Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovolval5lem3 Structured version   Visualization version   GIF version

Theorem ovolval5lem3 41532
Description: The value of the Lebesgue outer measure for subsets of the reals, using covers of left-closed right-open intervals are used, instead of open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
ovolval5lem3.m 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}
ovolval5lem3.q 𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}
Assertion
Ref Expression
ovolval5lem3 inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < )
Distinct variable groups:   𝐴,𝑓,𝑧,𝑦   𝑦,𝑀,𝑧   𝑄,𝑓,𝑦,𝑧
Allowed substitution hint:   𝑀(𝑓)

Proof of Theorem ovolval5lem3
Dummy variables 𝑔 𝑛 𝑤 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovolval5lem3.q . . . . 5 𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}
2 ssrab2 3849 . . . . 5 {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))} ⊆ ℝ*
31, 2eqsstri 3797 . . . 4 𝑄 ⊆ ℝ*
4 infxrcl 12370 . . . 4 (𝑄 ⊆ ℝ* → inf(𝑄, ℝ*, < ) ∈ ℝ*)
53, 4mp1i 13 . . 3 (⊤ → inf(𝑄, ℝ*, < ) ∈ ℝ*)
6 ovolval5lem3.m . . . . 5 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}
7 ssrab2 3849 . . . . 5 {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))} ⊆ ℝ*
86, 7eqsstri 3797 . . . 4 𝑀 ⊆ ℝ*
9 infxrcl 12370 . . . 4 (𝑀 ⊆ ℝ* → inf(𝑀, ℝ*, < ) ∈ ℝ*)
108, 9mp1i 13 . . 3 (⊤ → inf(𝑀, ℝ*, < ) ∈ ℝ*)
113a1i 11 . . . 4 (⊤ → 𝑄 ⊆ ℝ*)
128a1i 11 . . . 4 (⊤ → 𝑀 ⊆ ℝ*)
13 simpr 477 . . . . . 6 ((𝑦𝑀𝑤 ∈ ℝ+) → 𝑤 ∈ ℝ+)
146rabeq2i 3346 . . . . . . . . 9 (𝑦𝑀 ↔ (𝑦 ∈ ℝ* ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
1514biimpi 207 . . . . . . . 8 (𝑦𝑀 → (𝑦 ∈ ℝ* ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
1615simprd 489 . . . . . . 7 (𝑦𝑀 → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
1716adantr 472 . . . . . 6 ((𝑦𝑀𝑤 ∈ ℝ+) → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
18 coeq2 5451 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝑓 → ((,) ∘ 𝑔) = ((,) ∘ 𝑓))
1918rneqd 5523 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝑓))
2019unieqd 4606 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝑓))
2120sseq2d 3795 . . . . . . . . . . . . . 14 (𝑔 = 𝑓 → (𝐴 ran ((,) ∘ 𝑔) ↔ 𝐴 ran ((,) ∘ 𝑓)))
22 coeq2 5451 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → ((vol ∘ (,)) ∘ 𝑔) = ((vol ∘ (,)) ∘ 𝑓))
2322fveq2d 6383 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → (Σ^‘((vol ∘ (,)) ∘ 𝑔)) = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))
2423eqeq2d 2775 . . . . . . . . . . . . . 14 (𝑔 = 𝑓 → (𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)) ↔ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
2521, 24anbi12d 624 . . . . . . . . . . . . 13 (𝑔 = 𝑓 → ((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))) ↔ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
2625cbvrexv 3320 . . . . . . . . . . . 12 (∃𝑔 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))) ↔ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
2726rabbii 3334 . . . . . . . . . . 11 {𝑧 ∈ ℝ* ∣ ∃𝑔 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))} = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}
281, 27eqtr4i 2790 . . . . . . . . . 10 𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑔 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))}
29 simp3r 1259 . . . . . . . . . 10 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
30 eqid 2765 . . . . . . . . . 10 ^‘((vol ∘ (,)) ∘ (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓𝑚))⟩))) = (Σ^‘((vol ∘ (,)) ∘ (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓𝑚))⟩)))
31 elmapi 8086 . . . . . . . . . . 11 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
32313ad2ant2 1164 . . . . . . . . . 10 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑓:ℕ⟶(ℝ × ℝ))
33 simp3l 1258 . . . . . . . . . 10 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐴 ran ([,) ∘ 𝑓))
34 simp1 1166 . . . . . . . . . 10 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑤 ∈ ℝ+)
35 2fveq3 6384 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (1st ‘(𝑓𝑚)) = (1st ‘(𝑓𝑛)))
36 oveq2 6854 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
3736oveq2d 6862 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (𝑤 / (2↑𝑚)) = (𝑤 / (2↑𝑛)))
3835, 37oveq12d 6864 . . . . . . . . . . . 12 (𝑚 = 𝑛 → ((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))) = ((1st ‘(𝑓𝑛)) − (𝑤 / (2↑𝑛))))
39 2fveq3 6384 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (2nd ‘(𝑓𝑚)) = (2nd ‘(𝑓𝑛)))
4038, 39opeq12d 4569 . . . . . . . . . . 11 (𝑚 = 𝑛 → ⟨((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓𝑚))⟩ = ⟨((1st ‘(𝑓𝑛)) − (𝑤 / (2↑𝑛))), (2nd ‘(𝑓𝑛))⟩)
4140cbvmptv 4911 . . . . . . . . . 10 (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓𝑚))⟩) = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝑓𝑛)) − (𝑤 / (2↑𝑛))), (2nd ‘(𝑓𝑛))⟩)
4228, 29, 30, 32, 33, 34, 41ovolval5lem2 41531 . . . . . . . . 9 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
43423exp 1148 . . . . . . . 8 (𝑤 ∈ ℝ+ → (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))))
4443rexlimdv 3177 . . . . . . 7 (𝑤 ∈ ℝ+ → (∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤)))
4544imp 395 . . . . . 6 ((𝑤 ∈ ℝ+ ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
4613, 17, 45syl2anc 579 . . . . 5 ((𝑦𝑀𝑤 ∈ ℝ+) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
47463adant1 1160 . . . 4 ((⊤ ∧ 𝑦𝑀𝑤 ∈ ℝ+) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
4811, 12, 47infleinf 40250 . . 3 (⊤ → inf(𝑄, ℝ*, < ) ≤ inf(𝑀, ℝ*, < ))
49 eqeq1 2769 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
5049anbi2d 622 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
5150rexbidv 3199 . . . . . . . 8 (𝑧 = 𝑦 → (∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
5251cbvrabv 3348 . . . . . . 7 {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))} = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}
53 simpr 477 . . . . . . . . . . . . . 14 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → 𝐴 ran ((,) ∘ 𝑓))
54 ioossico 12470 . . . . . . . . . . . . . . . . . . . 20 ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))) ⊆ ((1st ‘(𝑓𝑛))[,)(2nd ‘(𝑓𝑛)))
5554a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))) ⊆ ((1st ‘(𝑓𝑛))[,)(2nd ‘(𝑓𝑛))))
5631adantr 472 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
57 simpr 477 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
5856, 57fvovco 40052 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) = ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))))
5956, 57fvovco 40052 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (([,) ∘ 𝑓)‘𝑛) = ((1st ‘(𝑓𝑛))[,)(2nd ‘(𝑓𝑛))))
6058, 59sseq12d 3796 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛) ↔ ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))) ⊆ ((1st ‘(𝑓𝑛))[,)(2nd ‘(𝑓𝑛)))))
6155, 60mpbird 248 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛))
6261ralrimiva 3113 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ∀𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛))
63 ss2iun 4694 . . . . . . . . . . . . . . . . 17 (∀𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛) → 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛))
6462, 63syl 17 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛))
65 ioof 12479 . . . . . . . . . . . . . . . . . . . . 21 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
6665a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (,):(ℝ* × ℝ*)⟶𝒫 ℝ)
67 rexpssxrxp 10342 . . . . . . . . . . . . . . . . . . . . . 22 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
6867a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
6931, 68fssd 6239 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ* × ℝ*))
70 fco 6242 . . . . . . . . . . . . . . . . . . . 20 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝑓:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝑓):ℕ⟶𝒫 ℝ)
7166, 69, 70syl2anc 579 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((,) ∘ 𝑓):ℕ⟶𝒫 ℝ)
7271ffnd 6226 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((,) ∘ 𝑓) Fn ℕ)
73 fniunfv 6701 . . . . . . . . . . . . . . . . . 18 (((,) ∘ 𝑓) Fn ℕ → 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) = ran ((,) ∘ 𝑓))
7472, 73syl 17 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) = ran ((,) ∘ 𝑓))
75 icof 40080 . . . . . . . . . . . . . . . . . . . . 21 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
7675a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
77 fco 6242 . . . . . . . . . . . . . . . . . . . 20 (([,):(ℝ* × ℝ*)⟶𝒫 ℝ*𝑓:ℕ⟶(ℝ* × ℝ*)) → ([,) ∘ 𝑓):ℕ⟶𝒫 ℝ*)
7876, 69, 77syl2anc 579 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ([,) ∘ 𝑓):ℕ⟶𝒫 ℝ*)
7978ffnd 6226 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ([,) ∘ 𝑓) Fn ℕ)
80 fniunfv 6701 . . . . . . . . . . . . . . . . . 18 (([,) ∘ 𝑓) Fn ℕ → 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛) = ran ([,) ∘ 𝑓))
8179, 80syl 17 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛) = ran ([,) ∘ 𝑓))
8274, 81sseq12d 3796 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ( 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛) ↔ ran ((,) ∘ 𝑓) ⊆ ran ([,) ∘ 𝑓)))
8364, 82mpbid 223 . . . . . . . . . . . . . . 15 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ran ((,) ∘ 𝑓) ⊆ ran ([,) ∘ 𝑓))
8483adantr 472 . . . . . . . . . . . . . 14 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → ran ((,) ∘ 𝑓) ⊆ ran ([,) ∘ 𝑓))
8553, 84sstrd 3773 . . . . . . . . . . . . 13 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → 𝐴 ran ([,) ∘ 𝑓))
8685adantrr 708 . . . . . . . . . . . 12 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → 𝐴 ran ([,) ∘ 𝑓))
87 simpr 477 . . . . . . . . . . . . . 14 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))
8831voliooicof 40874 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ [,)) ∘ 𝑓))
8988fveq2d 6383 . . . . . . . . . . . . . . 15 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
9089adantr 472 . . . . . . . . . . . . . 14 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
9187, 90eqtrd 2799 . . . . . . . . . . . . 13 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
9291adantrl 707 . . . . . . . . . . . 12 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
9386, 92jca 507 . . . . . . . . . . 11 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
9493ex 401 . . . . . . . . . 10 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
9594reximia 3155 . . . . . . . . 9 (∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
9695rgenw 3071 . . . . . . . 8 𝑦 ∈ ℝ* (∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
97 ss2rab 3840 . . . . . . . 8 ({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))} ⊆ {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))} ↔ ∀𝑦 ∈ ℝ* (∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
9896, 97mpbir 222 . . . . . . 7 {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))} ⊆ {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}
9952, 98eqsstri 3797 . . . . . 6 {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))} ⊆ {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}
1001, 6sseq12i 3793 . . . . . 6 (𝑄𝑀 ↔ {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))} ⊆ {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))})
10199, 100mpbir 222 . . . . 5 𝑄𝑀
102 infxrss 12376 . . . . 5 ((𝑄𝑀𝑀 ⊆ ℝ*) → inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < ))
103101, 8, 102mp2an 683 . . . 4 inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < )
104103a1i 11 . . 3 (⊤ → inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < ))
1055, 10, 48, 104xrletrid 12193 . 2 (⊤ → inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < ))
106105mptru 1660 1 inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wtru 1653  wcel 2155  wral 3055  wrex 3056  {crab 3059  wss 3734  𝒫 cpw 4317  cop 4342   cuni 4596   ciun 4678   class class class wbr 4811  cmpt 4890   × cxp 5277  ran crn 5280  ccom 5283   Fn wfn 6065  wf 6066  cfv 6070  (class class class)co 6846  1st c1st 7368  2nd c2nd 7369  𝑚 cmap 8064  infcinf 8558  cr 10192  *cxr 10331   < clt 10332  cle 10333  cmin 10524   / cdiv 10942  cn 11278  2c2 11331  +crp 12033   +𝑒 cxad 12149  (,)cioo 12382  [,)cico 12384  cexp 13072  volcvol 23535  Σ^csumge0 41240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-of 7099  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-2o 7769  df-oadd 7772  df-er 7951  df-map 8066  df-pm 8067  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-fi 8528  df-sup 8559  df-inf 8560  df-oi 8626  df-card 9020  df-cda 9247  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-div 10943  df-nn 11279  df-2 11339  df-3 11340  df-n0 11543  df-z 11629  df-uz 11892  df-q 11995  df-rp 12034  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-ioo 12386  df-ico 12388  df-icc 12389  df-fz 12539  df-fzo 12679  df-fl 12806  df-seq 13014  df-exp 13073  df-hash 13327  df-cj 14138  df-re 14139  df-im 14140  df-sqrt 14274  df-abs 14275  df-clim 14518  df-rlim 14519  df-sum 14716  df-rest 16363  df-topgen 16384  df-psmet 20025  df-xmet 20026  df-met 20027  df-bl 20028  df-mopn 20029  df-top 20992  df-topon 21009  df-bases 21044  df-cmp 21484  df-ovol 23536  df-vol 23537  df-sumge0 41241
This theorem is referenced by:  ovolval5  41533
  Copyright terms: Public domain W3C validator