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Theorem ovolval5lem3 46610
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 4092 . . . 4 𝑄 ⊆ ℝ*
3 infxrcl 13372 . . . 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 4092 . . . 4 𝑀 ⊆ ℝ*
7 infxrcl 13372 . . . 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 3457 . . . . . . 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 5872 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → ((,) ∘ 𝑔) = ((,) ∘ 𝑓))
1413rneqd 5952 . . . . . . . . . . . . . 14 (𝑔 = 𝑓 → ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝑓))
1514unieqd 4925 . . . . . . . . . . . . 13 (𝑔 = 𝑓 ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝑓))
1615sseq2d 4028 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (𝐴 ran ((,) ∘ 𝑔) ↔ 𝐴 ran ((,) ∘ 𝑓)))
17 coeq2 5872 . . . . . . . . . . . . . 14 (𝑔 = 𝑓 → ((vol ∘ (,)) ∘ 𝑔) = ((vol ∘ (,)) ∘ 𝑓))
1817fveq2d 6911 . . . . . . . . . . . . 13 (𝑔 = 𝑓 → (Σ^‘((vol ∘ (,)) ∘ 𝑔)) = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))
1918eqeq2d 2746 . . . . . . . . . . . 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 3236 . . . . . . . . . 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 3439 . . . . . . . . 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 2766 . . . . . . . 8 𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑔 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))}
24 simp3r 1201 . . . . . . . 8 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
25 eqid 2735 . . . . . . . 8 ^‘((vol ∘ (,)) ∘ (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓𝑚))⟩))) = (Σ^‘((vol ∘ (,)) ∘ (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓𝑚))⟩)))
26 elmapi 8888 . . . . . . . . 9 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
27263ad2ant2 1133 . . . . . . . 8 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑓:ℕ⟶(ℝ × ℝ))
28 simp3l 1200 . . . . . . . 8 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐴 ran ([,) ∘ 𝑓))
29 simp1 1135 . . . . . . . 8 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑤 ∈ ℝ+)
30 2fveq3 6912 . . . . . . . . . . 11 (𝑚 = 𝑛 → (1st ‘(𝑓𝑚)) = (1st ‘(𝑓𝑛)))
31 oveq2 7439 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
3231oveq2d 7447 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑤 / (2↑𝑚)) = (𝑤 / (2↑𝑛)))
3330, 32oveq12d 7449 . . . . . . . . . 10 (𝑚 = 𝑛 → ((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))) = ((1st ‘(𝑓𝑛)) − (𝑤 / (2↑𝑛))))
34 2fveq3 6912 . . . . . . . . . 10 (𝑚 = 𝑛 → (2nd ‘(𝑓𝑚)) = (2nd ‘(𝑓𝑛)))
3533, 34opeq12d 4886 . . . . . . . . 9 (𝑚 = 𝑛 → ⟨((1st ‘(𝑓𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓𝑚))⟩ = ⟨((1st ‘(𝑓𝑛)) − (𝑤 / (2↑𝑛))), (2nd ‘(𝑓𝑛))⟩)
3635cbvmptv 5261 . . . . . . . 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 46609 . . . . . . 7 ((𝑤 ∈ ℝ+𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
3837rexlimdv3a 3157 . . . . . 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 1129 . . . 4 ((⊤ ∧ 𝑦𝑀𝑤 ∈ ℝ+) → ∃𝑧𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
419, 10, 40infleinf 45322 . . 3 (⊤ → inf(𝑄, ℝ*, < ) ≤ inf(𝑀, ℝ*, < ))
42 eqeq1 2739 . . . . . . . . . 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 3177 . . . . . . . 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 3444 . . . . . . 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 13475 . . . . . . . . . . . . . . . . . . 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 45136 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) = ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))))
5249, 50fvovco 45136 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (([,) ∘ 𝑓)‘𝑛) = ((1st ‘(𝑓𝑛))[,)(2nd ‘(𝑓𝑛))))
5348, 51, 523sstr4d 4043 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛))
5453ralrimiva 3144 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ∀𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛))
55 ss2iun 5015 . . . . . . . . . . . . . . . 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 13484 . . . . . . . . . . . . . . . . . . 19 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
5857a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (,):(ℝ* × ℝ*)⟶𝒫 ℝ)
59 rexpssxrxp 11304 . . . . . . . . . . . . . . . . . . 19 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
6059a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
6158, 60, 26fcoss 45153 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((,) ∘ 𝑓):ℕ⟶𝒫 ℝ)
6261ffnd 6738 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((,) ∘ 𝑓) Fn ℕ)
63 fniunfv 7267 . . . . . . . . . . . . . . . 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 45162 . . . . . . . . . . . . . . . . . . 19 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
6665a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
6766, 60, 26fcoss 45153 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ([,) ∘ 𝑓):ℕ⟶𝒫 ℝ*)
6867ffnd 6738 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ([,) ∘ 𝑓) Fn ℕ)
69 fniunfv 7267 . . . . . . . . . . . . . . . 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 4042 . . . . . . . . . . . . . 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 4006 . . . . . . . . . . . 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 45952 . . . . . . . . . . . . . . 15 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ [,)) ∘ 𝑓))
7675fveq2d 6911 . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . 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 3079 . . . . . . . . 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 4087 . . . . . . 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 4030 . . . . . 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 4039 . . . . 5 𝑄𝑀
86 infxrss 13378 . . . . 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 13194 . 2 (⊤ → inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < ))
9089mptru 1544 1 inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wtru 1538  wcel 2106  wral 3059  wrex 3068  {crab 3433  wss 3963  𝒫 cpw 4605  cop 4637   cuni 4912   ciun 4996   class class class wbr 5148  cmpt 5231   × cxp 5687  ran crn 5690  ccom 5693   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  m cmap 8865  infcinf 9479  cr 11152  *cxr 11292   < clt 11293  cle 11294  cmin 11490   / cdiv 11918  cn 12264  2c2 12319  +crp 13032   +𝑒 cxad 13150  (,)cioo 13384  [,)cico 13386  cexp 14099  volcvol 25512  Σ^csumge0 46318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-rlim 15522  df-sum 15720  df-rest 17469  df-topgen 17490  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-top 22916  df-topon 22933  df-bases 22969  df-cmp 23411  df-ovol 25513  df-vol 25514  df-sumge0 46319
This theorem is referenced by:  ovolval5  46611
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