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Theorem ovollb2lem 23302
Description: Lemma for ovollb2 23303. (Contributed by Mario Carneiro, 24-Mar-2015.)
Hypotheses
Ref Expression
ovollb2.1 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
ovollb2.2 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩)
ovollb2.3 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
ovollb2.4 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
ovollb2.5 (𝜑𝐴 ran ([,] ∘ 𝐹))
ovollb2.6 (𝜑𝐵 ∈ ℝ+)
ovollb2.7 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
Assertion
Ref Expression
ovollb2lem (𝜑 → (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
Distinct variable groups:   𝐴,𝑛   𝑛,𝐹   𝐵,𝑛   𝜑,𝑛   𝑆,𝑛
Allowed substitution hints:   𝑇(𝑛)   𝐺(𝑛)

Proof of Theorem ovollb2lem
Dummy variables 𝑚 𝑦 𝑧 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovollb2.5 . . . 4 (𝜑𝐴 ran ([,] ∘ 𝐹))
2 ovollb2.4 . . . . 5 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3 ovolficcss 23284 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
42, 3syl 17 . . . 4 (𝜑 ran ([,] ∘ 𝐹) ⊆ ℝ)
51, 4sstrd 3646 . . 3 (𝜑𝐴 ⊆ ℝ)
6 ovolcl 23292 . . 3 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
75, 6syl 17 . 2 (𝜑 → (vol*‘𝐴) ∈ ℝ*)
8 ovolfcl 23281 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
92, 8sylan 487 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
109simp1d 1093 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
11 ovollb2.6 . . . . . . . . . . . . . . 15 (𝜑𝐵 ∈ ℝ+)
1211rphalfcld 11922 . . . . . . . . . . . . . 14 (𝜑 → (𝐵 / 2) ∈ ℝ+)
1312adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐵 / 2) ∈ ℝ+)
14 2nn 11223 . . . . . . . . . . . . . . 15 2 ∈ ℕ
15 nnnn0 11337 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
1615adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
17 nnexpcl 12913 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
1814, 16, 17sylancr 696 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℕ)
1918nnrpd 11908 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ+)
2013, 19rpdivcld 11927 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑛)) ∈ ℝ+)
2120rpred 11910 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑛)) ∈ ℝ)
2210, 21resubcld 10496 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) ∈ ℝ)
239simp2d 1094 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
2423, 21readdcld 10107 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))) ∈ ℝ)
2510, 20ltsubrpd 11942 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) < (1st ‘(𝐹𝑛)))
269simp3d 1095 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
2723, 20ltaddrpd 11943 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) < ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
2810, 23, 24, 26, 27lelttrd 10233 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) < ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
2922, 10, 24, 25, 28lttrd 10236 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) < ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
3022, 24, 29ltled 10223 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) ≤ ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
31 df-br 4686 . . . . . . . . 9 (((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) ≤ ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))) ↔ ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ ∈ ≤ )
3230, 31sylib 208 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ ∈ ≤ )
33 opelxpi 5182 . . . . . . . . 9 ((((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) ∈ ℝ ∧ ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))) ∈ ℝ) → ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ ∈ (ℝ × ℝ))
3422, 24, 33syl2anc 694 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ ∈ (ℝ × ℝ))
3532, 34elind 3831 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
36 ovollb2.2 . . . . . . 7 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩)
3735, 36fmptd 6425 . . . . . 6 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
38 eqid 2651 . . . . . . 7 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
39 ovollb2.3 . . . . . . 7 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
4038, 39ovolsf 23287 . . . . . 6 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
4137, 40syl 17 . . . . 5 (𝜑𝑇:ℕ⟶(0[,)+∞))
42 frn 6091 . . . . 5 (𝑇:ℕ⟶(0[,)+∞) → ran 𝑇 ⊆ (0[,)+∞))
4341, 42syl 17 . . . 4 (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
44 icossxr 12296 . . . 4 (0[,)+∞) ⊆ ℝ*
4543, 44syl6ss 3648 . . 3 (𝜑 → ran 𝑇 ⊆ ℝ*)
46 supxrcl 12183 . . 3 (ran 𝑇 ⊆ ℝ* → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
4745, 46syl 17 . 2 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
48 ovollb2.7 . . . 4 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
4911rpred 11910 . . . 4 (𝜑𝐵 ∈ ℝ)
5048, 49readdcld 10107 . . 3 (𝜑 → (sup(ran 𝑆, ℝ*, < ) + 𝐵) ∈ ℝ)
5150rexrd 10127 . 2 (𝜑 → (sup(ran 𝑆, ℝ*, < ) + 𝐵) ∈ ℝ*)
52 fveq2 6229 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
5352fveq2d 6233 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (1st ‘(𝐹𝑛)) = (1st ‘(𝐹𝑚)))
54 oveq2 6698 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (2↑𝑛) = (2↑𝑚))
5554oveq2d 6706 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → ((𝐵 / 2) / (2↑𝑛)) = ((𝐵 / 2) / (2↑𝑚)))
5653, 55oveq12d 6708 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) = ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))))
5752fveq2d 6233 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (2nd ‘(𝐹𝑛)) = (2nd ‘(𝐹𝑚)))
5857, 55oveq12d 6708 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
5956, 58opeq12d 4441 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ = ⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩)
60 opex 4962 . . . . . . . . . . . . . . 15 ⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩ ∈ V
6159, 36, 60fvmpt 6321 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → (𝐺𝑚) = ⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩)
6261adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) = ⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩)
6362fveq2d 6233 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) = (1st ‘⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩))
64 ovex 6718 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))) ∈ V
65 ovex 6718 . . . . . . . . . . . . 13 ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))) ∈ V
6664, 65op1st 7218 . . . . . . . . . . . 12 (1st ‘⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩) = ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))
6763, 66syl6eq 2701 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) = ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))))
68 ovolfcl 23281 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) ∈ ℝ ∧ (2nd ‘(𝐹𝑚)) ∈ ℝ ∧ (1st ‘(𝐹𝑚)) ≤ (2nd ‘(𝐹𝑚))))
692, 68sylan 487 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) ∈ ℝ ∧ (2nd ‘(𝐹𝑚)) ∈ ℝ ∧ (1st ‘(𝐹𝑚)) ≤ (2nd ‘(𝐹𝑚))))
7069simp1d 1093 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐹𝑚)) ∈ ℝ)
7112adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (𝐵 / 2) ∈ ℝ+)
72 nnnn0 11337 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
7372adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℕ0)
74 nnexpcl 12913 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ0) → (2↑𝑚) ∈ ℕ)
7514, 73, 74sylancr 696 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (2↑𝑚) ∈ ℕ)
7675nnrpd 11908 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (2↑𝑚) ∈ ℝ+)
7771, 76rpdivcld 11927 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑚)) ∈ ℝ+)
7870, 77ltsubrpd 11942 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))) < (1st ‘(𝐹𝑚)))
7967, 78eqbrtrd 4707 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) < (1st ‘(𝐹𝑚)))
8079adantlr 751 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) < (1st ‘(𝐹𝑚)))
81 ovolfcl 23281 . . . . . . . . . . . . 13 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐺𝑚)) ∈ ℝ ∧ (2nd ‘(𝐺𝑚)) ∈ ℝ ∧ (1st ‘(𝐺𝑚)) ≤ (2nd ‘(𝐺𝑚))))
8237, 81sylan 487 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝐺𝑚)) ∈ ℝ ∧ (2nd ‘(𝐺𝑚)) ∈ ℝ ∧ (1st ‘(𝐺𝑚)) ≤ (2nd ‘(𝐺𝑚))))
8382simp1d 1093 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) ∈ ℝ)
8483adantlr 751 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) ∈ ℝ)
8570adantlr 751 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐹𝑚)) ∈ ℝ)
865sselda 3636 . . . . . . . . . . 11 ((𝜑𝑧𝐴) → 𝑧 ∈ ℝ)
8786adantr 480 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → 𝑧 ∈ ℝ)
88 ltletr 10167 . . . . . . . . . 10 (((1st ‘(𝐺𝑚)) ∈ ℝ ∧ (1st ‘(𝐹𝑚)) ∈ ℝ ∧ 𝑧 ∈ ℝ) → (((1st ‘(𝐺𝑚)) < (1st ‘(𝐹𝑚)) ∧ (1st ‘(𝐹𝑚)) ≤ 𝑧) → (1st ‘(𝐺𝑚)) < 𝑧))
8984, 85, 87, 88syl3anc 1366 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (((1st ‘(𝐺𝑚)) < (1st ‘(𝐹𝑚)) ∧ (1st ‘(𝐹𝑚)) ≤ 𝑧) → (1st ‘(𝐺𝑚)) < 𝑧))
9080, 89mpand 711 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) ≤ 𝑧 → (1st ‘(𝐺𝑚)) < 𝑧))
9169simp2d 1094 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) ∈ ℝ)
9291, 77ltaddrpd 11943 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) < ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
9362fveq2d 6233 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐺𝑚)) = (2nd ‘⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩))
9464, 65op2nd 7219 . . . . . . . . . . . 12 (2nd ‘⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))
9593, 94syl6eq 2701 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐺𝑚)) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
9692, 95breqtrrd 4713 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) < (2nd ‘(𝐺𝑚)))
9796adantlr 751 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) < (2nd ‘(𝐺𝑚)))
9891adantlr 751 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) ∈ ℝ)
9982simp2d 1094 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐺𝑚)) ∈ ℝ)
10099adantlr 751 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐺𝑚)) ∈ ℝ)
101 lelttr 10166 . . . . . . . . . 10 ((𝑧 ∈ ℝ ∧ (2nd ‘(𝐹𝑚)) ∈ ℝ ∧ (2nd ‘(𝐺𝑚)) ∈ ℝ) → ((𝑧 ≤ (2nd ‘(𝐹𝑚)) ∧ (2nd ‘(𝐹𝑚)) < (2nd ‘(𝐺𝑚))) → 𝑧 < (2nd ‘(𝐺𝑚))))
10287, 98, 100, 101syl3anc 1366 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → ((𝑧 ≤ (2nd ‘(𝐹𝑚)) ∧ (2nd ‘(𝐹𝑚)) < (2nd ‘(𝐺𝑚))) → 𝑧 < (2nd ‘(𝐺𝑚))))
10397, 102mpan2d 710 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (𝑧 ≤ (2nd ‘(𝐹𝑚)) → 𝑧 < (2nd ‘(𝐺𝑚))))
10490, 103anim12d 585 . . . . . . 7 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚))) → ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
105104reximdva 3046 . . . . . 6 ((𝜑𝑧𝐴) → (∃𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚))) → ∃𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
106105ralimdva 2991 . . . . 5 (𝜑 → (∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚))) → ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
107 ovolficc 23283 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ([,] ∘ 𝐹) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚)))))
1085, 2, 107syl2anc 694 . . . . 5 (𝜑 → (𝐴 ran ([,] ∘ 𝐹) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚)))))
109 ovolfioo 23282 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐺) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
1105, 37, 109syl2anc 694 . . . . 5 (𝜑 → (𝐴 ran ((,) ∘ 𝐺) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
111106, 108, 1103imtr4d 283 . . . 4 (𝜑 → (𝐴 ran ([,] ∘ 𝐹) → 𝐴 ran ((,) ∘ 𝐺)))
1121, 111mpd 15 . . 3 (𝜑𝐴 ran ((,) ∘ 𝐺))
11339ovollb 23293 . . 3 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ((,) ∘ 𝐺)) → (vol*‘𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
11437, 112, 113syl2anc 694 . 2 (𝜑 → (vol*‘𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
11539fveq1i 6230 . . . . . . 7 (𝑇𝑘) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘)
116 fzfid 12812 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (1...𝑘) ∈ Fin)
117 rge0ssre 12318 . . . . . . . . . . 11 (0[,)+∞) ⊆ ℝ
118 eqid 2651 . . . . . . . . . . . . . . 15 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
119118ovolfsf 23286 . . . . . . . . . . . . . 14 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
1202, 119syl 17 . . . . . . . . . . . . 13 (𝜑 → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
121120adantr 480 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
122 elfznn 12408 . . . . . . . . . . . 12 (𝑚 ∈ (1...𝑘) → 𝑚 ∈ ℕ)
123 ffvelrn 6397 . . . . . . . . . . . 12 ((((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ (0[,)+∞))
124121, 122, 123syl2an 493 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ (0[,)+∞))
125117, 124sseldi 3634 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ ℝ)
126125recnd 10106 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ ℂ)
12711adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → 𝐵 ∈ ℝ+)
128127, 76rpdivcld 11927 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝐵 / (2↑𝑚)) ∈ ℝ+)
129128rpcnd 11912 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (𝐵 / (2↑𝑚)) ∈ ℂ)
130122, 129sylan2 490 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...𝑘)) → (𝐵 / (2↑𝑚)) ∈ ℂ)
131130adantlr 751 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (𝐵 / (2↑𝑚)) ∈ ℂ)
132116, 126, 131fsumadd 14514 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = (Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) + Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚))))
13338ovolfsval 23285 . . . . . . . . . . . . 13 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((2nd ‘(𝐺𝑚)) − (1st ‘(𝐺𝑚))))
13437, 133sylan 487 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((2nd ‘(𝐺𝑚)) − (1st ‘(𝐺𝑚))))
13591recnd 10106 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) ∈ ℂ)
13677rpcnd 11912 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑚)) ∈ ℂ)
13770recnd 10106 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐹𝑚)) ∈ ℂ)
138137, 136subcld 10430 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))) ∈ ℂ)
139135, 136, 138addsubassd 10450 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))) = ((2nd ‘(𝐹𝑚)) + (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))))))
14095, 67oveq12d 6708 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((2nd ‘(𝐺𝑚)) − (1st ‘(𝐺𝑚))) = (((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))))
141135, 137, 129subadd23d 10452 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (((2nd ‘(𝐹𝑚)) − (1st ‘(𝐹𝑚))) + (𝐵 / (2↑𝑚))) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹𝑚)))))
142118ovolfsval 23285 . . . . . . . . . . . . . . . 16 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) = ((2nd ‘(𝐹𝑚)) − (1st ‘(𝐹𝑚))))
1432, 142sylan 487 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) = ((2nd ‘(𝐹𝑚)) − (1st ‘(𝐹𝑚))))
144143oveq1d 6705 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = (((2nd ‘(𝐹𝑚)) − (1st ‘(𝐹𝑚))) + (𝐵 / (2↑𝑚))))
145136, 137, 136subsub3d 10460 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))) = ((((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − (1st ‘(𝐹𝑚))))
14671rpcnd 11912 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → (𝐵 / 2) ∈ ℂ)
14775nncnd 11074 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → (2↑𝑚) ∈ ℂ)
14875nnne0d 11103 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → (2↑𝑚) ≠ 0)
149146, 146, 147, 148divdird 10877 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) + (𝐵 / 2)) / (2↑𝑚)) = (((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
150127rpcnd 11912 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚 ∈ ℕ) → 𝐵 ∈ ℂ)
1511502halvesd 11316 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → ((𝐵 / 2) + (𝐵 / 2)) = 𝐵)
152151oveq1d 6705 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) + (𝐵 / 2)) / (2↑𝑚)) = (𝐵 / (2↑𝑚)))
153149, 152eqtr3d 2687 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) = (𝐵 / (2↑𝑚)))
154153oveq1d 6705 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → ((((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − (1st ‘(𝐹𝑚))) = ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹𝑚))))
155145, 154eqtrd 2685 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))) = ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹𝑚))))
156155oveq2d 6706 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((2nd ‘(𝐹𝑚)) + (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))))) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹𝑚)))))
157141, 144, 1563eqtr4d 2695 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = ((2nd ‘(𝐹𝑚)) + (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))))))
158139, 140, 1573eqtr4d 2695 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((2nd ‘(𝐺𝑚)) − (1st ‘(𝐺𝑚))) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
159134, 158eqtrd 2685 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
160122, 159sylan2 490 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
161160adantlr 751 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
162 simpr 476 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
163 nnuz 11761 . . . . . . . . . 10 ℕ = (ℤ‘1)
164162, 163syl6eleq 2740 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
165126, 131addcld 10097 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) ∈ ℂ)
166161, 164, 165fsumser 14505 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘))
167 eqidd 2652 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) = (((abs ∘ − ) ∘ 𝐹)‘𝑚))
168167, 164, 126fsumser 14505 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘))
169 ovollb2.1 . . . . . . . . . . 11 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
170169fveq1i 6230 . . . . . . . . . 10 (𝑆𝑘) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘)
171168, 170syl6eqr 2703 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) = (𝑆𝑘))
17211adantr 480 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → 𝐵 ∈ ℝ+)
173172rpcnd 11912 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝐵 ∈ ℂ)
174 geo2sum 14648 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ 𝐵 ∈ ℂ) → Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚)) = (𝐵 − (𝐵 / (2↑𝑘))))
175162, 173, 174syl2anc 694 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚)) = (𝐵 − (𝐵 / (2↑𝑘))))
176171, 175oveq12d 6708 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) + Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚))) = ((𝑆𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))))
177132, 166, 1763eqtr3d 2693 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) = ((𝑆𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))))
178115, 177syl5eq 2697 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝑇𝑘) = ((𝑆𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))))
179118, 169ovolsf 23287 . . . . . . . . . 10 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
1802, 179syl 17 . . . . . . . . 9 (𝜑𝑆:ℕ⟶(0[,)+∞))
181180ffvelrnda 6399 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ (0[,)+∞))
182117, 181sseldi 3634 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ℝ)
183172rpred 11910 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐵 ∈ ℝ)
184 nnnn0 11337 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
185184adantl 481 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ0)
186 nnexpcl 12913 . . . . . . . . . . . 12 ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ)
18714, 185, 186sylancr 696 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (2↑𝑘) ∈ ℕ)
188187nnrpd 11908 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (2↑𝑘) ∈ ℝ+)
189172, 188rpdivcld 11927 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐵 / (2↑𝑘)) ∈ ℝ+)
190189rpred 11910 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐵 / (2↑𝑘)) ∈ ℝ)
191183, 190resubcld 10496 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) ∈ ℝ)
19248adantr 480 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
193 frn 6091 . . . . . . . . . . 11 (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
194180, 193syl 17 . . . . . . . . . 10 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
195194, 44syl6ss 3648 . . . . . . . . 9 (𝜑 → ran 𝑆 ⊆ ℝ*)
196195adantr 480 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → ran 𝑆 ⊆ ℝ*)
197 ffn 6083 . . . . . . . . . 10 (𝑆:ℕ⟶(0[,)+∞) → 𝑆 Fn ℕ)
198180, 197syl 17 . . . . . . . . 9 (𝜑𝑆 Fn ℕ)
199 fnfvelrn 6396 . . . . . . . . 9 ((𝑆 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ran 𝑆)
200198, 199sylan 487 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ran 𝑆)
201 supxrub 12192 . . . . . . . 8 ((ran 𝑆 ⊆ ℝ* ∧ (𝑆𝑘) ∈ ran 𝑆) → (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
202196, 200, 201syl2anc 694 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
203183, 189ltsubrpd 11942 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) < 𝐵)
204191, 183, 203ltled 10223 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) ≤ 𝐵)
205182, 191, 192, 183, 202, 204le2addd 10684 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ((𝑆𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
206178, 205eqbrtrd 4707 . . . . 5 ((𝜑𝑘 ∈ ℕ) → (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
207206ralrimiva 2995 . . . 4 (𝜑 → ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
208 ffn 6083 . . . . 5 (𝑇:ℕ⟶(0[,)+∞) → 𝑇 Fn ℕ)
209 breq1 4688 . . . . . 6 (𝑦 = (𝑇𝑘) → (𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
210209ralrn 6402 . . . . 5 (𝑇 Fn ℕ → (∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
21141, 208, 2103syl 18 . . . 4 (𝜑 → (∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
212207, 211mpbird 247 . . 3 (𝜑 → ∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
213 supxrleub 12194 . . . 4 ((ran 𝑇 ⊆ ℝ* ∧ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ∈ ℝ*) → (sup(ran 𝑇, ℝ*, < ) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ ∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
21445, 51, 213syl2anc 694 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ ∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
215212, 214mpbird 247 . 2 (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
2167, 47, 51, 114, 215xrletrd 12031 1 (𝜑 → (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  cin 3606  wss 3607  cop 4216   cuni 4468   class class class wbr 4685  cmpt 4762   × cxp 5141  ran crn 5144  ccom 5147   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  supcsup 8387  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  2c2 11108  0cn0 11330  cuz 11725  +crp 11870  (,)cioo 12213  [,)cico 12215  [,]cicc 12216  ...cfz 12364  seqcseq 12841  cexp 12900  abscabs 14018  Σcsu 14460  vol*covol 23277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-ovol 23279
This theorem is referenced by:  ovollb2  23303
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