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Theorem ovnhoilem1 41313
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 1984 . . . . 5 𝑘𝜑
5 ovnhoilem1.a . . . . . 6 (𝜑𝐴:𝑋⟶ℝ)
65ffvelrnda 6514 . . . . 5 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
7 ovnhoilem1.b . . . . . . 7 (𝜑𝐵:𝑋⟶ℝ)
87ffvelrnda 6514 . . . . . 6 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
98rexrd 10273 . . . . 5 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ*)
104, 6, 9hoissrrn2 41290 . . . 4 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ (ℝ ↑𝑚 𝑋))
113, 10eqsstrd 3772 . . 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 41257 . 2 (𝜑 → ((voln*‘𝑋)‘𝐼) = if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )))
14 iftrue 4228 . . . . 5 (𝑋 = ∅ → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = 0)
1514adantl 473 . . . 4 ((𝜑𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = 0)
16 0xr 10270 . . . . . . 7 0 ∈ ℝ*
1716a1i 11 . . . . . 6 (𝜑 → 0 ∈ ℝ*)
18 pnfxr 10276 . . . . . . 7 +∞ ∈ ℝ*
1918a1i 11 . . . . . 6 (𝜑 → +∞ ∈ ℝ*)
204, 1, 6, 8hoiprodcl3 41292 . . . . . 6 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ (0[,)+∞))
21 icogelb 12410 . . . . . 6 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ (0[,)+∞)) → 0 ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
2217, 19, 20, 21syl3anc 1473 . . . . 5 (𝜑 → 0 ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
2322adantr 472 . . . 4 ((𝜑𝑋 = ∅) → 0 ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
2415, 23eqbrtrd 4818 . . 3 ((𝜑𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
25 iffalse 4231 . . . . 5 𝑋 = ∅ → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = inf(𝑀, ℝ*, < ))
2625adantl 473 . . . 4 ((𝜑 ∧ ¬ 𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = inf(𝑀, ℝ*, < ))
27 ssrab2 3820 . . . . . . 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 3768 . . . . . 6 𝑀 ⊆ ℝ*
2928a1i 11 . . . . 5 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑀 ⊆ ℝ*)
30 icossxr 12443 . . . . . . . . . 10 (0[,)+∞) ⊆ ℝ*
3130, 20sseldi 3734 . . . . . . . . 9 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ*)
3231adantr 472 . . . . . . . 8 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ*)
33 opelxpi 5297 . . . . . . . . . . . . . . . . 17 (((𝐴𝑘) ∈ ℝ ∧ (𝐵𝑘) ∈ ℝ) → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ (ℝ × ℝ))
346, 8, 33syl2anc 696 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ (ℝ × ℝ))
35 0re 10224 . . . . . . . . . . . . . . . . . 18 0 ∈ ℝ
36 opelxpi 5297 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℝ ∧ 0 ∈ ℝ) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
3735, 35, 36mp2an 710 . . . . . . . . . . . . . . . . 17 ⟨0, 0⟩ ∈ (ℝ × ℝ)
3837a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
3934, 38ifcld 4267 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑋) → if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩) ∈ (ℝ × ℝ))
40 eqid 2752 . . . . . . . . . . . . . . 15 (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) = (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩))
4139, 40fmptd 6540 . . . . . . . . . . . . . 14 (𝜑 → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)):𝑋⟶(ℝ × ℝ))
42 reex 10211 . . . . . . . . . . . . . . . . 17 ℝ ∈ V
4342, 42xpex 7119 . . . . . . . . . . . . . . . 16 (ℝ × ℝ) ∈ V
441, 43jctil 561 . . . . . . . . . . . . . . 15 (𝜑 → ((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin))
45 elmapg 8028 . . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . 13 (𝜑 → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
4847adantr 472 . . . . . . . . . . . 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 6540 . . . . . . . . . . 11 (𝜑𝐻:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
51 ovex 6833 . . . . . . . . . . . 12 ((ℝ × ℝ) ↑𝑚 𝑋) ∈ V
52 nnex 11210 . . . . . . . . . . . 12 ℕ ∈ V
5351, 52elmap 8044 . . . . . . . . . . 11 (𝐻 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ↔ 𝐻:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
5450, 53sylibr 224 . . . . . . . . . 10 (𝜑𝐻 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
5554adantr 472 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐻 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
56 eqidd 2753 . . . . . . . . . . . . 13 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
57 eqid 2752 . . . . . . . . . . . . . . . . . . 19 (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) = (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩)
5834, 57fmptd 6540 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩):𝑋⟶(ℝ × ℝ))
5949a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐻 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩))))
60 iftrue 4228 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 1 → if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩) = ⟨(𝐴𝑘), (𝐵𝑘)⟩)
6160mpteq2dv 4889 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 1 → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) = (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩))
6261adantl 473 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 = 1) → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) = (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩))
63 1nn 11215 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℕ
6463a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 1 ∈ ℕ)
65 mptexg 6640 . . . . . . . . . . . . . . . . . . . . 21 (𝑋 ∈ Fin → (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) ∈ V)
661, 65syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) ∈ V)
6759, 62, 64, 66fvmptd 6442 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐻‘1) = (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩))
6867feq1d 6183 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝐻‘1):𝑋⟶(ℝ × ℝ) ↔ (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩):𝑋⟶(ℝ × ℝ)))
6958, 68mpbird 247 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐻‘1):𝑋⟶(ℝ × ℝ))
7069adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → (𝐻‘1):𝑋⟶(ℝ × ℝ))
71 simpr 479 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → 𝑘𝑋)
7270, 71fvovco 39872 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑋) → (([,) ∘ (𝐻‘1))‘𝑘) = ((1st ‘((𝐻‘1)‘𝑘))[,)(2nd ‘((𝐻‘1)‘𝑘))))
7334elexd 3346 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑋) → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V)
7467, 73fvmpt2d 6447 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝑋) → ((𝐻‘1)‘𝑘) = ⟨(𝐴𝑘), (𝐵𝑘)⟩)
7574fveq2d 6348 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (1st ‘((𝐻‘1)‘𝑘)) = (1st ‘⟨(𝐴𝑘), (𝐵𝑘)⟩))
76 fvex 6354 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑘) ∈ V
77 fvex 6354 . . . . . . . . . . . . . . . . . . 19 (𝐵𝑘) ∈ V
7876, 77op1st 7333 . . . . . . . . . . . . . . . . . 18 (1st ‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = (𝐴𝑘)
7978a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (1st ‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = (𝐴𝑘))
8075, 79eqtrd 2786 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → (1st ‘((𝐻‘1)‘𝑘)) = (𝐴𝑘))
8174fveq2d 6348 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (2nd ‘((𝐻‘1)‘𝑘)) = (2nd ‘⟨(𝐴𝑘), (𝐵𝑘)⟩))
8276, 77op2nd 7334 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = (𝐵𝑘)
8382a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (2nd ‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = (𝐵𝑘))
8481, 83eqtrd 2786 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → (2nd ‘((𝐻‘1)‘𝑘)) = (𝐵𝑘))
8580, 84oveq12d 6823 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑋) → ((1st ‘((𝐻‘1)‘𝑘))[,)(2nd ‘((𝐻‘1)‘𝑘))) = ((𝐴𝑘)[,)(𝐵𝑘)))
8672, 85eqtrd 2786 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑋) → (([,) ∘ (𝐻‘1))‘𝑘) = ((𝐴𝑘)[,)(𝐵𝑘)))
8786ixpeq2dva 8081 . . . . . . . . . . . . 13 (𝜑X𝑘𝑋 (([,) ∘ (𝐻‘1))‘𝑘) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
8856, 3, 873eqtr4d 2796 . . . . . . . . . . . 12 (𝜑𝐼 = X𝑘𝑋 (([,) ∘ (𝐻‘1))‘𝑘))
89 fveq2 6344 . . . . . . . . . . . . . . . . 17 (𝑗 = 1 → (𝐻𝑗) = (𝐻‘1))
9089coeq2d 5432 . . . . . . . . . . . . . . . 16 (𝑗 = 1 → ([,) ∘ (𝐻𝑗)) = ([,) ∘ (𝐻‘1)))
9190fveq1d 6346 . . . . . . . . . . . . . . 15 (𝑗 = 1 → (([,) ∘ (𝐻𝑗))‘𝑘) = (([,) ∘ (𝐻‘1))‘𝑘))
9291ixpeq2dv 8082 . . . . . . . . . . . . . 14 (𝑗 = 1 → X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝐻‘1))‘𝑘))
9392ssiun2s 4708 . . . . . . . . . . . . 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 3788 . . . . . . . . . . 11 (𝜑𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘))
9695adantr 472 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘))
9786fveq2d 6348 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑋) → (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
9897eqcomd 2758 . . . . . . . . . . . . 13 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
9998prodeq2dv 14844 . . . . . . . . . . . 12 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
10099adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
101 1red 10239 . . . . . . . . . . . . . 14 (𝜑 → 1 ∈ ℝ)
102 icossicc 12445 . . . . . . . . . . . . . . 15 (0[,)+∞) ⊆ (0[,]+∞)
1034, 1, 69hoiprodcl 41259 . . . . . . . . . . . . . . 15 (𝜑 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) ∈ (0[,)+∞))
104102, 103sseldi 3734 . . . . . . . . . . . . . 14 (𝜑 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) ∈ (0[,]+∞))
10591fveq2d 6348 . . . . . . . . . . . . . . 15 (𝑗 = 1 → (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
106105prodeq2ad 40319 . . . . . . . . . . . . . 14 (𝑗 = 1 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
107101, 104, 106sge0snmpt 41095 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
108107eqcomd 2758 . . . . . . . . . . . 12 (𝜑 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) = (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))))
109108adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) = (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))))
110 nfv 1984 . . . . . . . . . . . 12 𝑗(𝜑 ∧ ¬ 𝑋 = ∅)
11152a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝑋 = ∅) → ℕ ∈ V)
112 snssi 4476 . . . . . . . . . . . . . 14 (1 ∈ ℕ → {1} ⊆ ℕ)
11363, 112ax-mp 5 . . . . . . . . . . . . 13 {1} ⊆ ℕ
114113a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝑋 = ∅) → {1} ⊆ ℕ)
115 nfv 1984 . . . . . . . . . . . . . 14 𝑘((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1})
1161ad2antrr 764 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → 𝑋 ∈ Fin)
117 simpl 474 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ {1}) → 𝜑)
118 elsni 4330 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ {1} → 𝑗 = 1)
119118adantl 473 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ {1}) → 𝑗 = 1)
12069adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 = 1) → (𝐻‘1):𝑋⟶(ℝ × ℝ))
12189adantl 473 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 = 1) → (𝐻𝑗) = (𝐻‘1))
122121feq1d 6183 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 = 1) → ((𝐻𝑗):𝑋⟶(ℝ × ℝ) ↔ (𝐻‘1):𝑋⟶(ℝ × ℝ)))
123120, 122mpbird 247 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 = 1) → (𝐻𝑗):𝑋⟶(ℝ × ℝ))
124117, 119, 123syl2anc 696 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ {1}) → (𝐻𝑗):𝑋⟶(ℝ × ℝ))
125124adantlr 753 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → (𝐻𝑗):𝑋⟶(ℝ × ℝ))
126115, 116, 125hoiprodcl 41259 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) ∈ (0[,)+∞))
127102, 126sseldi 3734 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) ∈ (0[,]+∞))
128 eqid 2752 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘𝑋 ↦ ⟨0, 0⟩) = (𝑘𝑋 ↦ ⟨0, 0⟩)
12938, 128fmptd 6540 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑘𝑋 ↦ ⟨0, 0⟩):𝑋⟶(ℝ × ℝ))
130129adantr 472 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → (𝑘𝑋 ↦ ⟨0, 0⟩):𝑋⟶(ℝ × ℝ))
131 simpl 474 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → 𝜑)
132 eldifi 3867 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (ℕ ∖ {1}) → 𝑗 ∈ ℕ)
133132adantl 473 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → 𝑗 ∈ ℕ)
13448elexd 3346 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) ∈ V)
13559, 134fvmpt2d 6447 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ ℕ) → (𝐻𝑗) = (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))
136131, 133, 135syl2anc 696 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → (𝐻𝑗) = (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))
137 eldifsni 4458 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (ℕ ∖ {1}) → 𝑗 ≠ 1)
138137neneqd 2929 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (ℕ ∖ {1}) → ¬ 𝑗 = 1)
139138iffalsed 4233 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (ℕ ∖ {1}) → if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩) = ⟨0, 0⟩)
140139mpteq2dv 4889 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (ℕ ∖ {1}) → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) = (𝑘𝑋 ↦ ⟨0, 0⟩))
141140adantl 473 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) = (𝑘𝑋 ↦ ⟨0, 0⟩))
142136, 141eqtrd 2786 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → (𝐻𝑗) = (𝑘𝑋 ↦ ⟨0, 0⟩))
143142feq1d 6183 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → ((𝐻𝑗):𝑋⟶(ℝ × ℝ) ↔ (𝑘𝑋 ↦ ⟨0, 0⟩):𝑋⟶(ℝ × ℝ)))
144130, 143mpbird 247 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → (𝐻𝑗):𝑋⟶(ℝ × ℝ))
145144adantr 472 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (𝐻𝑗):𝑋⟶(ℝ × ℝ))
146 simpr 479 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → 𝑘𝑋)
147145, 146fvovco 39872 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (([,) ∘ (𝐻𝑗))‘𝑘) = ((1st ‘((𝐻𝑗)‘𝑘))[,)(2nd ‘((𝐻𝑗)‘𝑘))))
14837elexi 3345 . . . . . . . . . . . . . . . . . . . . . . 23 ⟨0, 0⟩ ∈ V
149148a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → ⟨0, 0⟩ ∈ V)
150142, 149fvmpt2d 6447 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → ((𝐻𝑗)‘𝑘) = ⟨0, 0⟩)
151150fveq2d 6348 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (1st ‘((𝐻𝑗)‘𝑘)) = (1st ‘⟨0, 0⟩))
15216elexi 3345 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ V
153152, 152op1st 7333 . . . . . . . . . . . . . . . . . . . . 21 (1st ‘⟨0, 0⟩) = 0
154153a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (1st ‘⟨0, 0⟩) = 0)
155151, 154eqtrd 2786 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (1st ‘((𝐻𝑗)‘𝑘)) = 0)
156150fveq2d 6348 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (2nd ‘((𝐻𝑗)‘𝑘)) = (2nd ‘⟨0, 0⟩))
157152, 152op2nd 7334 . . . . . . . . . . . . . . . . . . . . 21 (2nd ‘⟨0, 0⟩) = 0
158157a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (2nd ‘⟨0, 0⟩) = 0)
159156, 158eqtrd 2786 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (2nd ‘((𝐻𝑗)‘𝑘)) = 0)
160155, 159oveq12d 6823 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → ((1st ‘((𝐻𝑗)‘𝑘))[,)(2nd ‘((𝐻𝑗)‘𝑘))) = (0[,)0))
161 0le0 11294 . . . . . . . . . . . . . . . . . . . 20 0 ≤ 0
162 ico0 12406 . . . . . . . . . . . . . . . . . . . . 21 ((0 ∈ ℝ* ∧ 0 ∈ ℝ*) → ((0[,)0) = ∅ ↔ 0 ≤ 0))
16316, 16, 162mp2an 710 . . . . . . . . . . . . . . . . . . . 20 ((0[,)0) = ∅ ↔ 0 ≤ 0)
164161, 163mpbir 221 . . . . . . . . . . . . . . . . . . 19 (0[,)0) = ∅
165164a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (0[,)0) = ∅)
166147, 160, 1653eqtrd 2790 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (([,) ∘ (𝐻𝑗))‘𝑘) = ∅)
167166fveq2d 6348 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = (vol‘∅))
168 vol0 40670 . . . . . . . . . . . . . . . . 17 (vol‘∅) = 0
169168a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (vol‘∅) = 0)
170167, 169eqtrd 2786 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = 0)
171170prodeq2dv 14844 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = ∏𝑘𝑋 0)
172171adantlr 753 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = ∏𝑘𝑋 0)
173 0cnd 10217 . . . . . . . . . . . . . . 15 (𝜑 → 0 ∈ ℂ)
174 fprodconst 14899 . . . . . . . . . . . . . . 15 ((𝑋 ∈ Fin ∧ 0 ∈ ℂ) → ∏𝑘𝑋 0 = (0↑(♯‘𝑋)))
1751, 173, 174syl2anc 696 . . . . . . . . . . . . . 14 (𝜑 → ∏𝑘𝑋 0 = (0↑(♯‘𝑋)))
176175ad2antrr 764 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘𝑋 0 = (0↑(♯‘𝑋)))
177 neqne 2932 . . . . . . . . . . . . . . . . 17 𝑋 = ∅ → 𝑋 ≠ ∅)
178177adantl 473 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
1791adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin)
180 hashnncl 13341 . . . . . . . . . . . . . . . . 17 (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
181179, 180syl 17 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ¬ 𝑋 = ∅) → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
182178, 181mpbird 247 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ¬ 𝑋 = ∅) → (♯‘𝑋) ∈ ℕ)
183 0exp 13081 . . . . . . . . . . . . . . 15 ((♯‘𝑋) ∈ ℕ → (0↑(♯‘𝑋)) = 0)
184182, 183syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ 𝑋 = ∅) → (0↑(♯‘𝑋)) = 0)
185184adantr 472 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → (0↑(♯‘𝑋)) = 0)
186172, 176, 1853eqtrd 2790 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = 0)
187110, 111, 114, 127, 186sge0ss 41124 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝑋 = ∅) → (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))))
188100, 109, 1873eqtrd 2790 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))))
18996, 188jca 555 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘) ∧ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘))))))
190 nfcv 2894 . . . . . . . . . . . . . . 15 𝑘𝑖
191 nfcv 2894 . . . . . . . . . . . . . . . . 17 𝑘
192 nfmpt1 4891 . . . . . . . . . . . . . . . . 17 𝑘(𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩))
193191, 192nfmpt 4890 . . . . . . . . . . . . . . . 16 𝑘(𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))
19449, 193nfcxfr 2892 . . . . . . . . . . . . . . 15 𝑘𝐻
195190, 194nfeq 2906 . . . . . . . . . . . . . 14 𝑘 𝑖 = 𝐻
196 fveq1 6343 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝐻 → (𝑖𝑗) = (𝐻𝑗))
197196coeq2d 5432 . . . . . . . . . . . . . . . 16 (𝑖 = 𝐻 → ([,) ∘ (𝑖𝑗)) = ([,) ∘ (𝐻𝑗)))
198197fveq1d 6346 . . . . . . . . . . . . . . 15 (𝑖 = 𝐻 → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐻𝑗))‘𝑘))
199198adantr 472 . . . . . . . . . . . . . 14 ((𝑖 = 𝐻𝑘𝑋) → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐻𝑗))‘𝑘))
200195, 199ixpeq2d 39728 . . . . . . . . . . . . 13 (𝑖 = 𝐻X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘))
201200iuneq2d 4691 . . . . . . . . . . . 12 (𝑖 = 𝐻 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘))
202201sseq2d 3766 . . . . . . . . . . 11 (𝑖 = 𝐻 → (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ↔ 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘)))
203198fveq2d 6348 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝐻 → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))
204203a1d 25 . . . . . . . . . . . . . . . 16 (𝑖 = 𝐻 → (𝑘𝑋 → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻𝑗))‘𝑘))))
205195, 204ralrimi 3087 . . . . . . . . . . . . . . 15 (𝑖 = 𝐻 → ∀𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))
206205prodeq2d 14843 . . . . . . . . . . . . . 14 (𝑖 = 𝐻 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))
207206mpteq2dv 4889 . . . . . . . . . . . . 13 (𝑖 = 𝐻 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘))))
208207fveq2d 6348 . . . . . . . . . . . 12 (𝑖 = 𝐻 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))))
209208eqeq2d 2762 . . . . . . . . . . 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 749 . . . . . . . . . 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 3441 . . . . . . . . 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 696 . . . . . . . 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 555 . . . . . . 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 2756 . . . . . . . . . 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 742 . . . . . . . . 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 3182 . . . . . . . 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 3496 . . . . . . 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 224 . . . . . 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 2761 . . . . . . 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 2833 . . . . 5 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ 𝑀)
222 infxrlb 12349 . . . . 5 ((𝑀 ⊆ ℝ* ∧ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ 𝑀) → inf(𝑀, ℝ*, < ) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
22329, 221, 222syl2anc 696 . . . 4 ((𝜑 ∧ ¬ 𝑋 = ∅) → inf(𝑀, ℝ*, < ) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
22426, 223eqbrtrd 4818 . . 3 ((𝜑 ∧ ¬ 𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
22524, 224pm2.61dan 867 . 2 (𝜑 → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
22613, 225eqbrtrd 4818 1 (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  wne 2924  wrex 3043  {crab 3046  Vcvv 3332  cdif 3704  wss 3707  c0 4050  ifcif 4222  {csn 4313  cop 4319   ciun 4664   class class class wbr 4796  cmpt 4873   × cxp 5256  ccom 5262  wf 6037  cfv 6041  (class class class)co 6805  1st c1st 7323  2nd c2nd 7324  𝑚 cmap 8015  Xcixp 8066  Fincfn 8113  infcinf 8504  cc 10118  cr 10119  0cc0 10120  1c1 10121  +∞cpnf 10255  *cxr 10257   < clt 10258  cle 10259  cn 11204  [,)cico 12362  [,]cicc 12363  cexp 13046  chash 13303  cprod 14826  volcvol 23424  Σ^csumge0 41074  voln*covoln 41248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-inf2 8703  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197  ax-pre-sup 10198
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-of 7054  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-2o 7722  df-oadd 7725  df-er 7903  df-map 8017  df-pm 8018  df-ixp 8067  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-fi 8474  df-sup 8505  df-inf 8506  df-oi 8572  df-card 8947  df-cda 9174  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-div 10869  df-nn 11205  df-2 11263  df-3 11264  df-n0 11477  df-z 11562  df-uz 11872  df-q 11974  df-rp 12018  df-xneg 12131  df-xadd 12132  df-xmul 12133  df-ioo 12364  df-ico 12366  df-icc 12367  df-fz 12512  df-fzo 12652  df-fl 12779  df-seq 12988  df-exp 13047  df-hash 13304  df-cj 14030  df-re 14031  df-im 14032  df-sqrt 14166  df-abs 14167  df-clim 14410  df-rlim 14411  df-sum 14608  df-prod 14827  df-rest 16277  df-topgen 16298  df-psmet 19932  df-xmet 19933  df-met 19934  df-bl 19935  df-mopn 19936  df-top 20893  df-topon 20910  df-bases 20944  df-cmp 21384  df-ovol 23425  df-vol 23426  df-sumge0 41075  df-ovoln 41249
This theorem is referenced by:  ovnhoi  41315
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