Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovnhoilem1 Structured version   Visualization version   GIF version

Theorem ovnhoilem1 39491
 Description: The Lebesgue outer measure of a multidimensional half-open interval is less than or equal to the product of its length in each dimension. First part of the proof of Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Hypotheses
Ref Expression
ovnhoilem1.x (𝜑𝑋 ∈ Fin)
ovnhoilem1.a (𝜑𝐴:𝑋⟶ℝ)
ovnhoilem1.b (𝜑𝐵:𝑋⟶ℝ)
ovnhoilem1.c 𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘))
ovnhoilem1.m 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}
ovnhoilem1.h 𝐻 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))
Assertion
Ref Expression
ovnhoilem1 (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
Distinct variable groups:   𝐴,𝑖,𝑗,𝑧   𝐵,𝑖,𝑗,𝑧   𝑖,𝐻,𝑗   𝑖,𝐼,𝑧   𝑖,𝑋,𝑗,𝑘,𝑧   𝜑,𝑗,𝑘
Allowed substitution hints:   𝜑(𝑧,𝑖)   𝐴(𝑘)   𝐵(𝑘)   𝐻(𝑧,𝑘)   𝐼(𝑗,𝑘)   𝑀(𝑧,𝑖,𝑗,𝑘)

Proof of Theorem ovnhoilem1
StepHypRef Expression
1 ovnhoilem1.x . . 3 (𝜑𝑋 ∈ Fin)
2 ovnhoilem1.c . . . . 5 𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘))
32a1i 11 . . . 4 (𝜑𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
4 nfv 1830 . . . . 5 𝑘𝜑
5 ovnhoilem1.a . . . . . 6 (𝜑𝐴:𝑋⟶ℝ)
65ffvelrnda 6267 . . . . 5 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
7 ovnhoilem1.b . . . . . . 7 (𝜑𝐵:𝑋⟶ℝ)
87ffvelrnda 6267 . . . . . 6 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
98rexrd 9968 . . . . 5 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ*)
104, 6, 9hoissrrn2 39468 . . . 4 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ (ℝ ↑𝑚 𝑋))
113, 10eqsstrd 3602 . . 3 (𝜑𝐼 ⊆ (ℝ ↑𝑚 𝑋))
12 ovnhoilem1.m . . 3 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}
131, 11, 12ovnval2 39435 . 2 (𝜑 → ((voln*‘𝑋)‘𝐼) = if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )))
14 iftrue 4042 . . . . 5 (𝑋 = ∅ → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = 0)
1514adantl 481 . . . 4 ((𝜑𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = 0)
16 0xr 9965 . . . . . . 7 0 ∈ ℝ*
1716a1i 11 . . . . . 6 (𝜑 → 0 ∈ ℝ*)
18 pnfxr 9971 . . . . . . 7 +∞ ∈ ℝ*
1918a1i 11 . . . . . 6 (𝜑 → +∞ ∈ ℝ*)
204, 1, 6, 8hoiprodcl3 39470 . . . . . 6 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ (0[,)+∞))
21 icogelb 12096 . . . . . 6 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ (0[,)+∞)) → 0 ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
2217, 19, 20, 21syl3anc 1318 . . . . 5 (𝜑 → 0 ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
2322adantr 480 . . . 4 ((𝜑𝑋 = ∅) → 0 ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
2415, 23eqbrtrd 4605 . . 3 ((𝜑𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
25 iffalse 4045 . . . . 5 𝑋 = ∅ → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = inf(𝑀, ℝ*, < ))
2625adantl 481 . . . 4 ((𝜑 ∧ ¬ 𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = inf(𝑀, ℝ*, < ))
27 ssrab2 3650 . . . . . . 7 {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} ⊆ ℝ*
2812, 27eqsstri 3598 . . . . . 6 𝑀 ⊆ ℝ*
2928a1i 11 . . . . 5 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑀 ⊆ ℝ*)
30 icossxr 12129 . . . . . . . . . 10 (0[,)+∞) ⊆ ℝ*
3130, 20sseldi 3566 . . . . . . . . 9 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ*)
3231adantr 480 . . . . . . . 8 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ*)
33 opelxpi 5072 . . . . . . . . . . . . . . . . 17 (((𝐴𝑘) ∈ ℝ ∧ (𝐵𝑘) ∈ ℝ) → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ (ℝ × ℝ))
346, 8, 33syl2anc 691 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ (ℝ × ℝ))
35 0re 9919 . . . . . . . . . . . . . . . . . 18 0 ∈ ℝ
36 opelxpi 5072 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℝ ∧ 0 ∈ ℝ) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
3735, 35, 36mp2an 704 . . . . . . . . . . . . . . . . 17 ⟨0, 0⟩ ∈ (ℝ × ℝ)
3837a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
3934, 38ifcld 4081 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑋) → if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩) ∈ (ℝ × ℝ))
40 eqid 2610 . . . . . . . . . . . . . . 15 (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) = (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩))
4139, 40fmptd 6292 . . . . . . . . . . . . . 14 (𝜑 → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)):𝑋⟶(ℝ × ℝ))
42 reex 9906 . . . . . . . . . . . . . . . . 17 ℝ ∈ V
4342, 42xpex 6860 . . . . . . . . . . . . . . . 16 (ℝ × ℝ) ∈ V
441, 43jctil 558 . . . . . . . . . . . . . . 15 (𝜑 → ((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin))
45 elmapg 7757 . . . . . . . . . . . . . . 15 (((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↔ (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)):𝑋⟶(ℝ × ℝ)))
4644, 45syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↔ (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)):𝑋⟶(ℝ × ℝ)))
4741, 46mpbird 246 . . . . . . . . . . . . 13 (𝜑 → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
4847adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
49 ovnhoilem1.h . . . . . . . . . . . 12 𝐻 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))
5048, 49fmptd 6292 . . . . . . . . . . 11 (𝜑𝐻:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
51 ovex 6577 . . . . . . . . . . . 12 ((ℝ × ℝ) ↑𝑚 𝑋) ∈ V
52 nnex 10903 . . . . . . . . . . . 12 ℕ ∈ V
5351, 52elmap 7772 . . . . . . . . . . 11 (𝐻 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ↔ 𝐻:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
5450, 53sylibr 223 . . . . . . . . . 10 (𝜑𝐻 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
5554adantr 480 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐻 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
56 eqidd 2611 . . . . . . . . . . . . 13 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
57 eqid 2610 . . . . . . . . . . . . . . . . . . 19 (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) = (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩)
5834, 57fmptd 6292 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩):𝑋⟶(ℝ × ℝ))
5949a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐻 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩))))
60 iftrue 4042 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 1 → if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩) = ⟨(𝐴𝑘), (𝐵𝑘)⟩)
6160mpteq2dv 4673 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 1 → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) = (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩))
6261adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 = 1) → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) = (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩))
63 1nn 10908 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℕ
6463a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 1 ∈ ℕ)
65 mptexg 6389 . . . . . . . . . . . . . . . . . . . . 21 (𝑋 ∈ Fin → (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) ∈ V)
661, 65syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) ∈ V)
6759, 62, 64, 66fvmptd 6197 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐻‘1) = (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩))
6867feq1d 5943 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝐻‘1):𝑋⟶(ℝ × ℝ) ↔ (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩):𝑋⟶(ℝ × ℝ)))
6958, 68mpbird 246 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐻‘1):𝑋⟶(ℝ × ℝ))
7069adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → (𝐻‘1):𝑋⟶(ℝ × ℝ))
71 simpr 476 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → 𝑘𝑋)
7270, 71fvovco 38376 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑋) → (([,) ∘ (𝐻‘1))‘𝑘) = ((1st ‘((𝐻‘1)‘𝑘))[,)(2nd ‘((𝐻‘1)‘𝑘))))
7334elexd 3187 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑋) → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V)
7467, 73fvmpt2d 6202 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝑋) → ((𝐻‘1)‘𝑘) = ⟨(𝐴𝑘), (𝐵𝑘)⟩)
7574fveq2d 6107 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (1st ‘((𝐻‘1)‘𝑘)) = (1st ‘⟨(𝐴𝑘), (𝐵𝑘)⟩))
76 fvex 6113 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑘) ∈ V
77 fvex 6113 . . . . . . . . . . . . . . . . . . 19 (𝐵𝑘) ∈ V
7876, 77op1st 7067 . . . . . . . . . . . . . . . . . 18 (1st ‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = (𝐴𝑘)
7978a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (1st ‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = (𝐴𝑘))
8075, 79eqtrd 2644 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → (1st ‘((𝐻‘1)‘𝑘)) = (𝐴𝑘))
8174fveq2d 6107 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (2nd ‘((𝐻‘1)‘𝑘)) = (2nd ‘⟨(𝐴𝑘), (𝐵𝑘)⟩))
8276, 77op2nd 7068 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = (𝐵𝑘)
8382a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (2nd ‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = (𝐵𝑘))
8481, 83eqtrd 2644 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → (2nd ‘((𝐻‘1)‘𝑘)) = (𝐵𝑘))
8580, 84oveq12d 6567 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑋) → ((1st ‘((𝐻‘1)‘𝑘))[,)(2nd ‘((𝐻‘1)‘𝑘))) = ((𝐴𝑘)[,)(𝐵𝑘)))
8672, 85eqtrd 2644 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑋) → (([,) ∘ (𝐻‘1))‘𝑘) = ((𝐴𝑘)[,)(𝐵𝑘)))
8786ixpeq2dva 7809 . . . . . . . . . . . . 13 (𝜑X𝑘𝑋 (([,) ∘ (𝐻‘1))‘𝑘) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
8856, 3, 873eqtr4d 2654 . . . . . . . . . . . 12 (𝜑𝐼 = X𝑘𝑋 (([,) ∘ (𝐻‘1))‘𝑘))
89 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑗 = 1 → (𝐻𝑗) = (𝐻‘1))
9089coeq2d 5206 . . . . . . . . . . . . . . . 16 (𝑗 = 1 → ([,) ∘ (𝐻𝑗)) = ([,) ∘ (𝐻‘1)))
9190fveq1d 6105 . . . . . . . . . . . . . . 15 (𝑗 = 1 → (([,) ∘ (𝐻𝑗))‘𝑘) = (([,) ∘ (𝐻‘1))‘𝑘))
9291ixpeq2dv 7810 . . . . . . . . . . . . . 14 (𝑗 = 1 → X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝐻‘1))‘𝑘))
9392ssiun2s 4500 . . . . . . . . . . . . 13 (1 ∈ ℕ → X𝑘𝑋 (([,) ∘ (𝐻‘1))‘𝑘) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘))
9463, 93ax-mp 5 . . . . . . . . . . . 12 X𝑘𝑋 (([,) ∘ (𝐻‘1))‘𝑘) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘)
9588, 94syl6eqss 3618 . . . . . . . . . . 11 (𝜑𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘))
9695adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘))
9786fveq2d 6107 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑋) → (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
9897eqcomd 2616 . . . . . . . . . . . . 13 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
9998prodeq2dv 14492 . . . . . . . . . . . 12 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
10099adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
101 1red 9934 . . . . . . . . . . . . . 14 (𝜑 → 1 ∈ ℝ)
102 icossicc 12131 . . . . . . . . . . . . . . 15 (0[,)+∞) ⊆ (0[,]+∞)
1034, 1, 69hoiprodcl 39437 . . . . . . . . . . . . . . 15 (𝜑 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) ∈ (0[,)+∞))
104102, 103sseldi 3566 . . . . . . . . . . . . . 14 (𝜑 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) ∈ (0[,]+∞))
10591fveq2d 6107 . . . . . . . . . . . . . . 15 (𝑗 = 1 → (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
106105prodeq2ad 38659 . . . . . . . . . . . . . 14 (𝑗 = 1 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
107101, 104, 106sge0snmpt 39276 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
108107eqcomd 2616 . . . . . . . . . . . 12 (𝜑 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) = (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))))
109108adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) = (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))))
110 nfv 1830 . . . . . . . . . . . 12 𝑗(𝜑 ∧ ¬ 𝑋 = ∅)
11152a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝑋 = ∅) → ℕ ∈ V)
112 snssi 4280 . . . . . . . . . . . . . 14 (1 ∈ ℕ → {1} ⊆ ℕ)
11363, 112ax-mp 5 . . . . . . . . . . . . 13 {1} ⊆ ℕ
114113a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝑋 = ∅) → {1} ⊆ ℕ)
115 nfv 1830 . . . . . . . . . . . . . 14 𝑘((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1})
1161ad2antrr 758 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → 𝑋 ∈ Fin)
117 simpl 472 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ {1}) → 𝜑)
118 elsni 4142 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ {1} → 𝑗 = 1)
119118adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ {1}) → 𝑗 = 1)
12069adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 = 1) → (𝐻‘1):𝑋⟶(ℝ × ℝ))
12189adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 = 1) → (𝐻𝑗) = (𝐻‘1))
122121feq1d 5943 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 = 1) → ((𝐻𝑗):𝑋⟶(ℝ × ℝ) ↔ (𝐻‘1):𝑋⟶(ℝ × ℝ)))
123120, 122mpbird 246 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 = 1) → (𝐻𝑗):𝑋⟶(ℝ × ℝ))
124117, 119, 123syl2anc 691 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ {1}) → (𝐻𝑗):𝑋⟶(ℝ × ℝ))
125124adantlr 747 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → (𝐻𝑗):𝑋⟶(ℝ × ℝ))
126115, 116, 125hoiprodcl 39437 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) ∈ (0[,)+∞))
127102, 126sseldi 3566 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) ∈ (0[,]+∞))
128 eqid 2610 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘𝑋 ↦ ⟨0, 0⟩) = (𝑘𝑋 ↦ ⟨0, 0⟩)
12938, 128fmptd 6292 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑘𝑋 ↦ ⟨0, 0⟩):𝑋⟶(ℝ × ℝ))
130129adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → (𝑘𝑋 ↦ ⟨0, 0⟩):𝑋⟶(ℝ × ℝ))
131 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → 𝜑)
132 eldifi 3694 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (ℕ ∖ {1}) → 𝑗 ∈ ℕ)
133132adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → 𝑗 ∈ ℕ)
13448elexd 3187 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) ∈ V)
13559, 134fvmpt2d 6202 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ ℕ) → (𝐻𝑗) = (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))
136131, 133, 135syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → (𝐻𝑗) = (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))
137 eldifsni 4261 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (ℕ ∖ {1}) → 𝑗 ≠ 1)
138137neneqd 2787 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (ℕ ∖ {1}) → ¬ 𝑗 = 1)
139138iffalsed 4047 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (ℕ ∖ {1}) → if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩) = ⟨0, 0⟩)
140139mpteq2dv 4673 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (ℕ ∖ {1}) → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) = (𝑘𝑋 ↦ ⟨0, 0⟩))
141140adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) = (𝑘𝑋 ↦ ⟨0, 0⟩))
142136, 141eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → (𝐻𝑗) = (𝑘𝑋 ↦ ⟨0, 0⟩))
143142feq1d 5943 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → ((𝐻𝑗):𝑋⟶(ℝ × ℝ) ↔ (𝑘𝑋 ↦ ⟨0, 0⟩):𝑋⟶(ℝ × ℝ)))
144130, 143mpbird 246 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → (𝐻𝑗):𝑋⟶(ℝ × ℝ))
145144adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (𝐻𝑗):𝑋⟶(ℝ × ℝ))
146 simpr 476 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → 𝑘𝑋)
147145, 146fvovco 38376 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (([,) ∘ (𝐻𝑗))‘𝑘) = ((1st ‘((𝐻𝑗)‘𝑘))[,)(2nd ‘((𝐻𝑗)‘𝑘))))
14837elexi 3186 . . . . . . . . . . . . . . . . . . . . . . 23 ⟨0, 0⟩ ∈ V
149148a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → ⟨0, 0⟩ ∈ V)
150142, 149fvmpt2d 6202 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → ((𝐻𝑗)‘𝑘) = ⟨0, 0⟩)
151150fveq2d 6107 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (1st ‘((𝐻𝑗)‘𝑘)) = (1st ‘⟨0, 0⟩))
15216elexi 3186 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ V
153152, 152op1st 7067 . . . . . . . . . . . . . . . . . . . . 21 (1st ‘⟨0, 0⟩) = 0
154153a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (1st ‘⟨0, 0⟩) = 0)
155151, 154eqtrd 2644 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (1st ‘((𝐻𝑗)‘𝑘)) = 0)
156150fveq2d 6107 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (2nd ‘((𝐻𝑗)‘𝑘)) = (2nd ‘⟨0, 0⟩))
157152, 152op2nd 7068 . . . . . . . . . . . . . . . . . . . . 21 (2nd ‘⟨0, 0⟩) = 0
158157a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (2nd ‘⟨0, 0⟩) = 0)
159156, 158eqtrd 2644 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (2nd ‘((𝐻𝑗)‘𝑘)) = 0)
160155, 159oveq12d 6567 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → ((1st ‘((𝐻𝑗)‘𝑘))[,)(2nd ‘((𝐻𝑗)‘𝑘))) = (0[,)0))
161 0le0 10987 . . . . . . . . . . . . . . . . . . . 20 0 ≤ 0
162 ico0 12092 . . . . . . . . . . . . . . . . . . . . 21 ((0 ∈ ℝ* ∧ 0 ∈ ℝ*) → ((0[,)0) = ∅ ↔ 0 ≤ 0))
16316, 16, 162mp2an 704 . . . . . . . . . . . . . . . . . . . 20 ((0[,)0) = ∅ ↔ 0 ≤ 0)
164161, 163mpbir 220 . . . . . . . . . . . . . . . . . . 19 (0[,)0) = ∅
165164a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (0[,)0) = ∅)
166147, 160, 1653eqtrd 2648 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (([,) ∘ (𝐻𝑗))‘𝑘) = ∅)
167166fveq2d 6107 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = (vol‘∅))
168 vol0 38851 . . . . . . . . . . . . . . . . 17 (vol‘∅) = 0
169168a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (vol‘∅) = 0)
170167, 169eqtrd 2644 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = 0)
171170prodeq2dv 14492 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = ∏𝑘𝑋 0)
172171adantlr 747 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = ∏𝑘𝑋 0)
173 0cnd 9912 . . . . . . . . . . . . . . 15 (𝜑 → 0 ∈ ℂ)
174 fprodconst 14547 . . . . . . . . . . . . . . 15 ((𝑋 ∈ Fin ∧ 0 ∈ ℂ) → ∏𝑘𝑋 0 = (0↑(#‘𝑋)))
1751, 173, 174syl2anc 691 . . . . . . . . . . . . . 14 (𝜑 → ∏𝑘𝑋 0 = (0↑(#‘𝑋)))
176175ad2antrr 758 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘𝑋 0 = (0↑(#‘𝑋)))
177 neqne 2790 . . . . . . . . . . . . . . . . 17 𝑋 = ∅ → 𝑋 ≠ ∅)
178177adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
1791adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin)
180 hashnncl 13018 . . . . . . . . . . . . . . . . 17 (𝑋 ∈ Fin → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
181179, 180syl 17 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ¬ 𝑋 = ∅) → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
182178, 181mpbird 246 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ¬ 𝑋 = ∅) → (#‘𝑋) ∈ ℕ)
183 0exp 12757 . . . . . . . . . . . . . . 15 ((#‘𝑋) ∈ ℕ → (0↑(#‘𝑋)) = 0)
184182, 183syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ 𝑋 = ∅) → (0↑(#‘𝑋)) = 0)
185184adantr 480 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → (0↑(#‘𝑋)) = 0)
186172, 176, 1853eqtrd 2648 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = 0)
187110, 111, 114, 127, 186sge0ss 39305 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝑋 = ∅) → (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))))
188100, 109, 1873eqtrd 2648 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))))
18996, 188jca 553 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘) ∧ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘))))))
190 nfcv 2751 . . . . . . . . . . . . . . 15 𝑘𝑖
191 nfcv 2751 . . . . . . . . . . . . . . . . 17 𝑘
192 nfmpt1 4675 . . . . . . . . . . . . . . . . 17 𝑘(𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩))
193191, 192nfmpt 4674 . . . . . . . . . . . . . . . 16 𝑘(𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))
19449, 193nfcxfr 2749 . . . . . . . . . . . . . . 15 𝑘𝐻
195190, 194nfeq 2762 . . . . . . . . . . . . . 14 𝑘 𝑖 = 𝐻
196 fveq1 6102 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝐻 → (𝑖𝑗) = (𝐻𝑗))
197196coeq2d 5206 . . . . . . . . . . . . . . . 16 (𝑖 = 𝐻 → ([,) ∘ (𝑖𝑗)) = ([,) ∘ (𝐻𝑗)))
198197fveq1d 6105 . . . . . . . . . . . . . . 15 (𝑖 = 𝐻 → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐻𝑗))‘𝑘))
199198adantr 480 . . . . . . . . . . . . . 14 ((𝑖 = 𝐻𝑘𝑋) → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐻𝑗))‘𝑘))
200195, 199ixpeq2d 38262 . . . . . . . . . . . . 13 (𝑖 = 𝐻X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘))
201200iuneq2d 4483 . . . . . . . . . . . 12 (𝑖 = 𝐻 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘))
202201sseq2d 3596 . . . . . . . . . . 11 (𝑖 = 𝐻 → (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ↔ 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘)))
203198fveq2d 6107 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝐻 → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))
204203a1d 25 . . . . . . . . . . . . . . . 16 (𝑖 = 𝐻 → (𝑘𝑋 → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻𝑗))‘𝑘))))
205195, 204ralrimi 2940 . . . . . . . . . . . . . . 15 (𝑖 = 𝐻 → ∀𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))
206205prodeq2d 14491 . . . . . . . . . . . . . 14 (𝑖 = 𝐻 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))
207206mpteq2dv 4673 . . . . . . . . . . . . 13 (𝑖 = 𝐻 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘))))
208207fveq2d 6107 . . . . . . . . . . . 12 (𝑖 = 𝐻 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))))
209208eqeq2d 2620 . . . . . . . . . . 11 (𝑖 = 𝐻 → (∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) ↔ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘))))))
210202, 209anbi12d 743 . . . . . . . . . 10 (𝑖 = 𝐻 → ((𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) ↔ (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘) ∧ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))))))
211210rspcev 3282 . . . . . . . . 9 ((𝐻 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘) ∧ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))))) → ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
21255, 189, 211syl2anc 691 . . . . . . . 8 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
21332, 212jca 553 . . . . . . 7 ((𝜑 ∧ ¬ 𝑋 = ∅) → (∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
214 eqeq1 2614 . . . . . . . . . 10 (𝑧 = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) → (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) ↔ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
215214anbi2d 736 . . . . . . . . 9 (𝑧 = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) → ((𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) ↔ (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
216215rexbidv 3034 . . . . . . . 8 (𝑧 = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) → (∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) ↔ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
217216elrab 3331 . . . . . . 7 (∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} ↔ (∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
218213, 217sylibr 223 . . . . . 6 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))})
21912eqcomi 2619 . . . . . . 7 {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} = 𝑀
220219a1i 11 . . . . . 6 ((𝜑 ∧ ¬ 𝑋 = ∅) → {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} = 𝑀)
221218, 220eleqtrd 2690 . . . . 5 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ 𝑀)
222 infxrlb 12036 . . . . 5 ((𝑀 ⊆ ℝ* ∧ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ 𝑀) → inf(𝑀, ℝ*, < ) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
22329, 221, 222syl2anc 691 . . . 4 ((𝜑 ∧ ¬ 𝑋 = ∅) → inf(𝑀, ℝ*, < ) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
22426, 223eqbrtrd 4605 . . 3 ((𝜑 ∧ ¬ 𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
22524, 224pm2.61dan 828 . 2 (𝜑 → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
22613, 225eqbrtrd 4605 1 (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  ifcif 4036  {csn 4125  ⟨cop 4131  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058   ↑𝑚 cmap 7744  Xcixp 7794  Fincfn 7841  infcinf 8230  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816  +∞cpnf 9950  ℝ*cxr 9952   < clt 9953   ≤ cle 9954  ℕcn 10897  [,)cico 12048  [,]cicc 12049  ↑cexp 12722  #chash 12979  ∏cprod 14474  volcvol 23039  Σ^csumge0 39255  voln*covoln 39426 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-prod 14475  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041  df-sumge0 39256  df-ovoln 39427 This theorem is referenced by:  ovnhoi  39493
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