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Theorem ovolval5lem3 46674
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)))}
21ssrab3 4081 . . . 4 𝑄 ⊆ ℝ*
3 infxrcl 13376 . . . 4 (𝑄 ⊆ ℝ* → inf(𝑄, ℝ*, < ) ∈ ℝ*)
42, 3mp1i 13 . . 3 (⊤ → inf(𝑄, ℝ*, < ) ∈ ℝ*)
5 ovolval5lem3.m . . . . 5 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}
65ssrab3 4081 . . . 4 𝑀 ⊆ ℝ*
7 infxrcl 13376 . . . 4 (𝑀 ⊆ ℝ* → inf(𝑀, ℝ*, < ) ∈ ℝ*)
86, 7mp1i 13 . . 3 (⊤ → inf(𝑀, ℝ*, < ) ∈ ℝ*)
92a1i 11 . . . 4 (⊤ → 𝑄 ⊆ ℝ*)
106a1i 11 . . . 4 (⊤ → 𝑀 ⊆ ℝ*)
115reqabi 3459 . . . . . . 7 (𝑦𝑀 ↔ (𝑦 ∈ ℝ* ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
1211simprbi 496 . . . . . 6 (𝑦𝑀 → ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
13 coeq2 5868 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → ((,) ∘ 𝑔) = ((,) ∘ 𝑓))
1413rneqd 5948 . . . . . . . . . . . . . 14 (𝑔 = 𝑓 → ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝑓))
1514unieqd 4919 . . . . . . . . . . . . 13 (𝑔 = 𝑓 ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝑓))
1615sseq2d 4015 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (𝐴 ran ((,) ∘ 𝑔) ↔ 𝐴 ran ((,) ∘ 𝑓)))
17 coeq2 5868 . . . . . . . . . . . . . 14 (𝑔 = 𝑓 → ((vol ∘ (,)) ∘ 𝑔) = ((vol ∘ (,)) ∘ 𝑓))
1817fveq2d 6909 . . . . . . . . . . . . 13 (𝑔 = 𝑓 → (Σ^‘((vol ∘ (,)) ∘ 𝑔)) = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))
1918eqeq2d 2747 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)) ↔ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
2016, 19anbi12d 632 . . . . . . . . . . 11 (𝑔 = 𝑓 → ((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))) ↔ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
2120cbvrexvw 3237 . . . . . . . . . 10 (∃𝑔 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))) ↔ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
2221rabbii 3441 . . . . . . . . 9 {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)))}
231, 22eqtr4i 2767 . . . . . . . 8 𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑔 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))}
24 simp3r 1202 . . . . . . . 8 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
25 eqid 2736 . . . . . . . 8 ^‘((vol ∘ (,)) ∘ (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓𝑚))⟩))) = (Σ^‘((vol ∘ (,)) ∘ (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓𝑚))⟩)))
26 elmapi 8890 . . . . . . . . 9 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
27263ad2ant2 1134 . . . . . . . 8 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑓:ℕ⟶(ℝ × ℝ))
28 simp3l 1201 . . . . . . . 8 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐴 ran ([,) ∘ 𝑓))
29 simp1 1136 . . . . . . . 8 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑤 ∈ ℝ+)
30 2fveq3 6910 . . . . . . . . . . 11 (𝑚 = 𝑛 → (1st ‘(𝑓𝑚)) = (1st ‘(𝑓𝑛)))
31 oveq2 7440 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
3231oveq2d 7448 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑤 / (2↑𝑚)) = (𝑤 / (2↑𝑛)))
3330, 32oveq12d 7450 . . . . . . . . . 10 (𝑚 = 𝑛 → ((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))) = ((1st ‘(𝑓𝑛)) − (𝑤 / (2↑𝑛))))
34 2fveq3 6910 . . . . . . . . . 10 (𝑚 = 𝑛 → (2nd ‘(𝑓𝑚)) = (2nd ‘(𝑓𝑛)))
3533, 34opeq12d 4880 . . . . . . . . 9 (𝑚 = 𝑛 → ⟨((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓𝑚))⟩ = ⟨((1st ‘(𝑓𝑛)) − (𝑤 / (2↑𝑛))), (2nd ‘(𝑓𝑛))⟩)
3635cbvmptv 5254 . . . . . . . 8 (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓𝑚))⟩) = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝑓𝑛)) − (𝑤 / (2↑𝑛))), (2nd ‘(𝑓𝑛))⟩)
3723, 24, 25, 27, 28, 29, 36ovolval5lem2 46673 . . . . . . 7 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
3837rexlimdv3a 3158 . . . . . 6 (𝑤 ∈ ℝ+ → (∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤)))
3912, 38mpan9 506 . . . . 5 ((𝑦𝑀𝑤 ∈ ℝ+) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
40393adant1 1130 . . . 4 ((⊤ ∧ 𝑦𝑀𝑤 ∈ ℝ+) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
419, 10, 40infleinf 45388 . . 3 (⊤ → inf(𝑄, ℝ*, < ) ≤ inf(𝑀, ℝ*, < ))
42 eqeq1 2740 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
4342anbi2d 630 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
4443rexbidv 3178 . . . . . . . 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)))))
4544cbvrabv 3446 . . . . . . 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)))}
46 simpr 484 . . . . . . . . . . . . 13 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → 𝐴 ran ((,) ∘ 𝑓))
47 ioossico 13479 . . . . . . . . . . . . . . . . . . 19 ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))) ⊆ ((1st ‘(𝑓𝑛))[,)(2nd ‘(𝑓𝑛)))
4847a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))) ⊆ ((1st ‘(𝑓𝑛))[,)(2nd ‘(𝑓𝑛))))
4926adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
50 simpr 484 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
5149, 50fvovco 45203 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) = ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))))
5249, 50fvovco 45203 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (([,) ∘ 𝑓)‘𝑛) = ((1st ‘(𝑓𝑛))[,)(2nd ‘(𝑓𝑛))))
5348, 51, 523sstr4d 4038 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛))
5453ralrimiva 3145 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ∀𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛))
55 ss2iun 5009 . . . . . . . . . . . . . . . 16 (∀𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛) → 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛))
5654, 55syl 17 . . . . . . . . . . . . . . 15 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛))
57 ioof 13488 . . . . . . . . . . . . . . . . . . 19 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
5857a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (,):(ℝ* × ℝ*)⟶𝒫 ℝ)
59 rexpssxrxp 11307 . . . . . . . . . . . . . . . . . . 19 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
6059a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
6158, 60, 26fcoss 45220 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((,) ∘ 𝑓):ℕ⟶𝒫 ℝ)
6261ffnd 6736 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((,) ∘ 𝑓) Fn ℕ)
63 fniunfv 7268 . . . . . . . . . . . . . . . 16 (((,) ∘ 𝑓) Fn ℕ → 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) = ran ((,) ∘ 𝑓))
6462, 63syl 17 . . . . . . . . . . . . . . 15 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) = ran ((,) ∘ 𝑓))
65 icof 45229 . . . . . . . . . . . . . . . . . . 19 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
6665a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
6766, 60, 26fcoss 45220 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ([,) ∘ 𝑓):ℕ⟶𝒫 ℝ*)
6867ffnd 6736 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ([,) ∘ 𝑓) Fn ℕ)
69 fniunfv 7268 . . . . . . . . . . . . . . . 16 (([,) ∘ 𝑓) Fn ℕ → 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛) = ran ([,) ∘ 𝑓))
7068, 69syl 17 . . . . . . . . . . . . . . 15 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛) = ran ([,) ∘ 𝑓))
7156, 64, 703sstr3d 4037 . . . . . . . . . . . . . 14 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ran ((,) ∘ 𝑓) ⊆ ran ([,) ∘ 𝑓))
7271adantr 480 . . . . . . . . . . . . 13 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → ran ((,) ∘ 𝑓) ⊆ ran ([,) ∘ 𝑓))
7346, 72sstrd 3993 . . . . . . . . . . . 12 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → 𝐴 ran ([,) ∘ 𝑓))
74 simpr 484 . . . . . . . . . . . . 13 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))
7526voliooicof 46016 . . . . . . . . . . . . . . 15 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ [,)) ∘ 𝑓))
7675fveq2d 6909 . . . . . . . . . . . . . 14 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
7776adantr 480 . . . . . . . . . . . . 13 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
7874, 77eqtrd 2776 . . . . . . . . . . . 12 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
7973, 78anim12dan 619 . . . . . . . . . . 11 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
8079ex 412 . . . . . . . . . 10 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
8180reximia 3080 . . . . . . . . 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))))
8281a1i 11 . . . . . . . 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)))))
8382ss2rabi 4076 . . . . . . 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)))}
8445, 83eqsstri 4029 . . . . . 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)))}
8584, 1, 53sstr4i 4034 . . . . 5 𝑄𝑀
86 infxrss 13382 . . . . 5 ((𝑄𝑀𝑀 ⊆ ℝ*) → inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < ))
8785, 6, 86mp2an 692 . . . 4 inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < )
8887a1i 11 . . 3 (⊤ → inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < ))
894, 8, 41, 88xrletrid 13198 . 2 (⊤ → inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < ))
9089mptru 1546 1 inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wtru 1540  wcel 2107  wral 3060  wrex 3069  {crab 3435  wss 3950  𝒫 cpw 4599  cop 4631   cuni 4906   ciun 4990   class class class wbr 5142  cmpt 5224   × cxp 5682  ran crn 5685  ccom 5688   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432  1st c1st 8013  2nd c2nd 8014  m cmap 8867  infcinf 9482  cr 11155  *cxr 11295   < clt 11296  cle 11297  cmin 11493   / cdiv 11921  cn 12267  2c2 12322  +crp 13035   +𝑒 cxad 13153  (,)cioo 13388  [,)cico 13390  cexp 14103  volcvol 25499  Σ^csumge0 46382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-pm 8870  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fi 9452  df-sup 9483  df-inf 9484  df-oi 9551  df-dju 9942  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-n0 12529  df-z 12616  df-uz 12880  df-q 12992  df-rp 13036  df-xneg 13155  df-xadd 13156  df-xmul 13157  df-ioo 13392  df-ico 13394  df-icc 13395  df-fz 13549  df-fzo 13696  df-fl 13833  df-seq 14044  df-exp 14104  df-hash 14371  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-clim 15525  df-rlim 15526  df-sum 15724  df-rest 17468  df-topgen 17489  df-psmet 21357  df-xmet 21358  df-met 21359  df-bl 21360  df-mopn 21361  df-top 22901  df-topon 22918  df-bases 22954  df-cmp 23396  df-ovol 25500  df-vol 25501  df-sumge0 46383
This theorem is referenced by:  ovolval5  46675
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