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Theorem ovnhoilem1 41319
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 1990 . . . . 5 𝑘𝜑
5 ovnhoilem1.a . . . . . 6 (𝜑𝐴:𝑋⟶ℝ)
65ffvelrnda 6520 . . . . 5 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
7 ovnhoilem1.b . . . . . . 7 (𝜑𝐵:𝑋⟶ℝ)
87ffvelrnda 6520 . . . . . 6 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
98rexrd 10279 . . . . 5 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ*)
104, 6, 9hoissrrn2 41296 . . . 4 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ (ℝ ↑𝑚 𝑋))
113, 10eqsstrd 3778 . . 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 41263 . 2 (𝜑 → ((voln*‘𝑋)‘𝐼) = if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )))
14 iftrue 4234 . . . . 5 (𝑋 = ∅ → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = 0)
1514adantl 473 . . . 4 ((𝜑𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = 0)
16 0xr 10276 . . . . . . 7 0 ∈ ℝ*
1716a1i 11 . . . . . 6 (𝜑 → 0 ∈ ℝ*)
18 pnfxr 10282 . . . . . . 7 +∞ ∈ ℝ*
1918a1i 11 . . . . . 6 (𝜑 → +∞ ∈ ℝ*)
204, 1, 6, 8hoiprodcl3 41298 . . . . . 6 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ (0[,)+∞))
21 icogelb 12416 . . . . . 6 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ (0[,)+∞)) → 0 ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
2217, 19, 20, 21syl3anc 1477 . . . . 5 (𝜑 → 0 ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
2322adantr 472 . . . 4 ((𝜑𝑋 = ∅) → 0 ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
2415, 23eqbrtrd 4824 . . 3 ((𝜑𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
25 iffalse 4237 . . . . 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 3826 . . . . . . 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 3774 . . . . . 6 𝑀 ⊆ ℝ*
2928a1i 11 . . . . 5 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑀 ⊆ ℝ*)
30 icossxr 12449 . . . . . . . . . 10 (0[,)+∞) ⊆ ℝ*
3130, 20sseldi 3740 . . . . . . . . 9 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ*)
3231adantr 472 . . . . . . . 8 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ*)
33 opelxpi 5303 . . . . . . . . . . . . . . . . 17 (((𝐴𝑘) ∈ ℝ ∧ (𝐵𝑘) ∈ ℝ) → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ (ℝ × ℝ))
346, 8, 33syl2anc 696 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ (ℝ × ℝ))
35 0re 10230 . . . . . . . . . . . . . . . . . 18 0 ∈ ℝ
36 opelxpi 5303 . . . . . . . . . . . . . . . . . 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 4273 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑋) → if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩) ∈ (ℝ × ℝ))
40 eqid 2758 . . . . . . . . . . . . . . 15 (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) = (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩))
4139, 40fmptd 6546 . . . . . . . . . . . . . 14 (𝜑 → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)):𝑋⟶(ℝ × ℝ))
42 reex 10217 . . . . . . . . . . . . . . . . 17 ℝ ∈ V
4342, 42xpex 7125 . . . . . . . . . . . . . . . 16 (ℝ × ℝ) ∈ V
441, 43jctil 561 . . . . . . . . . . . . . . 15 (𝜑 → ((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin))
45 elmapg 8034 . . . . . . . . . . . . . . 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 6546 . . . . . . . . . . 11 (𝜑𝐻:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
51 ovex 6839 . . . . . . . . . . . 12 ((ℝ × ℝ) ↑𝑚 𝑋) ∈ V
52 nnex 11216 . . . . . . . . . . . 12 ℕ ∈ V
5351, 52elmap 8050 . . . . . . . . . . 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 2759 . . . . . . . . . . . . 13 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
57 eqid 2758 . . . . . . . . . . . . . . . . . . 19 (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) = (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩)
5834, 57fmptd 6546 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩):𝑋⟶(ℝ × ℝ))
5949a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐻 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩))))
60 iftrue 4234 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 1 → if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩) = ⟨(𝐴𝑘), (𝐵𝑘)⟩)
6160mpteq2dv 4895 . . . . . . . . . . . . . . . . . . . . 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 11221 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℕ
6463a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 1 ∈ ℕ)
65 mptexg 6646 . . . . . . . . . . . . . . . . . . . . 21 (𝑋 ∈ Fin → (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) ∈ V)
661, 65syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) ∈ V)
6759, 62, 64, 66fvmptd 6448 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐻‘1) = (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩))
6867feq1d 6189 . . . . . . . . . . . . . . . . . 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 39878 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑋) → (([,) ∘ (𝐻‘1))‘𝑘) = ((1st ‘((𝐻‘1)‘𝑘))[,)(2nd ‘((𝐻‘1)‘𝑘))))
7334elexd 3352 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑋) → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V)
7467, 73fvmpt2d 6453 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝑋) → ((𝐻‘1)‘𝑘) = ⟨(𝐴𝑘), (𝐵𝑘)⟩)
7574fveq2d 6354 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (1st ‘((𝐻‘1)‘𝑘)) = (1st ‘⟨(𝐴𝑘), (𝐵𝑘)⟩))
76 fvex 6360 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑘) ∈ V
77 fvex 6360 . . . . . . . . . . . . . . . . . . 19 (𝐵𝑘) ∈ V
7876, 77op1st 7339 . . . . . . . . . . . . . . . . . 18 (1st ‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = (𝐴𝑘)
7978a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (1st ‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = (𝐴𝑘))
8075, 79eqtrd 2792 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → (1st ‘((𝐻‘1)‘𝑘)) = (𝐴𝑘))
8174fveq2d 6354 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (2nd ‘((𝐻‘1)‘𝑘)) = (2nd ‘⟨(𝐴𝑘), (𝐵𝑘)⟩))
8276, 77op2nd 7340 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = (𝐵𝑘)
8382a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (2nd ‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = (𝐵𝑘))
8481, 83eqtrd 2792 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → (2nd ‘((𝐻‘1)‘𝑘)) = (𝐵𝑘))
8580, 84oveq12d 6829 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑋) → ((1st ‘((𝐻‘1)‘𝑘))[,)(2nd ‘((𝐻‘1)‘𝑘))) = ((𝐴𝑘)[,)(𝐵𝑘)))
8672, 85eqtrd 2792 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑋) → (([,) ∘ (𝐻‘1))‘𝑘) = ((𝐴𝑘)[,)(𝐵𝑘)))
8786ixpeq2dva 8087 . . . . . . . . . . . . 13 (𝜑X𝑘𝑋 (([,) ∘ (𝐻‘1))‘𝑘) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
8856, 3, 873eqtr4d 2802 . . . . . . . . . . . 12 (𝜑𝐼 = X𝑘𝑋 (([,) ∘ (𝐻‘1))‘𝑘))
89 fveq2 6350 . . . . . . . . . . . . . . . . 17 (𝑗 = 1 → (𝐻𝑗) = (𝐻‘1))
9089coeq2d 5438 . . . . . . . . . . . . . . . 16 (𝑗 = 1 → ([,) ∘ (𝐻𝑗)) = ([,) ∘ (𝐻‘1)))
9190fveq1d 6352 . . . . . . . . . . . . . . 15 (𝑗 = 1 → (([,) ∘ (𝐻𝑗))‘𝑘) = (([,) ∘ (𝐻‘1))‘𝑘))
9291ixpeq2dv 8088 . . . . . . . . . . . . . 14 (𝑗 = 1 → X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝐻‘1))‘𝑘))
9392ssiun2s 4714 . . . . . . . . . . . . 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 3794 . . . . . . . . . . 11 (𝜑𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘))
9695adantr 472 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘))
9786fveq2d 6354 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑋) → (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
9897eqcomd 2764 . . . . . . . . . . . . 13 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
9998prodeq2dv 14850 . . . . . . . . . . . 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 10245 . . . . . . . . . . . . . 14 (𝜑 → 1 ∈ ℝ)
102 icossicc 12451 . . . . . . . . . . . . . . 15 (0[,)+∞) ⊆ (0[,]+∞)
1034, 1, 69hoiprodcl 41265 . . . . . . . . . . . . . . 15 (𝜑 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) ∈ (0[,)+∞))
104102, 103sseldi 3740 . . . . . . . . . . . . . 14 (𝜑 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) ∈ (0[,]+∞))
10591fveq2d 6354 . . . . . . . . . . . . . . 15 (𝑗 = 1 → (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
106105prodeq2ad 40325 . . . . . . . . . . . . . 14 (𝑗 = 1 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
107101, 104, 106sge0snmpt 41101 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
108107eqcomd 2764 . . . . . . . . . . . 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 1990 . . . . . . . . . . . 12 𝑗(𝜑 ∧ ¬ 𝑋 = ∅)
11152a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝑋 = ∅) → ℕ ∈ V)
112 snssi 4482 . . . . . . . . . . . . . 14 (1 ∈ ℕ → {1} ⊆ ℕ)
11363, 112ax-mp 5 . . . . . . . . . . . . 13 {1} ⊆ ℕ
114113a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝑋 = ∅) → {1} ⊆ ℕ)
115 nfv 1990 . . . . . . . . . . . . . 14 𝑘((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1})
1161ad2antrr 764 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → 𝑋 ∈ Fin)
117 simpl 474 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ {1}) → 𝜑)
118 elsni 4336 . . . . . . . . . . . . . . . . 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 6189 . . . . . . . . . . . . . . . . 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 41265 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) ∈ (0[,)+∞))
127102, 126sseldi 3740 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) ∈ (0[,]+∞))
128 eqid 2758 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘𝑋 ↦ ⟨0, 0⟩) = (𝑘𝑋 ↦ ⟨0, 0⟩)
12938, 128fmptd 6546 . . . . . . . . . . . . . . . . . . . . . 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 3873 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (ℕ ∖ {1}) → 𝑗 ∈ ℕ)
133132adantl 473 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → 𝑗 ∈ ℕ)
13448elexd 3352 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) ∈ V)
13559, 134fvmpt2d 6453 . . . . . . . . . . . . . . . . . . . . . . . 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 4464 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (ℕ ∖ {1}) → 𝑗 ≠ 1)
138137neneqd 2935 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (ℕ ∖ {1}) → ¬ 𝑗 = 1)
139138iffalsed 4239 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (ℕ ∖ {1}) → if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩) = ⟨0, 0⟩)
140139mpteq2dv 4895 . . . . . . . . . . . . . . . . . . . . . . . 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 2792 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → (𝐻𝑗) = (𝑘𝑋 ↦ ⟨0, 0⟩))
143142feq1d 6189 . . . . . . . . . . . . . . . . . . . . 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 39878 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (([,) ∘ (𝐻𝑗))‘𝑘) = ((1st ‘((𝐻𝑗)‘𝑘))[,)(2nd ‘((𝐻𝑗)‘𝑘))))
14837elexi 3351 . . . . . . . . . . . . . . . . . . . . . . 23 ⟨0, 0⟩ ∈ V
149148a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → ⟨0, 0⟩ ∈ V)
150142, 149fvmpt2d 6453 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → ((𝐻𝑗)‘𝑘) = ⟨0, 0⟩)
151150fveq2d 6354 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (1st ‘((𝐻𝑗)‘𝑘)) = (1st ‘⟨0, 0⟩))
15216elexi 3351 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ V
153152, 152op1st 7339 . . . . . . . . . . . . . . . . . . . . 21 (1st ‘⟨0, 0⟩) = 0
154153a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (1st ‘⟨0, 0⟩) = 0)
155151, 154eqtrd 2792 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (1st ‘((𝐻𝑗)‘𝑘)) = 0)
156150fveq2d 6354 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (2nd ‘((𝐻𝑗)‘𝑘)) = (2nd ‘⟨0, 0⟩))
157152, 152op2nd 7340 . . . . . . . . . . . . . . . . . . . . 21 (2nd ‘⟨0, 0⟩) = 0
158157a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (2nd ‘⟨0, 0⟩) = 0)
159156, 158eqtrd 2792 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (2nd ‘((𝐻𝑗)‘𝑘)) = 0)
160155, 159oveq12d 6829 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → ((1st ‘((𝐻𝑗)‘𝑘))[,)(2nd ‘((𝐻𝑗)‘𝑘))) = (0[,)0))
161 0le0 11300 . . . . . . . . . . . . . . . . . . . 20 0 ≤ 0
162 ico0 12412 . . . . . . . . . . . . . . . . . . . . 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 2796 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (([,) ∘ (𝐻𝑗))‘𝑘) = ∅)
167166fveq2d 6354 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = (vol‘∅))
168 vol0 40676 . . . . . . . . . . . . . . . . 17 (vol‘∅) = 0
169168a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (vol‘∅) = 0)
170167, 169eqtrd 2792 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = 0)
171170prodeq2dv 14850 . . . . . . . . . . . . . 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 10223 . . . . . . . . . . . . . . 15 (𝜑 → 0 ∈ ℂ)
174 fprodconst 14905 . . . . . . . . . . . . . . 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 2938 . . . . . . . . . . . . . . . . 17 𝑋 = ∅ → 𝑋 ≠ ∅)
178177adantl 473 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
1791adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin)
180 hashnncl 13347 . . . . . . . . . . . . . . . . 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 13087 . . . . . . . . . . . . . . 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 2796 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = 0)
187110, 111, 114, 127, 186sge0ss 41130 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝑋 = ∅) → (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))))
188100, 109, 1873eqtrd 2796 . . . . . . . . . 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 2900 . . . . . . . . . . . . . . 15 𝑘𝑖
191 nfcv 2900 . . . . . . . . . . . . . . . . 17 𝑘
192 nfmpt1 4897 . . . . . . . . . . . . . . . . 17 𝑘(𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩))
193191, 192nfmpt 4896 . . . . . . . . . . . . . . . 16 𝑘(𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))
19449, 193nfcxfr 2898 . . . . . . . . . . . . . . 15 𝑘𝐻
195190, 194nfeq 2912 . . . . . . . . . . . . . 14 𝑘 𝑖 = 𝐻
196 fveq1 6349 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝐻 → (𝑖𝑗) = (𝐻𝑗))
197196coeq2d 5438 . . . . . . . . . . . . . . . 16 (𝑖 = 𝐻 → ([,) ∘ (𝑖𝑗)) = ([,) ∘ (𝐻𝑗)))
198197fveq1d 6352 . . . . . . . . . . . . . . 15 (𝑖 = 𝐻 → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐻𝑗))‘𝑘))
199198adantr 472 . . . . . . . . . . . . . 14 ((𝑖 = 𝐻𝑘𝑋) → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐻𝑗))‘𝑘))
200195, 199ixpeq2d 39734 . . . . . . . . . . . . 13 (𝑖 = 𝐻X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘))
201200iuneq2d 4697 . . . . . . . . . . . 12 (𝑖 = 𝐻 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘))
202201sseq2d 3772 . . . . . . . . . . 11 (𝑖 = 𝐻 → (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ↔ 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘)))
203198fveq2d 6354 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝐻 → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))
204203a1d 25 . . . . . . . . . . . . . . . 16 (𝑖 = 𝐻 → (𝑘𝑋 → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻𝑗))‘𝑘))))
205195, 204ralrimi 3093 . . . . . . . . . . . . . . 15 (𝑖 = 𝐻 → ∀𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))
206205prodeq2d 14849 . . . . . . . . . . . . . 14 (𝑖 = 𝐻 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))
207206mpteq2dv 4895 . . . . . . . . . . . . 13 (𝑖 = 𝐻 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘))))
208207fveq2d 6354 . . . . . . . . . . . 12 (𝑖 = 𝐻 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))))
209208eqeq2d 2768 . . . . . . . . . . 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 3447 . . . . . . . . 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 2762 . . . . . . . . . 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 3188 . . . . . . . 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 3502 . . . . . . 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 2767 . . . . . . 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 2839 . . . . 5 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ 𝑀)
222 infxrlb 12355 . . . . 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 4824 . . 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 4824 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 1630  wcel 2137  wne 2930  wrex 3049  {crab 3052  Vcvv 3338  cdif 3710  wss 3713  c0 4056  ifcif 4228  {csn 4319  cop 4325   ciun 4670   class class class wbr 4802  cmpt 4879   × cxp 5262  ccom 5268  wf 6043  cfv 6047  (class class class)co 6811  1st c1st 7329  2nd c2nd 7330  𝑚 cmap 8021  Xcixp 8072  Fincfn 8119  infcinf 8510  cc 10124  cr 10125  0cc0 10126  1c1 10127  +∞cpnf 10261  *cxr 10263   < clt 10264  cle 10265  cn 11210  [,)cico 12368  [,]cicc 12369  cexp 13052  chash 13309  cprod 14832  volcvol 23430  Σ^csumge0 41080  voln*covoln 41254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-inf2 8709  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202  ax-pre-mulgt0 10203  ax-pre-sup 10204
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-se 5224  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-isom 6056  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-of 7060  df-om 7229  df-1st 7331  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-2o 7728  df-oadd 7731  df-er 7909  df-map 8023  df-pm 8024  df-ixp 8073  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123  df-fi 8480  df-sup 8511  df-inf 8512  df-oi 8578  df-card 8953  df-cda 9180  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-sub 10458  df-neg 10459  df-div 10875  df-nn 11211  df-2 11269  df-3 11270  df-n0 11483  df-z 11568  df-uz 11878  df-q 11980  df-rp 12024  df-xneg 12137  df-xadd 12138  df-xmul 12139  df-ioo 12370  df-ico 12372  df-icc 12373  df-fz 12518  df-fzo 12658  df-fl 12785  df-seq 12994  df-exp 13053  df-hash 13310  df-cj 14036  df-re 14037  df-im 14038  df-sqrt 14172  df-abs 14173  df-clim 14416  df-rlim 14417  df-sum 14614  df-prod 14833  df-rest 16283  df-topgen 16304  df-psmet 19938  df-xmet 19939  df-met 19940  df-bl 19941  df-mopn 19942  df-top 20899  df-topon 20916  df-bases 20950  df-cmp 21390  df-ovol 23431  df-vol 23432  df-sumge0 41081  df-ovoln 41255
This theorem is referenced by:  ovnhoi  41321
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