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Theorem ovolval5lem3 42365
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 3947 . . . . 5 {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))} ⊆ ℝ*
31, 2eqsstri 3892 . . . 4 𝑄 ⊆ ℝ*
4 infxrcl 12542 . . . 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 3947 . . . . 5 {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))} ⊆ ℝ*
86, 7eqsstri 3892 . . . 4 𝑀 ⊆ ℝ*
9 infxrcl 12542 . . . 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 3411 . . . . . . . . 9 (𝑦𝑀 ↔ (𝑦 ∈ ℝ* ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
1514biimpi 208 . . . . . . . 8 (𝑦𝑀 → (𝑦 ∈ ℝ* ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
1615simprd 488 . . . . . . 7 (𝑦𝑀 → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
1716adantr 473 . . . . . 6 ((𝑦𝑀𝑤 ∈ ℝ+) → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
18 coeq2 5579 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝑓 → ((,) ∘ 𝑔) = ((,) ∘ 𝑓))
1918rneqd 5651 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝑓))
2019unieqd 4722 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝑓))
2120sseq2d 3890 . . . . . . . . . . . . . 14 (𝑔 = 𝑓 → (𝐴 ran ((,) ∘ 𝑔) ↔ 𝐴 ran ((,) ∘ 𝑓)))
22 coeq2 5579 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → ((vol ∘ (,)) ∘ 𝑔) = ((vol ∘ (,)) ∘ 𝑓))
2322fveq2d 6503 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → (Σ^‘((vol ∘ (,)) ∘ 𝑔)) = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))
2423eqeq2d 2789 . . . . . . . . . . . . . 14 (𝑔 = 𝑓 → (𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)) ↔ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
2521, 24anbi12d 621 . . . . . . . . . . . . 13 (𝑔 = 𝑓 → ((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))) ↔ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
2625cbvrexv 3385 . . . . . . . . . . . 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 3400 . . . . . . . . . . 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 2806 . . . . . . . . . 10 𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑔 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))}
29 simp3r 1182 . . . . . . . . . 10 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
30 eqid 2779 . . . . . . . . . 10 ^‘((vol ∘ (,)) ∘ (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓𝑚))⟩))) = (Σ^‘((vol ∘ (,)) ∘ (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓𝑚))⟩)))
31 elmapi 8228 . . . . . . . . . . 11 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
32313ad2ant2 1114 . . . . . . . . . 10 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑓:ℕ⟶(ℝ × ℝ))
33 simp3l 1181 . . . . . . . . . 10 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐴 ran ([,) ∘ 𝑓))
34 simp1 1116 . . . . . . . . . 10 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑤 ∈ ℝ+)
35 2fveq3 6504 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (1st ‘(𝑓𝑚)) = (1st ‘(𝑓𝑛)))
36 oveq2 6984 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
3736oveq2d 6992 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (𝑤 / (2↑𝑚)) = (𝑤 / (2↑𝑛)))
3835, 37oveq12d 6994 . . . . . . . . . . . 12 (𝑚 = 𝑛 → ((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))) = ((1st ‘(𝑓𝑛)) − (𝑤 / (2↑𝑛))))
39 2fveq3 6504 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (2nd ‘(𝑓𝑚)) = (2nd ‘(𝑓𝑛)))
4038, 39opeq12d 4685 . . . . . . . . . . 11 (𝑚 = 𝑛 → ⟨((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓𝑚))⟩ = ⟨((1st ‘(𝑓𝑛)) − (𝑤 / (2↑𝑛))), (2nd ‘(𝑓𝑛))⟩)
4140cbvmptv 5028 . . . . . . . . . 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 42364 . . . . . . . . 9 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
43423exp 1099 . . . . . . . 8 (𝑤 ∈ ℝ+ → (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))))
4443rexlimdv 3229 . . . . . . 7 (𝑤 ∈ ℝ+ → (∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤)))
4544imp 398 . . . . . 6 ((𝑤 ∈ ℝ+ ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
4613, 17, 45syl2anc 576 . . . . 5 ((𝑦𝑀𝑤 ∈ ℝ+) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
47463adant1 1110 . . . 4 ((⊤ ∧ 𝑦𝑀𝑤 ∈ ℝ+) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
4811, 12, 47infleinf 41067 . . 3 (⊤ → inf(𝑄, ℝ*, < ) ≤ inf(𝑀, ℝ*, < ))
49 eqeq1 2783 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
5049anbi2d 619 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
5150rexbidv 3243 . . . . . . . 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 3413 . . . . . . 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 12642 . . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . . . . 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 40877 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) = ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))))
5956, 57fvovco 40877 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (([,) ∘ 𝑓)‘𝑛) = ((1st ‘(𝑓𝑛))[,)(2nd ‘(𝑓𝑛))))
6058, 59sseq12d 3891 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛) ↔ ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))) ⊆ ((1st ‘(𝑓𝑛))[,)(2nd ‘(𝑓𝑛)))))
6155, 60mpbird 249 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛))
6261ralrimiva 3133 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ∀𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛))
63 ss2iun 4809 . . . . . . . . . . . . . . . . 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 12651 . . . . . . . . . . . . . . . . . . . . 21 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
6665a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (,):(ℝ* × ℝ*)⟶𝒫 ℝ)
67 rexpssxrxp 10485 . . . . . . . . . . . . . . . . . . . . . 22 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
6867a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
6931, 68fssd 6358 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ* × ℝ*))
70 fco 6361 . . . . . . . . . . . . . . . . . . . 20 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝑓:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝑓):ℕ⟶𝒫 ℝ)
7166, 69, 70syl2anc 576 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((,) ∘ 𝑓):ℕ⟶𝒫 ℝ)
7271ffnd 6345 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((,) ∘ 𝑓) Fn ℕ)
73 fniunfv 6831 . . . . . . . . . . . . . . . . . 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 40905 . . . . . . . . . . . . . . . . . . . . 21 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
7675a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
77 fco 6361 . . . . . . . . . . . . . . . . . . . 20 (([,):(ℝ* × ℝ*)⟶𝒫 ℝ*𝑓:ℕ⟶(ℝ* × ℝ*)) → ([,) ∘ 𝑓):ℕ⟶𝒫 ℝ*)
7876, 69, 77syl2anc 576 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ([,) ∘ 𝑓):ℕ⟶𝒫 ℝ*)
7978ffnd 6345 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ([,) ∘ 𝑓) Fn ℕ)
80 fniunfv 6831 . . . . . . . . . . . . . . . . . 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 3891 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ( 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛) ↔ ran ((,) ∘ 𝑓) ⊆ ran ([,) ∘ 𝑓)))
8364, 82mpbid 224 . . . . . . . . . . . . . . 15 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ran ((,) ∘ 𝑓) ⊆ ran ([,) ∘ 𝑓))
8483adantr 473 . . . . . . . . . . . . . 14 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → ran ((,) ∘ 𝑓) ⊆ ran ([,) ∘ 𝑓))
8553, 84sstrd 3869 . . . . . . . . . . . . 13 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → 𝐴 ran ([,) ∘ 𝑓))
8685adantrr 704 . . . . . . . . . . . 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 41710 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ [,)) ∘ 𝑓))
8988fveq2d 6503 . . . . . . . . . . . . . . 15 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
9089adantr 473 . . . . . . . . . . . . . 14 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
9187, 90eqtrd 2815 . . . . . . . . . . . . 13 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
9291adantrl 703 . . . . . . . . . . . 12 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
9386, 92jca 504 . . . . . . . . . . 11 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
9493ex 405 . . . . . . . . . 10 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
9594reximia 3190 . . . . . . . . 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 3101 . . . . . . . 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 3938 . . . . . . . 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 223 . . . . . . 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 3892 . . . . . 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 3888 . . . . . 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 223 . . . . 5 𝑄𝑀
102 infxrss 12548 . . . . 5 ((𝑄𝑀𝑀 ⊆ ℝ*) → inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < ))
103101, 8, 102mp2an 679 . . . 4 inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < )
104103a1i 11 . . 3 (⊤ → inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < ))
1055, 10, 48, 104xrletrid 12365 . 2 (⊤ → inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < ))
106105mptru 1514 1 inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068   = wceq 1507  wtru 1508  wcel 2050  wral 3089  wrex 3090  {crab 3093  wss 3830  𝒫 cpw 4422  cop 4447   cuni 4712   ciun 4792   class class class wbr 4929  cmpt 5008   × cxp 5405  ran crn 5408  ccom 5411   Fn wfn 6183  wf 6184  cfv 6188  (class class class)co 6976  1st c1st 7499  2nd c2nd 7500  𝑚 cmap 8206  infcinf 8700  cr 10334  *cxr 10473   < clt 10474  cle 10475  cmin 10670   / cdiv 11098  cn 11439  2c2 11495  +crp 12204   +𝑒 cxad 12322  (,)cioo 12554  [,)cico 12556  cexp 13244  volcvol 23767  Σ^csumge0 42073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-inf2 8898  ax-cnex 10391  ax-resscn 10392  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-addrcl 10396  ax-mulcl 10397  ax-mulrcl 10398  ax-mulcom 10399  ax-addass 10400  ax-mulass 10401  ax-distr 10402  ax-i2m1 10403  ax-1ne0 10404  ax-1rid 10405  ax-rnegex 10406  ax-rrecex 10407  ax-cnre 10408  ax-pre-lttri 10409  ax-pre-lttrn 10410  ax-pre-ltadd 10411  ax-pre-mulgt0 10412  ax-pre-sup 10413
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-pss 3846  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-se 5367  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-isom 6197  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-of 7227  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-1o 7905  df-2o 7906  df-oadd 7909  df-er 8089  df-map 8208  df-pm 8209  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-fi 8670  df-sup 8701  df-inf 8702  df-oi 8769  df-dju 9124  df-card 9162  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-sub 10672  df-neg 10673  df-div 11099  df-nn 11440  df-2 11503  df-3 11504  df-n0 11708  df-z 11794  df-uz 12059  df-q 12163  df-rp 12205  df-xneg 12324  df-xadd 12325  df-xmul 12326  df-ioo 12558  df-ico 12560  df-icc 12561  df-fz 12709  df-fzo 12850  df-fl 12977  df-seq 13185  df-exp 13245  df-hash 13506  df-cj 14319  df-re 14320  df-im 14321  df-sqrt 14455  df-abs 14456  df-clim 14706  df-rlim 14707  df-sum 14904  df-rest 16552  df-topgen 16573  df-psmet 20239  df-xmet 20240  df-met 20241  df-bl 20242  df-mopn 20243  df-top 21206  df-topon 21223  df-bases 21258  df-cmp 21699  df-ovol 23768  df-vol 23769  df-sumge0 42074
This theorem is referenced by:  ovolval5  42366
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