Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovnlecvr2 Structured version   Visualization version   GIF version

Theorem ovnlecvr2 44841
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 (𝜑𝑋 ∈ Fin)
ovnlecvr2.c (𝜑𝐶:ℕ⟶(ℝ ↑m 𝑋))
ovnlecvr2.d (𝜑𝐷:ℕ⟶(ℝ ↑m 𝑋))
ovnlecvr2.s (𝜑𝐴 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
ovnlecvr2.l 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
Assertion
Ref Expression
ovnlecvr2 (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))
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.
StepHypRef Expression
1 fveq2 6842 . . . . . 6 (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅))
21fveq1d 6844 . . . . 5 (𝑋 = ∅ → ((voln*‘𝑋)‘𝐴) = ((voln*‘∅)‘𝐴))
32adantl 482 . . . 4 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) = ((voln*‘∅)‘𝐴))
4 ovnlecvr2.s . . . . . . 7 (𝜑𝐴 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
54adantr 481 . . . . . 6 ((𝜑𝑋 = ∅) → 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
6 1nn 12164 . . . . . . . . . . 11 1 ∈ ℕ
7 ne0i 4294 . . . . . . . . . . 11 (1 ∈ ℕ → ℕ ≠ ∅)
86, 7ax-mp 5 . . . . . . . . . 10 ℕ ≠ ∅
98a1i 11 . . . . . . . . 9 (𝜑 → ℕ ≠ ∅)
10 iunconst 4963 . . . . . . . . 9 (ℕ ≠ ∅ → 𝑗 ∈ ℕ {∅} = {∅})
119, 10syl 17 . . . . . . . 8 (𝜑 𝑗 ∈ ℕ {∅} = {∅})
1211adantr 481 . . . . . . 7 ((𝜑𝑋 = ∅) → 𝑗 ∈ ℕ {∅} = {∅})
13 ixpeq1 8846 . . . . . . . . . . 11 (𝑋 = ∅ → X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = X𝑘 ∈ ∅ (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
14 ixp0x 8864 . . . . . . . . . . . 12 X𝑘 ∈ ∅ (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = {∅}
1514a1i 11 . . . . . . . . . . 11 (𝑋 = ∅ → X𝑘 ∈ ∅ (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = {∅})
1613, 15eqtrd 2776 . . . . . . . . . 10 (𝑋 = ∅ → X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = {∅})
1716adantr 481 . . . . . . . . 9 ((𝑋 = ∅ ∧ 𝑗 ∈ ℕ) → X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = {∅})
1817iuneq2dv 4978 . . . . . . . 8 (𝑋 = ∅ → 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = 𝑗 ∈ ℕ {∅})
1918adantl 482 . . . . . . 7 ((𝜑𝑋 = ∅) → 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = 𝑗 ∈ ℕ {∅})
20 reex 11142 . . . . . . . . 9 ℝ ∈ V
21 mapdm0 8780 . . . . . . . . 9 (ℝ ∈ V → (ℝ ↑m ∅) = {∅})
2220, 21ax-mp 5 . . . . . . . 8 (ℝ ↑m ∅) = {∅}
2322a1i 11 . . . . . . 7 ((𝜑𝑋 = ∅) → (ℝ ↑m ∅) = {∅})
2412, 19, 233eqtr4d 2786 . . . . . 6 ((𝜑𝑋 = ∅) → 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = (ℝ ↑m ∅))
255, 24sseqtrd 3984 . . . . 5 ((𝜑𝑋 = ∅) → 𝐴 ⊆ (ℝ ↑m ∅))
2625ovn0val 44781 . . . 4 ((𝜑𝑋 = ∅) → ((voln*‘∅)‘𝐴) = 0)
273, 26eqtrd 2776 . . 3 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) = 0)
28 nfv 1917 . . . . 5 𝑗𝜑
29 nnex 12159 . . . . . 6 ℕ ∈ V
3029a1i 11 . . . . 5 (𝜑 → ℕ ∈ V)
31 icossicc 13353 . . . . . 6 (0[,)+∞) ⊆ (0[,]+∞)
32 ovnlecvr2.l . . . . . . 7 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
33 ovnlecvr2.x . . . . . . . 8 (𝜑𝑋 ∈ Fin)
3433adantr 481 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
35 ovnlecvr2.c . . . . . . . . 9 (𝜑𝐶:ℕ⟶(ℝ ↑m 𝑋))
3635ffvelcdmda 7035 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ (ℝ ↑m 𝑋))
37 elmapi 8787 . . . . . . . 8 ((𝐶𝑗) ∈ (ℝ ↑m 𝑋) → (𝐶𝑗):𝑋⟶ℝ)
3836, 37syl 17 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗):𝑋⟶ℝ)
39 ovnlecvr2.d . . . . . . . . 9 (𝜑𝐷:ℕ⟶(ℝ ↑m 𝑋))
4039ffvelcdmda 7035 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ (ℝ ↑m 𝑋))
41 elmapi 8787 . . . . . . . 8 ((𝐷𝑗) ∈ (ℝ ↑m 𝑋) → (𝐷𝑗):𝑋⟶ℝ)
4240, 41syl 17 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗):𝑋⟶ℝ)
4332, 34, 38, 42hoidmvcl 44813 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)) ∈ (0[,)+∞))
4431, 43sselid 3942 . . . . 5 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)) ∈ (0[,]+∞))
4528, 30, 44sge0ge0mpt 44669 . . . 4 (𝜑 → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))
4645adantr 481 . . 3 ((𝜑𝑋 = ∅) → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))
4727, 46eqbrtrd 5127 . 2 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))
48 simpl 483 . . 3 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝜑)
49 neqne 2951 . . . 4 𝑋 = ∅ → 𝑋 ≠ ∅)
5049adantl 482 . . 3 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
5133adantr 481 . . . . 5 ((𝜑𝑋 ≠ ∅) → 𝑋 ∈ Fin)
52 simpr 485 . . . . 5 ((𝜑𝑋 ≠ ∅) → 𝑋 ≠ ∅)
5338ffvelcdmda 7035 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶𝑗)‘𝑘) ∈ ℝ)
5442ffvelcdmda 7035 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐷𝑗)‘𝑘) ∈ ℝ)
5554rexrd 11205 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐷𝑗)‘𝑘) ∈ ℝ*)
56 icossre 13345 . . . . . . . . . . . . 13 ((((𝐶𝑗)‘𝑘) ∈ ℝ ∧ ((𝐷𝑗)‘𝑘) ∈ ℝ*) → (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) ⊆ ℝ)
5753, 55, 56syl2anc 584 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) ⊆ ℝ)
5857ralrimiva 3143 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → ∀𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) ⊆ ℝ)
59 ss2ixp 8848 . . . . . . . . . . 11 (∀𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) ⊆ ℝ → X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) ⊆ X𝑘𝑋 ℝ)
6058, 59syl 17 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) ⊆ X𝑘𝑋 ℝ)
6120a1i 11 . . . . . . . . . . . 12 (𝜑 → ℝ ∈ V)
62 ixpconstg 8844 . . . . . . . . . . . 12 ((𝑋 ∈ Fin ∧ ℝ ∈ V) → X𝑘𝑋 ℝ = (ℝ ↑m 𝑋))
6333, 61, 62syl2anc 584 . . . . . . . . . . 11 (𝜑X𝑘𝑋 ℝ = (ℝ ↑m 𝑋))
6463adantr 481 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → X𝑘𝑋 ℝ = (ℝ ↑m 𝑋))
6560, 64sseqtrd 3984 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) ⊆ (ℝ ↑m 𝑋))
6665ralrimiva 3143 . . . . . . . 8 (𝜑 → ∀𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) ⊆ (ℝ ↑m 𝑋))
67 iunss 5005 . . . . . . . 8 ( 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) ⊆ (ℝ ↑m 𝑋) ↔ ∀𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) ⊆ (ℝ ↑m 𝑋))
6866, 67sylibr 233 . . . . . . 7 (𝜑 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) ⊆ (ℝ ↑m 𝑋))
694, 68sstrd 3954 . . . . . 6 (𝜑𝐴 ⊆ (ℝ ↑m 𝑋))
7069adantr 481 . . . . 5 ((𝜑𝑋 ≠ ∅) → 𝐴 ⊆ (ℝ ↑m 𝑋))
71 eqid 2736 . . . . 5 {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}
7251, 52, 70, 71ovnn0val 44782 . . . 4 ((𝜑𝑋 ≠ ∅) → ((voln*‘𝑋)‘𝐴) = inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ))
73 ssrab2 4037 . . . . . 6 {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} ⊆ ℝ*
7473a1i 11 . . . . 5 ((𝜑𝑋 ≠ ∅) → {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} ⊆ ℝ*)
7528, 30, 44sge0xrclmpt 44659 . . . . . . . 8 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) ∈ ℝ*)
7675adantr 481 . . . . . . 7 ((𝜑𝑋 ≠ ∅) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) ∈ ℝ*)
77 opelxpi 5670 . . . . . . . . . . . . . 14 ((((𝐶𝑗)‘𝑘) ∈ ℝ ∧ ((𝐷𝑗)‘𝑘) ∈ ℝ) → ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩ ∈ (ℝ × ℝ))
7853, 54, 77syl2anc 584 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩ ∈ (ℝ × ℝ))
7978fmpttd 7063 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩):𝑋⟶(ℝ × ℝ))
8020, 20xpex 7687 . . . . . . . . . . . . . 14 (ℝ × ℝ) ∈ V
8180a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (ℝ × ℝ) ∈ V)
82 elmapg 8778 . . . . . . . . . . . . 13 (((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) ∈ ((ℝ × ℝ) ↑m 𝑋) ↔ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩):𝑋⟶(ℝ × ℝ)))
8381, 34, 82syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) ∈ ((ℝ × ℝ) ↑m 𝑋) ↔ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩):𝑋⟶(ℝ × ℝ)))
8479, 83mpbird 256 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) ∈ ((ℝ × ℝ) ↑m 𝑋))
8584fmpttd 7063 . . . . . . . . . 10 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)):ℕ⟶((ℝ × ℝ) ↑m 𝑋))
86 ovexd 7392 . . . . . . . . . . 11 (𝜑 → ((ℝ × ℝ) ↑m 𝑋) ∈ V)
87 elmapg 8778 . . . . . . . . . . 11 ((((ℝ × ℝ) ↑m 𝑋) ∈ V ∧ ℕ ∈ V) → ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↔ (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)):ℕ⟶((ℝ × ℝ) ↑m 𝑋)))
8886, 30, 87syl2anc 584 . . . . . . . . . 10 (𝜑 → ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↔ (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)):ℕ⟶((ℝ × ℝ) ↑m 𝑋)))
8985, 88mpbird 256 . . . . . . . . 9 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
9089adantr 481 . . . . . . . 8 ((𝜑𝑋 ≠ ∅) → (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
91 simpr 485 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
92 mptexg 7171 . . . . . . . . . . . . . . . . . . . 20 (𝑋 ∈ Fin → (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) ∈ V)
9333, 92syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) ∈ V)
9493adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) ∈ V)
95 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))
9695fvmpt2 6959 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ℕ ∧ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) ∈ V) → ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗) = (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))
9791, 94, 96syl2anc 584 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗) = (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))
9897coeq2d 5818 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗)) = ([,) ∘ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)))
9998fveq1d 6844 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘) = (([,) ∘ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑘))
10099adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘) = (([,) ∘ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑘))
10179adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩):𝑋⟶(ℝ × ℝ))
102 simpr 485 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → 𝑘𝑋)
103101, 102fvovco 43403 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑘) = ((1st ‘((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘))[,)(2nd ‘((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘))))
104 simpr 485 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑋) → 𝑘𝑋)
105 opex 5421 . . . . . . . . . . . . . . . . . . . 20 ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩ ∈ V
106105a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑋) → ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩ ∈ V)
107 eqid 2736 . . . . . . . . . . . . . . . . . . . 20 (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) = (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)
108107fvmpt2 6959 . . . . . . . . . . . . . . . . . . 19 ((𝑘𝑋 ∧ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩ ∈ V) → ((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘) = ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)
109104, 106, 108syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝑋) → ((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘) = ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)
110109fveq2d 6846 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (1st ‘((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘)) = (1st ‘⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))
111 fvex 6855 . . . . . . . . . . . . . . . . . . 19 ((𝐶𝑗)‘𝑘) ∈ V
112 fvex 6855 . . . . . . . . . . . . . . . . . . 19 ((𝐷𝑗)‘𝑘) ∈ V
113 op1stg 7933 . . . . . . . . . . . . . . . . . . 19 ((((𝐶𝑗)‘𝑘) ∈ V ∧ ((𝐷𝑗)‘𝑘) ∈ V) → (1st ‘⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) = ((𝐶𝑗)‘𝑘))
114111, 112, 113mp2an 690 . . . . . . . . . . . . . . . . . 18 (1st ‘⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) = ((𝐶𝑗)‘𝑘)
115114a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (1st ‘⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) = ((𝐶𝑗)‘𝑘))
116110, 115eqtrd 2776 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → (1st ‘((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘)) = ((𝐶𝑗)‘𝑘))
117109fveq2d 6846 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (2nd ‘((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘)) = (2nd ‘⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))
118111, 112op2nd 7930 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) = ((𝐷𝑗)‘𝑘)
119118a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (2nd ‘⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) = ((𝐷𝑗)‘𝑘))
120117, 119eqtrd 2776 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → (2nd ‘((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘)) = ((𝐷𝑗)‘𝑘))
121116, 120oveq12d 7375 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑋) → ((1st ‘((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘))[,)(2nd ‘((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘))) = (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
122121adantlr 713 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((1st ‘((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘))[,)(2nd ‘((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘))) = (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
123100, 103, 1223eqtrrd 2781 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
124123ixpeq2dva 8850 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
125124iuneq2dv 4978 . . . . . . . . . . 11 (𝜑 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
1264, 125sseqtrd 3984 . . . . . . . . . 10 (𝜑𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
127126adantr 481 . . . . . . . . 9 ((𝜑𝑋 ≠ ∅) → 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
128 eqidd 2737 . . . . . . . . . 10 ((𝜑𝑋 ≠ ∅) → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))))
12951adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
13052adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → 𝑋 ≠ ∅)
13138adantlr 713 . . . . . . . . . . . . 13 (((𝜑𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝐶𝑗):𝑋⟶ℝ)
13242adantlr 713 . . . . . . . . . . . . 13 (((𝜑𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝐷𝑗):𝑋⟶ℝ)
13332, 129, 130, 131, 132hoidmvn0val 44815 . . . . . . . . . . . 12 (((𝜑𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)) = ∏𝑘𝑋 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
134133mpteq2dva 5205 . . . . . . . . . . 11 ((𝜑𝑋 ≠ ∅) → (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))))
135134fveq2d 6846 . . . . . . . . . 10 ((𝜑𝑋 ≠ ∅) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))))
136123eqcomd 2742 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘) = (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
137136fveq2d 6846 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)) = (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
138137prodeq2dv 15806 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
139138mpteq2dva 5205 . . . . . . . . . . . 12 (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))))
140139fveq2d 6846 . . . . . . . . . . 11 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))))
141140adantr 481 . . . . . . . . . 10 ((𝜑𝑋 ≠ ∅) → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))))
142128, 135, 1413eqtr4d 2786 . . . . . . . . 9 ((𝜑𝑋 ≠ ∅) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))))
143127, 142jca 512 . . . . . . . 8 ((𝜑𝑋 ≠ ∅) → (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))))))
144 nfcv 2907 . . . . . . . . . . . . 13 𝑗𝑖
145 nfmpt1 5213 . . . . . . . . . . . . 13 𝑗(𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))
146144, 145nfeq 2920 . . . . . . . . . . . 12 𝑗 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))
147 nfcv 2907 . . . . . . . . . . . . . . 15 𝑘𝑖
148 nfcv 2907 . . . . . . . . . . . . . . . 16 𝑘
149 nfmpt1 5213 . . . . . . . . . . . . . . . 16 𝑘(𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)
150148, 149nfmpt 5212 . . . . . . . . . . . . . . 15 𝑘(𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))
151147, 150nfeq 2920 . . . . . . . . . . . . . 14 𝑘 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))
152 fveq1 6841 . . . . . . . . . . . . . . . . 17 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) → (𝑖𝑗) = ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))
153152coeq2d 5818 . . . . . . . . . . . . . . . 16 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) → ([,) ∘ (𝑖𝑗)) = ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗)))
154153fveq1d 6844 . . . . . . . . . . . . . . 15 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
155154adantr 481 . . . . . . . . . . . . . 14 ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) ∧ 𝑘𝑋) → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
156151, 155ixpeq2d 43266 . . . . . . . . . . . . 13 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) → X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
157156adantr 481 . . . . . . . . . . . 12 ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) ∧ 𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
158146, 157iuneq2df 43244 . . . . . . . . . . 11 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) → 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
159158sseq2d 3976 . . . . . . . . . 10 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) → (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ↔ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))
160 nfv 1917 . . . . . . . . . . . . . . . 16 𝑘 𝑗 ∈ ℕ
161151, 160nfan 1902 . . . . . . . . . . . . . . 15 𝑘(𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) ∧ 𝑗 ∈ ℕ)
162154fveq2d 6846 . . . . . . . . . . . . . . . . 17 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))
163162a1d 25 . . . . . . . . . . . . . . . 16 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) → (𝑘𝑋 → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))))
164163adantr 481 . . . . . . . . . . . . . . 15 ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) ∧ 𝑗 ∈ ℕ) → (𝑘𝑋 → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))))
165161, 164ralrimi 3240 . . . . . . . . . . . . . 14 ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) ∧ 𝑗 ∈ ℕ) → ∀𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))
166165prodeq2d 15805 . . . . . . . . . . . . 13 ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) ∧ 𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))
167146, 166mpteq2da 5203 . . . . . . . . . . . 12 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))))
168167fveq2d 6846 . . . . . . . . . . 11 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))))
169168eqeq2d 2747 . . . . . . . . . 10 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) → ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))))))
170159, 169anbi12d 631 . . . . . . . . 9 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) → ((𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) ↔ (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))))))
171170rspcev 3581 . . . . . . . 8 (((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))))) → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
17290, 143, 171syl2anc 584 . . . . . . 7 ((𝜑𝑋 ≠ ∅) → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
17376, 172jca 512 . . . . . 6 ((𝜑𝑋 ≠ ∅) → ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
174 eqeq1 2740 . . . . . . . . 9 (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) → (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
175174anbi2d 629 . . . . . . . 8 (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) → ((𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) ↔ (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
176175rexbidv 3175 . . . . . . 7 (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) → (∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) ↔ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
177176elrab 3645 . . . . . 6 ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} ↔ ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
178173, 177sylibr 233 . . . . 5 ((𝜑𝑋 ≠ ∅) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))})
179 infxrlb 13253 . . . . 5 (({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} ⊆ ℝ* ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}) → inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))
18074, 178, 179syl2anc 584 . . . 4 ((𝜑𝑋 ≠ ∅) → inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))
18172, 180eqbrtrd 5127 . . 3 ((𝜑𝑋 ≠ ∅) → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))
18248, 50, 181syl2anc 584 . 2 ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))
18347, 182pm2.61dan 811 1 (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  wss 3910  c0 4282  ifcif 4486  {csn 4586  cop 4592   ciun 4954   class class class wbr 5105  cmpt 5188   × cxp 5631  ccom 5637  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  m cmap 8765  Xcixp 8835  Fincfn 8883  infcinf 9377  cr 11050  0cc0 11051  1c1 11052  +∞cpnf 11186  *cxr 11188   < clt 11189  cle 11190  cn 12153  [,)cico 13266  [,]cicc 13267  cprod 15788  volcvol 24827  Σ^csumge0 44593  voln*covoln 44767
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-prod 15789  df-rest 17304  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cmp 22738  df-ovol 24828  df-vol 24829  df-sumge0 44594  df-ovoln 44768
This theorem is referenced by:  hspmbllem2  44858
  Copyright terms: Public domain W3C validator