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

Theorem ovolval5lem3 39544
 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 3650 . . . . 5 {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))} ⊆ ℝ*
31, 2eqsstri 3598 . . . 4 𝑄 ⊆ ℝ*
4 infxrcl 12035 . . . 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 3650 . . . . 5 {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))} ⊆ ℝ*
86, 7eqsstri 3598 . . . 4 𝑀 ⊆ ℝ*
9 infxrcl 12035 . . . 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 476 . . . . . 6 ((𝑦𝑀𝑤 ∈ ℝ+) → 𝑤 ∈ ℝ+)
146rabid3 38285 . . . . . . . . 9 (𝑦𝑀 ↔ (𝑦 ∈ ℝ* ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
1514biimpi 205 . . . . . . . 8 (𝑦𝑀 → (𝑦 ∈ ℝ* ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
1615simprd 478 . . . . . . 7 (𝑦𝑀 → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
1716adantr 480 . . . . . 6 ((𝑦𝑀𝑤 ∈ ℝ+) → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
18 coeq2 5202 . . . . . . . . . . . . . . . . . 18 (𝑔 = 𝑓 → ((,) ∘ 𝑔) = ((,) ∘ 𝑓))
1918rneqd 5274 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝑓 → ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝑓))
2019unieqd 4382 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝑓))
2120sseq2d 3596 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → (𝐴 ran ((,) ∘ 𝑔) ↔ 𝐴 ran ((,) ∘ 𝑓)))
22 coeq2 5202 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝑓 → ((vol ∘ (,)) ∘ 𝑔) = ((vol ∘ (,)) ∘ 𝑓))
2322fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → (Σ^‘((vol ∘ (,)) ∘ 𝑔)) = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))
2423eqeq2d 2620 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → (𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)) ↔ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
2521, 24anbi12d 743 . . . . . . . . . . . . . 14 (𝑔 = 𝑓 → ((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))) ↔ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
2625cbvrexv 3148 . . . . . . . . . . . . 13 (∃𝑔 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))) ↔ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
2726a1i 11 . . . . . . . . . . . 12 (𝑧 ∈ ℝ* → (∃𝑔 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))) ↔ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
2827rabbiia 3161 . . . . . . . . . . 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)))}
291, 28eqtr4i 2635 . . . . . . . . . 10 𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑔 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))}
30 simp3r 1083 . . . . . . . . . 10 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
31 eqid 2610 . . . . . . . . . 10 ^‘((vol ∘ (,)) ∘ (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓𝑚))⟩))) = (Σ^‘((vol ∘ (,)) ∘ (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓𝑚))⟩)))
32 elmapi 7765 . . . . . . . . . . 11 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
33323ad2ant2 1076 . . . . . . . . . 10 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑓:ℕ⟶(ℝ × ℝ))
34 simp3l 1082 . . . . . . . . . 10 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐴 ran ([,) ∘ 𝑓))
35 simp1 1054 . . . . . . . . . 10 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑤 ∈ ℝ+)
36 fveq2 6103 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (𝑓𝑚) = (𝑓𝑛))
3736fveq2d 6107 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (1st ‘(𝑓𝑚)) = (1st ‘(𝑓𝑛)))
38 oveq2 6557 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
3938oveq2d 6565 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (𝑤 / (2↑𝑚)) = (𝑤 / (2↑𝑛)))
4037, 39oveq12d 6567 . . . . . . . . . . . 12 (𝑚 = 𝑛 → ((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))) = ((1st ‘(𝑓𝑛)) − (𝑤 / (2↑𝑛))))
4136fveq2d 6107 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (2nd ‘(𝑓𝑚)) = (2nd ‘(𝑓𝑛)))
4240, 41opeq12d 4348 . . . . . . . . . . 11 (𝑚 = 𝑛 → ⟨((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓𝑚))⟩ = ⟨((1st ‘(𝑓𝑛)) − (𝑤 / (2↑𝑛))), (2nd ‘(𝑓𝑛))⟩)
4342cbvmptv 4678 . . . . . . . . . 10 (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓𝑚))⟩) = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝑓𝑛)) − (𝑤 / (2↑𝑛))), (2nd ‘(𝑓𝑛))⟩)
4429, 30, 31, 33, 34, 35, 43ovolval5lem2 39543 . . . . . . . . 9 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
45443exp 1256 . . . . . . . 8 (𝑤 ∈ ℝ+ → (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))))
4645rexlimdv 3012 . . . . . . 7 (𝑤 ∈ ℝ+ → (∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤)))
4746imp 444 . . . . . 6 ((𝑤 ∈ ℝ+ ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
4813, 17, 47syl2anc 691 . . . . 5 ((𝑦𝑀𝑤 ∈ ℝ+) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
49483adant1 1072 . . . 4 ((⊤ ∧ 𝑦𝑀𝑤 ∈ ℝ+) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
5011, 12, 49infleinf 38529 . . 3 (⊤ → inf(𝑄, ℝ*, < ) ≤ inf(𝑀, ℝ*, < ))
51 eqeq1 2614 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
5251anbi2d 736 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
5352rexbidv 3034 . . . . . . . 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)))))
5453cbvrabv 3172 . . . . . . 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)))}
55 simpr 476 . . . . . . . . . . . . . 14 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → 𝐴 ran ((,) ∘ 𝑓))
56 ioossico 12133 . . . . . . . . . . . . . . . . . . . 20 ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))) ⊆ ((1st ‘(𝑓𝑛))[,)(2nd ‘(𝑓𝑛)))
5756a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))) ⊆ ((1st ‘(𝑓𝑛))[,)(2nd ‘(𝑓𝑛))))
5832adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
59 simpr 476 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
6058, 59fvovco 38376 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) = ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))))
6158, 59fvovco 38376 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (([,) ∘ 𝑓)‘𝑛) = ((1st ‘(𝑓𝑛))[,)(2nd ‘(𝑓𝑛))))
6260, 61sseq12d 3597 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛) ↔ ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))) ⊆ ((1st ‘(𝑓𝑛))[,)(2nd ‘(𝑓𝑛)))))
6357, 62mpbird 246 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛))
6463ralrimiva 2949 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ∀𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛))
65 ss2iun 4472 . . . . . . . . . . . . . . . . 17 (∀𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛) → 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛))
6664, 65syl 17 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛))
67 ioof 12142 . . . . . . . . . . . . . . . . . . . . 21 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
6867a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (,):(ℝ* × ℝ*)⟶𝒫 ℝ)
69 rexpssxrxp 9963 . . . . . . . . . . . . . . . . . . . . . 22 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
7069a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
7132, 70fssd 5970 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ* × ℝ*))
72 fco 5971 . . . . . . . . . . . . . . . . . . . 20 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝑓:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝑓):ℕ⟶𝒫 ℝ)
7368, 71, 72syl2anc 691 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((,) ∘ 𝑓):ℕ⟶𝒫 ℝ)
74 ffn 5958 . . . . . . . . . . . . . . . . . . 19 (((,) ∘ 𝑓):ℕ⟶𝒫 ℝ → ((,) ∘ 𝑓) Fn ℕ)
7573, 74syl 17 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((,) ∘ 𝑓) Fn ℕ)
76 fniunfv 6409 . . . . . . . . . . . . . . . . . 18 (((,) ∘ 𝑓) Fn ℕ → 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) = ran ((,) ∘ 𝑓))
7775, 76syl 17 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) = ran ((,) ∘ 𝑓))
78 icof 38406 . . . . . . . . . . . . . . . . . . . . 21 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
7978a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
80 fco 5971 . . . . . . . . . . . . . . . . . . . 20 (([,):(ℝ* × ℝ*)⟶𝒫 ℝ*𝑓:ℕ⟶(ℝ* × ℝ*)) → ([,) ∘ 𝑓):ℕ⟶𝒫 ℝ*)
8179, 71, 80syl2anc 691 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ([,) ∘ 𝑓):ℕ⟶𝒫 ℝ*)
82 ffn 5958 . . . . . . . . . . . . . . . . . . 19 (([,) ∘ 𝑓):ℕ⟶𝒫 ℝ* → ([,) ∘ 𝑓) Fn ℕ)
8381, 82syl 17 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ([,) ∘ 𝑓) Fn ℕ)
84 fniunfv 6409 . . . . . . . . . . . . . . . . . 18 (([,) ∘ 𝑓) Fn ℕ → 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛) = ran ([,) ∘ 𝑓))
8583, 84syl 17 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛) = ran ([,) ∘ 𝑓))
8677, 85sseq12d 3597 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ( 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛) ↔ ran ((,) ∘ 𝑓) ⊆ ran ([,) ∘ 𝑓)))
8766, 86mpbid 221 . . . . . . . . . . . . . . 15 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ran ((,) ∘ 𝑓) ⊆ ran ([,) ∘ 𝑓))
8887adantr 480 . . . . . . . . . . . . . 14 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → ran ((,) ∘ 𝑓) ⊆ ran ([,) ∘ 𝑓))
8955, 88sstrd 3578 . . . . . . . . . . . . 13 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → 𝐴 ran ([,) ∘ 𝑓))
9089adantrr 749 . . . . . . . . . . . 12 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → 𝐴 ran ([,) ∘ 𝑓))
91 simpr 476 . . . . . . . . . . . . . 14 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))
9232voliooicof 38889 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ [,)) ∘ 𝑓))
9392fveq2d 6107 . . . . . . . . . . . . . . 15 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
9493adantr 480 . . . . . . . . . . . . . 14 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
9591, 94eqtrd 2644 . . . . . . . . . . . . 13 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
9695adantrl 748 . . . . . . . . . . . 12 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
9790, 96jca 553 . . . . . . . . . . 11 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
9897ex 449 . . . . . . . . . 10 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
9998reximia 2992 . . . . . . . . 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))))
10099rgenw 2908 . . . . . . . 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))))
101 ss2rab 3641 . . . . . . . 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)))))
102100, 101mpbir 220 . . . . . . 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)))}
10354, 102eqsstri 3598 . . . . . 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)))}
1041, 6sseq12i 3594 . . . . . 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)))})
105103, 104mpbir 220 . . . . 5 𝑄𝑀
106 infxrss 12040 . . . . 5 ((𝑄𝑀𝑀 ⊆ ℝ*) → inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < ))
107105, 8, 106mp2an 704 . . . 4 inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < )
108107a1i 11 . . 3 (⊤ → inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < ))
1095, 10, 50, 108xrletrid 11862 . 2 (⊤ → inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < ))
110109trud 1484 1 inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900   ⊆ wss 3540  𝒫 cpw 4108  ⟨cop 4131  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ran crn 5039   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058   ↑𝑚 cmap 7744  infcinf 8230  ℝcr 9814  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145   / cdiv 10563  ℕcn 10897  2c2 10947  ℝ+crp 11708   +𝑒 cxad 11820  (,)cioo 12046  [,)cico 12048  ↑cexp 12722  volcvol 23039  Σ^csumge0 39255 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041  df-sumge0 39256 This theorem is referenced by:  ovolval5  39545
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