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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 (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 6229 . . . . . 6 (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅))
21fveq1d 6231 . . . . 5 (𝑋 = ∅ → ((voln*‘𝑋)‘𝐴) = ((voln*‘∅)‘𝐴))
32adantl 481 . . . 4 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) = ((voln*‘∅)‘𝐴))
4 ovnlecvr2.s . . . . . . 7 (𝜑𝐴 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
54adantr 480 . . . . . 6 ((𝜑𝑋 = ∅) → 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
6 1nn 11069 . . . . . . . . . . 11 1 ∈ ℕ
7 ne0i 3954 . . . . . . . . . . 11 (1 ∈ ℕ → ℕ ≠ ∅)
86, 7ax-mp 5 . . . . . . . . . 10 ℕ ≠ ∅
98a1i 11 . . . . . . . . 9 (𝜑 → ℕ ≠ ∅)
10 iunconst 4561 . . . . . . . . 9 (ℕ ≠ ∅ → 𝑗 ∈ ℕ {∅} = {∅})
119, 10syl 17 . . . . . . . 8 (𝜑 𝑗 ∈ ℕ {∅} = {∅})
1211adantr 480 . . . . . . 7 ((𝜑𝑋 = ∅) → 𝑗 ∈ ℕ {∅} = {∅})
13 ixpeq1 7961 . . . . . . . . . . 11 (𝑋 = ∅ → X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = X𝑘 ∈ ∅ (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
14 ixp0x 7978 . . . . . . . . . . . 12 X𝑘 ∈ ∅ (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = {∅}
1514a1i 11 . . . . . . . . . . 11 (𝑋 = ∅ → X𝑘 ∈ ∅ (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = {∅})
1613, 15eqtrd 2685 . . . . . . . . . 10 (𝑋 = ∅ → X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = {∅})
1716adantr 480 . . . . . . . . 9 ((𝑋 = ∅ ∧ 𝑗 ∈ ℕ) → X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = {∅})
1817iuneq2dv 4574 . . . . . . . 8 (𝑋 = ∅ → 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = 𝑗 ∈ ℕ {∅})
1918adantl 481 . . . . . . 7 ((𝜑𝑋 = ∅) → 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = 𝑗 ∈ ℕ {∅})
20 reex 10065 . . . . . . . . 9 ℝ ∈ V
21 mapdm0 7914 . . . . . . . . 9 (ℝ ∈ V → (ℝ ↑𝑚 ∅) = {∅})
2220, 21ax-mp 5 . . . . . . . 8 (ℝ ↑𝑚 ∅) = {∅}
2322a1i 11 . . . . . . 7 ((𝜑𝑋 = ∅) → (ℝ ↑𝑚 ∅) = {∅})
2412, 19, 233eqtr4d 2695 . . . . . 6 ((𝜑𝑋 = ∅) → 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = (ℝ ↑𝑚 ∅))
255, 24sseqtrd 3674 . . . . 5 ((𝜑𝑋 = ∅) → 𝐴 ⊆ (ℝ ↑𝑚 ∅))
2625ovn0val 41085 . . . 4 ((𝜑𝑋 = ∅) → ((voln*‘∅)‘𝐴) = 0)
273, 26eqtrd 2685 . . 3 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) = 0)
28 nfv 1883 . . . . 5 𝑗𝜑
29 nnex 11064 . . . . . 6 ℕ ∈ V
3029a1i 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, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
33 ovnlecvr2.x . . . . . . . 8 (𝜑𝑋 ∈ Fin)
3433adantr 480 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
35 ovnlecvr2.c . . . . . . . . 9 (𝜑𝐶:ℕ⟶(ℝ ↑𝑚 𝑋))
3635ffvelrnda 6399 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ (ℝ ↑𝑚 𝑋))
37 elmapi 7921 . . . . . . . 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))
4039ffvelrnda 6399 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ (ℝ ↑𝑚 𝑋))
41 elmapi 7921 . . . . . . . 8 ((𝐷𝑗) ∈ (ℝ ↑𝑚 𝑋) → (𝐷𝑗):𝑋⟶ℝ)
4240, 41syl 17 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗):𝑋⟶ℝ)
4332, 34, 38, 42hoidmvcl 41117 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)) ∈ (0[,)+∞))
4431, 43sseldi 3634 . . . . 5 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)) ∈ (0[,]+∞))
4528, 30, 44sge0ge0mpt 40973 . . . 4 (𝜑 → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))
4645adantr 480 . . 3 ((𝜑𝑋 = ∅) → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))
4727, 46eqbrtrd 4707 . 2 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))
48 simpl 472 . . 3 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝜑)
49 neqne 2831 . . . 4 𝑋 = ∅ → 𝑋 ≠ ∅)
5049adantl 481 . . 3 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
5133adantr 480 . . . . 5 ((𝜑𝑋 ≠ ∅) → 𝑋 ∈ Fin)
52 simpr 476 . . . . 5 ((𝜑𝑋 ≠ ∅) → 𝑋 ≠ ∅)
5338ffvelrnda 6399 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶𝑗)‘𝑘) ∈ ℝ)
5442ffvelrnda 6399 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐷𝑗)‘𝑘) ∈ ℝ)
5554rexrd 10127 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐷𝑗)‘𝑘) ∈ ℝ*)
56 icossre 12292 . . . . . . . . . . . . 13 ((((𝐶𝑗)‘𝑘) ∈ ℝ ∧ ((𝐷𝑗)‘𝑘) ∈ ℝ*) → (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) ⊆ ℝ)
5753, 55, 56syl2anc 694 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) ⊆ ℝ)
5857ralrimiva 2995 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → ∀𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) ⊆ ℝ)
59 ss2ixp 7963 . . . . . . . . . . 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 7959 . . . . . . . . . . . 12 ((𝑋 ∈ Fin ∧ ℝ ∈ V) → X𝑘𝑋 ℝ = (ℝ ↑𝑚 𝑋))
6333, 61, 62syl2anc 694 . . . . . . . . . . 11 (𝜑X𝑘𝑋 ℝ = (ℝ ↑𝑚 𝑋))
6463adantr 480 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → X𝑘𝑋 ℝ = (ℝ ↑𝑚 𝑋))
6560, 64sseqtrd 3674 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) ⊆ (ℝ ↑𝑚 𝑋))
6665ralrimiva 2995 . . . . . . . 8 (𝜑 → ∀𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) ⊆ (ℝ ↑𝑚 𝑋))
67 iunss 4593 . . . . . . . 8 ( 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) ⊆ (ℝ ↑𝑚 𝑋) ↔ ∀𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) ⊆ (ℝ ↑𝑚 𝑋))
6866, 67sylibr 224 . . . . . . 7 (𝜑 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) ⊆ (ℝ ↑𝑚 𝑋))
694, 68sstrd 3646 . . . . . 6 (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))
7069adantr 480 . . . . 5 ((𝜑𝑋 ≠ ∅) → 𝐴 ⊆ (ℝ ↑𝑚 𝑋))
71 eqid 2651 . . . . 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 41086 . . . 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 3720 . . . . . 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 40963 . . . . . . . 8 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) ∈ ℝ*)
7675adantr 480 . . . . . . 7 ((𝜑𝑋 ≠ ∅) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) ∈ ℝ*)
77 opelxpi 5182 . . . . . . . . . . . . . 14 ((((𝐶𝑗)‘𝑘) ∈ ℝ ∧ ((𝐷𝑗)‘𝑘) ∈ ℝ) → ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩ ∈ (ℝ × ℝ))
7853, 54, 77syl2anc 694 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩ ∈ (ℝ × ℝ))
79 eqid 2651 . . . . . . . . . . . . 13 (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) = (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)
8078, 79fmptd 6425 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩):𝑋⟶(ℝ × ℝ))
8120, 20xpex 7004 . . . . . . . . . . . . . 14 (ℝ × ℝ) ∈ V
8281a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (ℝ × ℝ) ∈ V)
83 elmapg 7912 . . . . . . . . . . . . 13 (((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↔ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩):𝑋⟶(ℝ × ℝ)))
8482, 34, 83syl2anc 694 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↔ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩):𝑋⟶(ℝ × ℝ)))
8580, 84mpbird 247 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
86 eqid 2651 . . . . . . . . . . 11 (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))
8785, 86fmptd 6425 . . . . . . . . . 10 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘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 ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ↔ (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)):ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋)))
9088, 30, 89syl2anc 694 . . . . . . . . . 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)))
9187, 90mpbird 247 . . . . . . . . 9 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
9291adantr 480 . . . . . . . 8 ((𝜑𝑋 ≠ ∅) → (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
93 simpr 476 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
94 mptexg 6525 . . . . . . . . . . . . . . . . . . . 20 (𝑋 ∈ Fin → (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) ∈ V)
9533, 94syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) ∈ V)
9695adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) ∈ V)
9786fvmpt2 6330 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ℕ ∧ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) ∈ V) → ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗) = (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))
9893, 96, 97syl2anc 694 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗) = (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))
9998coeq2d 5317 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗)) = ([,) ∘ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)))
10099fveq1d 6231 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘) = (([,) ∘ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑘))
101100adantr 480 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘) = (([,) ∘ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑘))
10280adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩):𝑋⟶(ℝ × ℝ))
103 simpr 476 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → 𝑘𝑋)
104102, 103fvovco 39695 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑘) = ((1st ‘((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘))[,)(2nd ‘((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘))))
105 simpr 476 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑋) → 𝑘𝑋)
106 opex 4962 . . . . . . . . . . . . . . . . . . . 20 ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩ ∈ V
107106a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑋) → ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩ ∈ V)
10879fvmpt2 6330 . . . . . . . . . . . . . . . . . . 19 ((𝑘𝑋 ∧ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩ ∈ V) → ((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘) = ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)
109105, 107, 108syl2anc 694 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝑋) → ((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘) = ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)
110109fveq2d 6233 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (1st ‘((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘)) = (1st ‘⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))
111 fvex 6239 . . . . . . . . . . . . . . . . . . 19 ((𝐶𝑗)‘𝑘) ∈ V
112 fvex 6239 . . . . . . . . . . . . . . . . . . 19 ((𝐷𝑗)‘𝑘) ∈ V
113 op1stg 7222 . . . . . . . . . . . . . . . . . . 19 ((((𝐶𝑗)‘𝑘) ∈ V ∧ ((𝐷𝑗)‘𝑘) ∈ V) → (1st ‘⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) = ((𝐶𝑗)‘𝑘))
114111, 112, 113mp2an 708 . . . . . . . . . . . . . . . . . 18 (1st ‘⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) = ((𝐶𝑗)‘𝑘)
115114a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (1st ‘⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) = ((𝐶𝑗)‘𝑘))
116110, 115eqtrd 2685 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → (1st ‘((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘)) = ((𝐶𝑗)‘𝑘))
117109fveq2d 6233 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (2nd ‘((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘)) = (2nd ‘⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))
118111, 112op2nd 7219 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) = ((𝐷𝑗)‘𝑘)
119118a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (2nd ‘⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩) = ((𝐷𝑗)‘𝑘))
120117, 119eqtrd 2685 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → (2nd ‘((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘)) = ((𝐷𝑗)‘𝑘))
121116, 120oveq12d 6708 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑋) → ((1st ‘((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘))[,)(2nd ‘((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘))) = (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
122121adantlr 751 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((1st ‘((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘))[,)(2nd ‘((𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)‘𝑘))) = (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
123101, 104, 1223eqtrrd 2690 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
124123ixpeq2dva 7965 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
125124iuneq2dv 4574 . . . . . . . . . . 11 (𝜑 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
1264, 125sseqtrd 3674 . . . . . . . . . 10 (𝜑𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
127126adantr 480 . . . . . . . . 9 ((𝜑𝑋 ≠ ∅) → 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
128 eqidd 2652 . . . . . . . . . 10 ((𝜑𝑋 ≠ ∅) → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))))
12951adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
13052adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → 𝑋 ≠ ∅)
13138adantlr 751 . . . . . . . . . . . . 13 (((𝜑𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝐶𝑗):𝑋⟶ℝ)
13242adantlr 751 . . . . . . . . . . . . 13 (((𝜑𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝐷𝑗):𝑋⟶ℝ)
13332, 129, 130, 131, 132hoidmvn0val 41119 . . . . . . . . . . . 12 (((𝜑𝑋 ≠ ∅) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)) = ∏𝑘𝑋 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
134133mpteq2dva 4777 . . . . . . . . . . 11 ((𝜑𝑋 ≠ ∅) → (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))))
135134fveq2d 6233 . . . . . . . . . 10 ((𝜑𝑋 ≠ ∅) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))))
136123eqcomd 2657 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘) = (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
137136fveq2d 6233 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)) = (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
138137prodeq2dv 14697 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘))))
139138mpteq2dva 4777 . . . . . . . . . . . 12 (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))))
140139fveq2d 6233 . . . . . . . . . . 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 480 . . . . . . . . . 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 2695 . . . . . . . . 9 ((𝜑𝑋 ≠ ∅) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘)))))
143127, 142jca 553 . . . . . . . 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 2793 . . . . . . . . . . . . 13 𝑗𝑖
145 nfmpt1 4780 . . . . . . . . . . . . 13 𝑗(𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))
146144, 145nfeq 2805 . . . . . . . . . . . 12 𝑗 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))
147 nfcv 2793 . . . . . . . . . . . . . . 15 𝑘𝑖
148 nfcv 2793 . . . . . . . . . . . . . . . 16 𝑘
149 nfmpt1 4780 . . . . . . . . . . . . . . . 16 𝑘(𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)
150148, 149nfmpt 4779 . . . . . . . . . . . . . . 15 𝑘(𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))
151147, 150nfeq 2805 . . . . . . . . . . . . . 14 𝑘 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))
152 fveq1 6228 . . . . . . . . . . . . . . . . 17 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) → (𝑖𝑗) = ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))
153152coeq2d 5317 . . . . . . . . . . . . . . . 16 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) → ([,) ∘ (𝑖𝑗)) = ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗)))
154153fveq1d 6231 . . . . . . . . . . . . . . 15 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
155154adantr 480 . . . . . . . . . . . . . 14 ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) ∧ 𝑘𝑋) → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
156151, 155ixpeq2d 39551 . . . . . . . . . . . . 13 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) → X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩))‘𝑗))‘𝑘))
157156adantr 480 . . . . . . . . . . . 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 39526 . . . . . . . . . . 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 3666 . . . . . . . . . 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 1883 . . . . . . . . . . . . . . . 16 𝑘 𝑗 ∈ ℕ
161151, 160nfan 1868 . . . . . . . . . . . . . . 15 𝑘(𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨((𝐶𝑗)‘𝑘), ((𝐷𝑗)‘𝑘)⟩)) ∧ 𝑗 ∈ ℕ)
162154fveq2d 6233 . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 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 2986 . . . . . . . . . . . . . 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 14696 . . . . . . . . . . . . 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 4776 . . . . . . . . . . . 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 6233 . . . . . . . . . . 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 2661 . . . . . . . . . 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 747 . . . . . . . . 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 3340 . . . . . . . 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))))))
17292, 143, 171syl2anc 694 . . . . . . 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 553 . . . . . 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 2655 . . . . . . . . 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 740 . . . . . . . 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 3081 . . . . . . 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 3396 . . . . . 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 224 . . . . 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 12202 . . . . 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 694 . . . 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 4707 . . 3 ((𝜑𝑋 ≠ ∅) → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))
18248, 50, 181syl2anc 694 . 2 ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))
18347, 182pm2.61dan 849 1 (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))
Colors of variables: wff setvar class
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  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  Xcixp 7950  Fincfn 7997  infcinf 8388  cr 9973  0cc0 9974  1c1 9975  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  cn 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
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