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Theorem ovnhoilem1 43237
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 1915 . . . . 5 𝑘𝜑
5 ovnhoilem1.a . . . . . 6 (𝜑𝐴:𝑋⟶ℝ)
65ffvelrnda 6832 . . . . 5 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
7 ovnhoilem1.b . . . . . . 7 (𝜑𝐵:𝑋⟶ℝ)
87ffvelrnda 6832 . . . . . 6 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
98rexrd 10684 . . . . 5 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ*)
104, 6, 9hoissrrn2 43214 . . . 4 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ (ℝ ↑m 𝑋))
113, 10eqsstrd 3956 . . 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 43181 . 2 (𝜑 → ((voln*‘𝑋)‘𝐼) = if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )))
14 iftrue 4434 . . . . 5 (𝑋 = ∅ → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = 0)
1514adantl 485 . . . 4 ((𝜑𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = 0)
16 0xr 10681 . . . . . . 7 0 ∈ ℝ*
1716a1i 11 . . . . . 6 (𝜑 → 0 ∈ ℝ*)
18 pnfxr 10688 . . . . . . 7 +∞ ∈ ℝ*
1918a1i 11 . . . . . 6 (𝜑 → +∞ ∈ ℝ*)
204, 1, 6, 8hoiprodcl3 43216 . . . . . 6 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ (0[,)+∞))
21 icogelb 12780 . . . . . 6 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ (0[,)+∞)) → 0 ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
2217, 19, 20, 21syl3anc 1368 . . . . 5 (𝜑 → 0 ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
2322adantr 484 . . . 4 ((𝜑𝑋 = ∅) → 0 ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
2415, 23eqbrtrd 5055 . . 3 ((𝜑𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
25 iffalse 4437 . . . . 5 𝑋 = ∅ → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = inf(𝑀, ℝ*, < ))
2625adantl 485 . . . 4 ((𝜑 ∧ ¬ 𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = inf(𝑀, ℝ*, < ))
27 ssrab2 4010 . . . . . . 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 3952 . . . . . 6 𝑀 ⊆ ℝ*
2928a1i 11 . . . . 5 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑀 ⊆ ℝ*)
30 icossxr 12814 . . . . . . . . . 10 (0[,)+∞) ⊆ ℝ*
3130, 20sseldi 3916 . . . . . . . . 9 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ*)
3231adantr 484 . . . . . . . 8 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ*)
33 opelxpi 5560 . . . . . . . . . . . . . . . . 17 (((𝐴𝑘) ∈ ℝ ∧ (𝐵𝑘) ∈ ℝ) → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ (ℝ × ℝ))
346, 8, 33syl2anc 587 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ (ℝ × ℝ))
35 0re 10636 . . . . . . . . . . . . . . . . . 18 0 ∈ ℝ
36 opelxpi 5560 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℝ ∧ 0 ∈ ℝ) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
3735, 35, 36mp2an 691 . . . . . . . . . . . . . . . . 17 ⟨0, 0⟩ ∈ (ℝ × ℝ)
3837a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
3934, 38ifcld 4473 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑋) → if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩) ∈ (ℝ × ℝ))
4039fmpttd 6860 . . . . . . . . . . . . . 14 (𝜑 → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)):𝑋⟶(ℝ × ℝ))
41 reex 10621 . . . . . . . . . . . . . . . . 17 ℝ ∈ V
4241, 41xpex 7460 . . . . . . . . . . . . . . . 16 (ℝ × ℝ) ∈ V
431, 42jctil 523 . . . . . . . . . . . . . . 15 (𝜑 → ((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin))
44 elmapg 8406 . . . . . . . . . . . . . . 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)))
4543, 44syl 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)))
4640, 45mpbird 260 . . . . . . . . . . . . 13 (𝜑 → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) ∈ ((ℝ × ℝ) ↑m 𝑋))
4746adantr 484 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) ∈ ((ℝ × ℝ) ↑m 𝑋))
48 ovnhoilem1.h . . . . . . . . . . . 12 𝐻 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))
4947, 48fmptd 6859 . . . . . . . . . . 11 (𝜑𝐻:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
50 ovex 7172 . . . . . . . . . . . 12 ((ℝ × ℝ) ↑m 𝑋) ∈ V
51 nnex 11635 . . . . . . . . . . . 12 ℕ ∈ V
5250, 51elmap 8422 . . . . . . . . . . 11 (𝐻 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↔ 𝐻:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
5349, 52sylibr 237 . . . . . . . . . 10 (𝜑𝐻 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
5453adantr 484 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐻 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
55 eqidd 2802 . . . . . . . . . . . . 13 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
5634fmpttd 6860 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩):𝑋⟶(ℝ × ℝ))
57 iftrue 4434 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 1 → if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩) = ⟨(𝐴𝑘), (𝐵𝑘)⟩)
5857mpteq2dv 5129 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 1 → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) = (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩))
59 1nn 11640 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℕ
6059a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 1 ∈ ℕ)
61 mptexg 6965 . . . . . . . . . . . . . . . . . . . . 21 (𝑋 ∈ Fin → (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) ∈ V)
621, 61syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩) ∈ V)
6348, 58, 60, 62fvmptd3 6772 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐻‘1) = (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩))
6463feq1d 6476 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝐻‘1):𝑋⟶(ℝ × ℝ) ↔ (𝑘𝑋 ↦ ⟨(𝐴𝑘), (𝐵𝑘)⟩):𝑋⟶(ℝ × ℝ)))
6556, 64mpbird 260 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐻‘1):𝑋⟶(ℝ × ℝ))
6665adantr 484 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → (𝐻‘1):𝑋⟶(ℝ × ℝ))
67 simpr 488 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → 𝑘𝑋)
6866, 67fvovco 41818 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑋) → (([,) ∘ (𝐻‘1))‘𝑘) = ((1st ‘((𝐻‘1)‘𝑘))[,)(2nd ‘((𝐻‘1)‘𝑘))))
6934elexd 3464 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑋) → ⟨(𝐴𝑘), (𝐵𝑘)⟩ ∈ V)
7063, 69fvmpt2d 6762 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝑋) → ((𝐻‘1)‘𝑘) = ⟨(𝐴𝑘), (𝐵𝑘)⟩)
7170fveq2d 6653 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (1st ‘((𝐻‘1)‘𝑘)) = (1st ‘⟨(𝐴𝑘), (𝐵𝑘)⟩))
72 fvex 6662 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑘) ∈ V
73 fvex 6662 . . . . . . . . . . . . . . . . . . 19 (𝐵𝑘) ∈ V
7472, 73op1st 7683 . . . . . . . . . . . . . . . . . 18 (1st ‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = (𝐴𝑘)
7574a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (1st ‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = (𝐴𝑘))
7671, 75eqtrd 2836 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → (1st ‘((𝐻‘1)‘𝑘)) = (𝐴𝑘))
7770fveq2d 6653 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (2nd ‘((𝐻‘1)‘𝑘)) = (2nd ‘⟨(𝐴𝑘), (𝐵𝑘)⟩))
7872, 73op2nd 7684 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = (𝐵𝑘)
7978a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → (2nd ‘⟨(𝐴𝑘), (𝐵𝑘)⟩) = (𝐵𝑘))
8077, 79eqtrd 2836 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → (2nd ‘((𝐻‘1)‘𝑘)) = (𝐵𝑘))
8176, 80oveq12d 7157 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑋) → ((1st ‘((𝐻‘1)‘𝑘))[,)(2nd ‘((𝐻‘1)‘𝑘))) = ((𝐴𝑘)[,)(𝐵𝑘)))
8268, 81eqtrd 2836 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑋) → (([,) ∘ (𝐻‘1))‘𝑘) = ((𝐴𝑘)[,)(𝐵𝑘)))
8382ixpeq2dva 8463 . . . . . . . . . . . . 13 (𝜑X𝑘𝑋 (([,) ∘ (𝐻‘1))‘𝑘) = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
8455, 3, 833eqtr4d 2846 . . . . . . . . . . . 12 (𝜑𝐼 = X𝑘𝑋 (([,) ∘ (𝐻‘1))‘𝑘))
85 fveq2 6649 . . . . . . . . . . . . . . . . 17 (𝑗 = 1 → (𝐻𝑗) = (𝐻‘1))
8685coeq2d 5701 . . . . . . . . . . . . . . . 16 (𝑗 = 1 → ([,) ∘ (𝐻𝑗)) = ([,) ∘ (𝐻‘1)))
8786fveq1d 6651 . . . . . . . . . . . . . . 15 (𝑗 = 1 → (([,) ∘ (𝐻𝑗))‘𝑘) = (([,) ∘ (𝐻‘1))‘𝑘))
8887ixpeq2dv 8464 . . . . . . . . . . . . . 14 (𝑗 = 1 → X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝐻‘1))‘𝑘))
8988ssiun2s 4938 . . . . . . . . . . . . 13 (1 ∈ ℕ → X𝑘𝑋 (([,) ∘ (𝐻‘1))‘𝑘) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘))
9059, 89ax-mp 5 . . . . . . . . . . . 12 X𝑘𝑋 (([,) ∘ (𝐻‘1))‘𝑘) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘)
9184, 90eqsstrdi 3972 . . . . . . . . . . 11 (𝜑𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘))
9291adantr 484 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘))
9382fveq2d 6653 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑋) → (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
9493eqcomd 2807 . . . . . . . . . . . . 13 ((𝜑𝑘𝑋) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
9594prodeq2dv 15273 . . . . . . . . . . . 12 (𝜑 → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
9695adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
97 1red 10635 . . . . . . . . . . . . . 14 (𝜑 → 1 ∈ ℝ)
98 icossicc 12818 . . . . . . . . . . . . . . 15 (0[,)+∞) ⊆ (0[,]+∞)
994, 1, 65hoiprodcl 43183 . . . . . . . . . . . . . . 15 (𝜑 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) ∈ (0[,)+∞))
10098, 99sseldi 3916 . . . . . . . . . . . . . 14 (𝜑 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) ∈ (0[,]+∞))
10187fveq2d 6653 . . . . . . . . . . . . . . 15 (𝑗 = 1 → (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
102101prodeq2ad 42231 . . . . . . . . . . . . . 14 (𝑗 = 1 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
10397, 100, 102sge0snmpt 43019 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
104103eqcomd 2807 . . . . . . . . . . . 12 (𝜑 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) = (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))))
105104adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) = (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))))
106 nfv 1915 . . . . . . . . . . . 12 𝑗(𝜑 ∧ ¬ 𝑋 = ∅)
10751a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝑋 = ∅) → ℕ ∈ V)
108 snssi 4704 . . . . . . . . . . . . . 14 (1 ∈ ℕ → {1} ⊆ ℕ)
10959, 108ax-mp 5 . . . . . . . . . . . . 13 {1} ⊆ ℕ
110109a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝑋 = ∅) → {1} ⊆ ℕ)
111 nfv 1915 . . . . . . . . . . . . . 14 𝑘((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1})
1121ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → 𝑋 ∈ Fin)
113 simpl 486 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ {1}) → 𝜑)
114 elsni 4545 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ {1} → 𝑗 = 1)
115114adantl 485 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ {1}) → 𝑗 = 1)
11665adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 = 1) → (𝐻‘1):𝑋⟶(ℝ × ℝ))
11785adantl 485 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 = 1) → (𝐻𝑗) = (𝐻‘1))
118117feq1d 6476 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 = 1) → ((𝐻𝑗):𝑋⟶(ℝ × ℝ) ↔ (𝐻‘1):𝑋⟶(ℝ × ℝ)))
119116, 118mpbird 260 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 = 1) → (𝐻𝑗):𝑋⟶(ℝ × ℝ))
120113, 115, 119syl2anc 587 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ {1}) → (𝐻𝑗):𝑋⟶(ℝ × ℝ))
121120adantlr 714 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → (𝐻𝑗):𝑋⟶(ℝ × ℝ))
122111, 112, 121hoiprodcl 43183 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) ∈ (0[,)+∞))
12398, 122sseldi 3916 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) ∈ (0[,]+∞))
12438fmpttd 6860 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑘𝑋 ↦ ⟨0, 0⟩):𝑋⟶(ℝ × ℝ))
125124adantr 484 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → (𝑘𝑋 ↦ ⟨0, 0⟩):𝑋⟶(ℝ × ℝ))
126 simpl 486 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → 𝜑)
127 eldifi 4057 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (ℕ ∖ {1}) → 𝑗 ∈ ℕ)
128127adantl 485 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → 𝑗 ∈ ℕ)
12948a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐻 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩))))
13047elexd 3464 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) ∈ V)
131129, 130fvmpt2d 6762 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ ℕ) → (𝐻𝑗) = (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))
132126, 128, 131syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → (𝐻𝑗) = (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))
133 eldifsni 4686 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (ℕ ∖ {1}) → 𝑗 ≠ 1)
134133neneqd 2995 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (ℕ ∖ {1}) → ¬ 𝑗 = 1)
135134iffalsed 4439 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (ℕ ∖ {1}) → if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩) = ⟨0, 0⟩)
136135mpteq2dv 5129 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (ℕ ∖ {1}) → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) = (𝑘𝑋 ↦ ⟨0, 0⟩))
137136adantl 485 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) = (𝑘𝑋 ↦ ⟨0, 0⟩))
138132, 137eqtrd 2836 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → (𝐻𝑗) = (𝑘𝑋 ↦ ⟨0, 0⟩))
139138feq1d 6476 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → ((𝐻𝑗):𝑋⟶(ℝ × ℝ) ↔ (𝑘𝑋 ↦ ⟨0, 0⟩):𝑋⟶(ℝ × ℝ)))
140125, 139mpbird 260 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → (𝐻𝑗):𝑋⟶(ℝ × ℝ))
141140adantr 484 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (𝐻𝑗):𝑋⟶(ℝ × ℝ))
142 simpr 488 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → 𝑘𝑋)
143141, 142fvovco 41818 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (([,) ∘ (𝐻𝑗))‘𝑘) = ((1st ‘((𝐻𝑗)‘𝑘))[,)(2nd ‘((𝐻𝑗)‘𝑘))))
14437elexi 3463 . . . . . . . . . . . . . . . . . . . . . . 23 ⟨0, 0⟩ ∈ V
145144a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → ⟨0, 0⟩ ∈ V)
146138, 145fvmpt2d 6762 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → ((𝐻𝑗)‘𝑘) = ⟨0, 0⟩)
147146fveq2d 6653 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (1st ‘((𝐻𝑗)‘𝑘)) = (1st ‘⟨0, 0⟩))
14816elexi 3463 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ V
149148, 148op1st 7683 . . . . . . . . . . . . . . . . . . . . 21 (1st ‘⟨0, 0⟩) = 0
150149a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (1st ‘⟨0, 0⟩) = 0)
151147, 150eqtrd 2836 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (1st ‘((𝐻𝑗)‘𝑘)) = 0)
152146fveq2d 6653 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (2nd ‘((𝐻𝑗)‘𝑘)) = (2nd ‘⟨0, 0⟩))
153148, 148op2nd 7684 . . . . . . . . . . . . . . . . . . . . 21 (2nd ‘⟨0, 0⟩) = 0
154153a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (2nd ‘⟨0, 0⟩) = 0)
155152, 154eqtrd 2836 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (2nd ‘((𝐻𝑗)‘𝑘)) = 0)
156151, 155oveq12d 7157 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → ((1st ‘((𝐻𝑗)‘𝑘))[,)(2nd ‘((𝐻𝑗)‘𝑘))) = (0[,)0))
157 0le0 11730 . . . . . . . . . . . . . . . . . . . 20 0 ≤ 0
158 ico0 12776 . . . . . . . . . . . . . . . . . . . . 21 ((0 ∈ ℝ* ∧ 0 ∈ ℝ*) → ((0[,)0) = ∅ ↔ 0 ≤ 0))
15916, 16, 158mp2an 691 . . . . . . . . . . . . . . . . . . . 20 ((0[,)0) = ∅ ↔ 0 ≤ 0)
160157, 159mpbir 234 . . . . . . . . . . . . . . . . . . 19 (0[,)0) = ∅
161160a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (0[,)0) = ∅)
162143, 156, 1613eqtrd 2840 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (([,) ∘ (𝐻𝑗))‘𝑘) = ∅)
163162fveq2d 6653 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = (vol‘∅))
164 vol0 42598 . . . . . . . . . . . . . . . . 17 (vol‘∅) = 0
165164a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (vol‘∅) = 0)
166163, 165eqtrd 2836 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘𝑋) → (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = 0)
167166prodeq2dv 15273 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = ∏𝑘𝑋 0)
168167adantlr 714 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = ∏𝑘𝑋 0)
169 0cnd 10627 . . . . . . . . . . . . . . 15 (𝜑 → 0 ∈ ℂ)
170 fprodconst 15328 . . . . . . . . . . . . . . 15 ((𝑋 ∈ Fin ∧ 0 ∈ ℂ) → ∏𝑘𝑋 0 = (0↑(♯‘𝑋)))
1711, 169, 170syl2anc 587 . . . . . . . . . . . . . 14 (𝜑 → ∏𝑘𝑋 0 = (0↑(♯‘𝑋)))
172171ad2antrr 725 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘𝑋 0 = (0↑(♯‘𝑋)))
173 neqne 2998 . . . . . . . . . . . . . . . . 17 𝑋 = ∅ → 𝑋 ≠ ∅)
174173adantl 485 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
1751adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin)
176 hashnncl 13727 . . . . . . . . . . . . . . . . 17 (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
177175, 176syl 17 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ¬ 𝑋 = ∅) → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
178174, 177mpbird 260 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ¬ 𝑋 = ∅) → (♯‘𝑋) ∈ ℕ)
179 0exp 13464 . . . . . . . . . . . . . . 15 ((♯‘𝑋) ∈ ℕ → (0↑(♯‘𝑋)) = 0)
180178, 179syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ 𝑋 = ∅) → (0↑(♯‘𝑋)) = 0)
181180adantr 484 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → (0↑(♯‘𝑋)) = 0)
182168, 172, 1813eqtrd 2840 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)) = 0)
183106, 107, 110, 123, 182sge0ss 43048 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝑋 = ∅) → (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))))
18496, 105, 1833eqtrd 2840 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))))
18592, 184jca 515 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘) ∧ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘))))))
186 nfcv 2958 . . . . . . . . . . . . . . 15 𝑘𝑖
187 nfcv 2958 . . . . . . . . . . . . . . . . 17 𝑘
188 nfmpt1 5131 . . . . . . . . . . . . . . . . 17 𝑘(𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩))
189187, 188nfmpt 5130 . . . . . . . . . . . . . . . 16 𝑘(𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))
19048, 189nfcxfr 2956 . . . . . . . . . . . . . . 15 𝑘𝐻
191186, 190nfeq 2971 . . . . . . . . . . . . . 14 𝑘 𝑖 = 𝐻
192 fveq1 6648 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝐻 → (𝑖𝑗) = (𝐻𝑗))
193192coeq2d 5701 . . . . . . . . . . . . . . . 16 (𝑖 = 𝐻 → ([,) ∘ (𝑖𝑗)) = ([,) ∘ (𝐻𝑗)))
194193fveq1d 6651 . . . . . . . . . . . . . . 15 (𝑖 = 𝐻 → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐻𝑗))‘𝑘))
195194adantr 484 . . . . . . . . . . . . . 14 ((𝑖 = 𝐻𝑘𝑋) → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐻𝑗))‘𝑘))
196191, 195ixpeq2d 41699 . . . . . . . . . . . . 13 (𝑖 = 𝐻X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘))
197196iuneq2d 4913 . . . . . . . . . . . 12 (𝑖 = 𝐻 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘))
198197sseq2d 3950 . . . . . . . . . . 11 (𝑖 = 𝐻 → (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ↔ 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐻𝑗))‘𝑘)))
199194fveq2d 6653 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝐻 → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))
200199a1d 25 . . . . . . . . . . . . . . . 16 (𝑖 = 𝐻 → (𝑘𝑋 → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻𝑗))‘𝑘))))
201191, 200ralrimi 3183 . . . . . . . . . . . . . . 15 (𝑖 = 𝐻 → ∀𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))
202201prodeq2d 15272 . . . . . . . . . . . . . 14 (𝑖 = 𝐻 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))
203202mpteq2dv 5129 . . . . . . . . . . . . 13 (𝑖 = 𝐻 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘))))
204203fveq2d 6653 . . . . . . . . . . . 12 (𝑖 = 𝐻 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐻𝑗))‘𝑘)))))
205204eqeq2d 2812 . . . . . . . . . . 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))))))
206198, 205anbi12d 633 . . . . . . . . . 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)))))))
207206rspcev 3574 . . . . . . . . 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))))))
20854, 185, 207syl2anc 587 . . . . . . . 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))))))
20932, 208jca 515 . . . . . . 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)))))))
210 eqeq1 2805 . . . . . . . . . 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))))))
211210anbi2d 631 . . . . . . . . 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)))))))
212211rexbidv 3259 . . . . . . . 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)))))))
213212elrab 3631 . . . . . . 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)))))))
214209, 213sylibr 237 . . . . . 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)))))})
21512eqcomi 2810 . . . . . . 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
216215a1i 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)
217214, 216eleqtrd 2895 . . . . 5 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ 𝑀)
218 infxrlb 12719 . . . . 5 ((𝑀 ⊆ ℝ* ∧ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) ∈ 𝑀) → inf(𝑀, ℝ*, < ) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
21929, 217, 218syl2anc 587 . . . 4 ((𝜑 ∧ ¬ 𝑋 = ∅) → inf(𝑀, ℝ*, < ) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
22026, 219eqbrtrd 5055 . . 3 ((𝜑 ∧ ¬ 𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
22124, 220pm2.61dan 812 . 2 (𝜑 → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
22213, 221eqbrtrd 5055 1 (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wne 2990  wrex 3110  {crab 3113  Vcvv 3444  cdif 3881  wss 3884  c0 4246  ifcif 4428  {csn 4528  cop 4534   ciun 4884   class class class wbr 5033  cmpt 5113   × cxp 5521  ccom 5527  wf 6324  cfv 6328  (class class class)co 7139  1st c1st 7673  2nd c2nd 7674  m cmap 8393  Xcixp 8448  Fincfn 8496  infcinf 8893  cc 10528  cr 10529  0cc0 10530  1c1 10531  +∞cpnf 10665  *cxr 10667   < clt 10668  cle 10669  cn 11629  [,)cico 12732  [,]cicc 12733  cexp 13429  chash 13690  cprod 15255  volcvol 24071  Σ^csumge0 42998  voln*covoln 43172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ioo 12734  df-ico 12736  df-icc 12737  df-fz 12890  df-fzo 13033  df-fl 13161  df-seq 13369  df-exp 13430  df-hash 13691  df-cj 14454  df-re 14455  df-im 14456  df-sqrt 14590  df-abs 14591  df-clim 14841  df-rlim 14842  df-sum 15039  df-prod 15256  df-rest 16692  df-topgen 16713  df-psmet 20087  df-xmet 20088  df-met 20089  df-bl 20090  df-mopn 20091  df-top 21503  df-topon 21520  df-bases 21555  df-cmp 21996  df-ovol 24072  df-vol 24073  df-sumge0 42999  df-ovoln 43173
This theorem is referenced by:  ovnhoi  43239
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