Theorem ovnlecvr2 41145

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 6229	. . . . . 6 ⊢ (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅))
2	1	fveq1d 6231	. . . . 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 11069	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
7		ne0i 3954	. . . . . . . . . . 11 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
8	6, 7	ax-mp 5	. . . . . . . . . 10 ⊢ ℕ ≠ ∅
9	8	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℕ ≠ ∅)
10		iunconst 4561	. . . . . . . . 9 ⊢ (ℕ ≠ ∅ → ∪ 𝑗 ∈ ℕ {∅} = {∅})
11	9, 10	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ {∅} = {∅})
12	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑗 ∈ ℕ {∅} = {∅})
13		ixpeq1 7961	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = X𝑘 ∈ ∅ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
14		ixp0x 7978	. . . . . . . . . . . 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 2685	. . . . . . . . . 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 4574	. . . . . . . 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 10065	. . . . . . . . 9 ⊢ ℝ ∈ V
21		mapdm0 7914	. . . . . . . . 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 2695	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = (ℝ ↑𝑚 ∅))
25	5, 24	sseqtrd 3674	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ⊆ (ℝ ↑𝑚 ∅))
26	25	ovn0val 41085	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘𝐴) = 0)
27	3, 26	eqtrd 2685	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) = 0)
28		nfv 1883	. . . . 5 ⊢ Ⅎ𝑗𝜑
29		nnex 11064	. . . . . 6 ⊢ ℕ ∈ V
30	29	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ∈ V)
31		icossicc 12298	. . . . . 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	ffvelrnda 6399	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑𝑚 𝑋))
37		elmapi 7921	. . . . . . . 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	ffvelrnda 6399	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑𝑚 𝑋))
41		elmapi 7921	. . . . . . . 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 41117	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)) ∈ (0[,)+∞))
44	31, 43	sseldi 3634	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)) ∈ (0[,]+∞))
45	28, 30, 44	sge0ge0mpt 40973	. . . 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 4707	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
48		simpl 472	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝜑)
49		neqne 2831	. . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
50	49	adantl 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
51	33	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ∈ Fin)
52		simpr 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ≠ ∅)
53	38	ffvelrnda 6399	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑗)‘𝑘) ∈ ℝ)
54	42	ffvelrnda 6399	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐷‘𝑗)‘𝑘) ∈ ℝ)
55	54	rexrd 10127	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐷‘𝑗)‘𝑘) ∈ ℝ*)
56		icossre 12292	. . . . . . . . . . . . 13 ⊢ ((((𝐶‘𝑗)‘𝑘) ∈ ℝ ∧ ((𝐷‘𝑗)‘𝑘) ∈ ℝ*) → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ ℝ)
57	53, 55, 56	syl2anc 694	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ ℝ)
58	57	ralrimiva 2995	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∀𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ ℝ)
59		ss2ixp 7963	. . . . . . . . . . 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 7959	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ Fin ∧ ℝ ∈ V) → X𝑘 ∈ 𝑋 ℝ = (ℝ ↑𝑚 𝑋))
63	33, 61, 62	syl2anc 694	. . . . . . . . . . 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 3674	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ (ℝ ↑𝑚 𝑋))
66	65	ralrimiva 2995	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ (ℝ ↑𝑚 𝑋))
67		iunss 4593	. . . . . . . 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 224	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ (ℝ ↑𝑚 𝑋))
69	4, 68	sstrd 3646	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑𝑚 𝑋))
70	69	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐴 ⊆ (ℝ ↑𝑚 𝑋))
71		eqid 2651	. . . . 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 41086	. . . 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 3720	. . . . . 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 40963	. . . . . . . 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 5182	. . . . . . . . . . . . . 14 ⊢ ((((𝐶‘𝑗)‘𝑘) ∈ ℝ ∧ ((𝐷‘𝑗)‘𝑘) ∈ ℝ) → ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩ ∈ (ℝ × ℝ))
78	53, 54, 77	syl2anc 694	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩ ∈ (ℝ × ℝ))
79		eqid 2651	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) = (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)
80	78, 79	fmptd 6425	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩):𝑋⟶(ℝ × ℝ))
81	20, 20	xpex 7004	. . . . . . . . . . . . . 14 ⊢ (ℝ × ℝ) ∈ V
82	81	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (ℝ × ℝ) ∈ V)
83		elmapg 7912	. . . . . . . . . . . . 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)))
84	82, 34, 83	syl2anc 694	. . . . . . . . . . . 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)))
85	80, 84	mpbird 247	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
86		eqid 2651	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
87	85, 86	fmptd 6425	. . . . . . . . . 10 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)):ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
88		ovexd 6720	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℝ × ℝ) ↑𝑚 𝑋) ∈ V)
89		elmapg 7912	. . . . . . . . . . 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)))
90	88, 30, 89	syl2anc 694	. . . . . . . . . 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)))
91	87, 90	mpbird 247	. . . . . . . . 9 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
92	91	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
93		simpr 476	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
94		mptexg 6525	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑋 ∈ Fin → (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) ∈ V)
95	33, 94	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) ∈ V)
96	95	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) ∈ V)
97	86	fvmpt2 6330	. . . . . . . . . . . . . . . . . 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)⟩))
98	93, 96, 97	syl2anc 694	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗) = (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
99	98	coeq2d 5317	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑗)) = ([,) ∘ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)))
100	99	fveq1d 6231	. . . . . . . . . . . . . . 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))
101	100	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))
102	80	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩):𝑋⟶(ℝ × ℝ))
103		simpr 476	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
104	102, 103	fvovco 39695	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))‘𝑘) = ((1st ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘))[,)(2nd ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘))))
105		simpr 476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
106		opex 4962	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩ ∈ V
107	106	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩ ∈ V)
108	79	fvmpt2 6330	. . . . . . . . . . . . . . . . . . 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	105, 107, 108	syl2anc 694	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘) = ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)
110	109	fveq2d 6233	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘)) = (1st ‘⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
111		fvex 6239	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶‘𝑗)‘𝑘) ∈ V
112		fvex 6239	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐷‘𝑗)‘𝑘) ∈ V
113		op1stg 7222	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐶‘𝑗)‘𝑘) ∈ V ∧ ((𝐷‘𝑗)‘𝑘) ∈ V) → (1st ‘⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) = ((𝐶‘𝑗)‘𝑘))
114	111, 112, 113	mp2an 708	. . . . . . . . . . . . . . . . . 18 ⊢ (1st ‘⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) = ((𝐶‘𝑗)‘𝑘)
115	114	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) = ((𝐶‘𝑗)‘𝑘))
116	110, 115	eqtrd 2685	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘)) = ((𝐶‘𝑗)‘𝑘))
117	109	fveq2d 6233	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘)) = (2nd ‘⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
118	111, 112	op2nd 7219	. . . . . . . . . . . . . . . . . 18 ⊢ (2nd ‘⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) = ((𝐷‘𝑗)‘𝑘)
119	118	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩) = ((𝐷‘𝑗)‘𝑘))
120	117, 119	eqtrd 2685	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘)) = ((𝐷‘𝑗)‘𝑘))
121	116, 120	oveq12d 6708	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((1st ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘))[,)(2nd ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘))) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
122	121	adantlr 751	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((1st ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘))[,)(2nd ‘((𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)‘𝑘))) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
123	101, 104, 122	3eqtrrd 2690	. . . . . . . . . . . . 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 7965	. . . . . . . . . . . 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 4574	. . . . . . . . . . 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 3674	. . . . . . . . . 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 2652	. . . . . . . . . 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 751	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑋⟶ℝ)
132	42	adantlr 751	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑋⟶ℝ)
133	32, 129, 130, 131, 132	hoidmvn0val 41119	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)) = ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
134	133	mpteq2dva 4777	. . . . . . . . . . 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 6233	. . . . . . . . . 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 2657	. . . . . . . . . . . . . . 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 6233	. . . . . . . . . . . . . 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 14697	. . . . . . . . . . . . 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 4777	. . . . . . . . . . . 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 6233	. . . . . . . . . . 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 2695	. . . . . . . . 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 553	. . . . . . . 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 2793	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑖
145		nfmpt1 4780	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
146	144, 145	nfeq 2805	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
147		nfcv 2793	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝑖
148		nfcv 2793	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘ℕ
149		nfmpt1 4780	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)
150	148, 149	nfmpt 4779	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
151	147, 150	nfeq 2805	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩))
152		fveq1 6228	. . . . . . . . . . . . . . . . 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 5317	. . . . . . . . . . . . . . . 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 6231	. . . . . . . . . . . . . . 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 39551	. . . . . . . . . . . . 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 39526	. . . . . . . . . . 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 3666	. . . . . . . . . 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 1883	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 𝑗 ∈ ℕ
161	151, 160	nfan 1868	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨((𝐶‘𝑗)‘𝑘), ((𝐷‘𝑗)‘𝑘)⟩)) ∧ 𝑗 ∈ ℕ)
162	154	fveq2d 6233	. . . . . . . . . . . . . . . . 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 2986	. . . . . . . . . . . . . 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 14696	. . . . . . . . . . . . 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 4776	. . . . . . . . . . . 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 6233	. . . . . . . . . . 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 2661	. . . . . . . . . 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 747	. . . . . . . . 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 3340	. . . . . . . 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	92, 143, 171	syl2anc 694	. . . . . . 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 553	. . . . . 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 2655	. . . . . . . . 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 740	. . . . . . . 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 3081	. . . . . . 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 3396	. . . . . 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 224	. . . . 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 12202	. . . . 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 694	. . . 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 4707	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
182	48, 50, 181	syl2anc 694	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
183	47, 182	pm2.61dan 849	1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  {crab 2945  Vcvv 3231   ⊆ wss 3607  ∅c0 3948  ifcif 4119  {csn 4210  ⟨cop 4216  ∪ ciun 4552   class class class wbr 4685   ↦ cmpt 4762   × cxp 5141   ∘ ccom 5147  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ↦ cmpt2 6692  1^st c1st 7208  2^nd c2nd 7209   ↑_𝑚 cmap 7899  Xcixp 7950  Fincfn 7997  infcinf 8388  ℝcr 9973  0cc0 9974  1c1 9975  +∞cpnf 10109  ℝ^*cxr 10111   < clt 10112   ≤ cle 10113  ℕcn 11058  [,)cico 12215  [,]cicc 12216  ∏cprod 14679  volcvol 23278  Σ^{^}csumge0 40897  voln*covoln 41071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-prod 14680  df-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-ovol 23279  df-vol 23280  df-sumge0 40898  df-ovoln 41072

This theorem is referenced by:  hspmbllem2  41162