
27	3, 26	eqtrd 2777	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) = 0)
28		nfv 1914	. . . . 5 ⊢ Ⅎ𝑗𝜑
29		nnex 12272	. . . . . 6 ⊢ ℕ ∈ V
30	29	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ∈ V)
31		icossicc 13476	. . . . . 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 7104	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑_m 𝑋))
37		elmapi 8889	. . . . . . . 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 7104	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑_m 𝑋))
41		elmapi 8889	. . . . . . . 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 46597	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)) ∈ (0[,)+∞))
44	31, 43	sselid 3981	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)) ∈ (0[,]+∞))
45	28, 30, 44	sge0ge0mpt 46453	. . . 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 5165	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
48		simpl 482	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝜑)
49		neqne 2948	. . . 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 7104	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑗)‘𝑘) ∈ ℝ)
54	42	ffvelcdmda 7104	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐷‘𝑗)‘𝑘) ∈ ℝ)
55	54	rexrd 11311	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐷‘𝑗)‘𝑘) ∈ ℝ^*)
56		icossre 13468	. . . . . . . . . . . . 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 3146	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∀𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ ℝ)
59		ss2ixp 8950	. . . . . . . . . . 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 8946	. . . . . . . . . . . 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 4020	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ (ℝ ↑_m 𝑋))
66	65	ralrimiva 3146	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ (ℝ ↑_m 𝑋))
67		iunss 5045	. . . . . . . 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 3994	. . . . . 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 46566	. . . 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 4080	. . . . . 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 46443	. . . . . . . 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 5722	. . . . . . . . . . . . . 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 7135	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩):𝑋⟶(ℝ × ℝ))
80	20, 20	xpex 7773	. . . . . . . . . . . . . 14 ⊢ (ℝ × ℝ) ∈ V
81	80	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (ℝ × ℝ) ∈ V)
82		elmapg 8879	. . . . . . . . . . . . 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 7135	. . . . . . . . . 10 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)):ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
86		ovexd 7466	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℝ × ℝ) ↑_m 𝑋) ∈ V)
87		elmapg 8879	. . . . . . . . . . 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 7241	. . . . . . . . . . . . . . . . . . . 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 7027	. . . . . . . . . . . . . . . . . 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 5873	. . . . . . . . . . . . . . . 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 6908	. . . . . . . . . . . . . . 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 45198	. . . . . . . . . . . . . 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 5469	. . . . . . . . . . . . . . . . . . . 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 7027	. . . . . . . . . . . . . . . . . . 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 6910	. . . . . . . . . . . . . . . . 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 6919	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶‘𝑗)‘𝑘) ∈ V
112		fvex 6919	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐷‘𝑗)‘𝑘) ∈ V
113		op1stg 8026	. . . . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1^st ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘)) = ((𝐶‘𝑗)‘𝑘))
117	109	fveq2d 6910	. . . . . . . . . . . . . . . . 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 8023	. . . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2^nd ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘)) = ((𝐷‘𝑗)‘𝑘))
121	116, 120	oveq12d 7449	. . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . 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 8952	. . . . . . . . . . . 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 5016	. . . . . . . . . . 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 4020	. . . . . . . . . 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 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 46599	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)) = ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
134	133	mpteq2dva 5242	. . . . . . . . . . 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 6910	. . . . . . . . . 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 6910	. . . . . . . . . . . . . 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 15958	. . . . . . . . . . . . 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 5242	. . . . . . . . . . . 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 6910	. . . . . . . . . . 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 2787	. . . . . . . . 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 2905	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑖
145		nfmpt1 5250	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
146	144, 145	nfeq 2919	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
147		nfcv 2905	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝑖
148		nfcv 2905	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘ℕ
149		nfmpt1 5250	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)
150	148, 149	nfmpt 5249	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
151	147, 150	nfeq 2919	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
152		fveq1 6905	. . . . . . . . . . . . . . . . 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 5873	. . . . . . . . . . . . . . . 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 6908	. . . . . . . . . . . . . . 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 45073	. . . . . . . . . . . . 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 45052	. . . . . . . . . . 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 4016	. . . . . . . . . 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 1914	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 𝑗 ∈ ℕ
161	151, 160	nfan 1899	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) ∧ 𝑗 ∈ ℕ)
162	154	fveq2d 6910	. . . . . . . . . . . . . . . . 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 3257	. . . . . . . . . . . . . 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 15957	. . . . . . . . . . . . 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 5240	. . . . . . . . . . . 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 6910	. . . . . . . . . . 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 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 3622	. . . . . . . 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 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 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 3179	. . . . . . 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 3692	. . . . . 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 13376	. . . . 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 5165	. . 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 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 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3436 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 ifcif 4525 {csn 4626 ⟨cop 4632 ∪ ciun 4991 class class class wbr 5143 ↦ cmpt 5225 × cxp 5683 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 1^st c1st 8012 2^nd c2nd 8013 ↑_m cmap 8866 Xcixp 8937 Fincfn 8985 infcinf 9481 ℝcr 11154 0cc0 11155 1c1 11156 +∞cpnf 11292 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 ℕcn 12266 [,)cico 13389 [,]cicc 13390 ∏cprod 15939 volcvol 25498 Σ^{^}csumge0 46377 voln*covoln 46551

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-prod 15940 df-rest 17467 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953 df-cmp 23395 df-ovol 25499 df-vol 25500 df-sumge0 46378 df-ovoln 46552

This theorem is referenced by: hspmbllem2 46642

W3C validator