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Theorem ovollb2lem 23007
Description: Lemma for ovollb2 23008. (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 22989 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
42, 3syl 17 . . . 4 (𝜑 ran ([,] ∘ 𝐹) ⊆ ℝ)
51, 4sstrd 3577 . . 3 (𝜑𝐴 ⊆ ℝ)
6 ovolcl 22997 . . 3 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
75, 6syl 17 . 2 (𝜑 → (vol*‘𝐴) ∈ ℝ*)
8 ovolfcl 22986 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
92, 8sylan 486 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
109simp1d 1065 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
11 ovollb2.6 . . . . . . . . . . . . . . 15 (𝜑𝐵 ∈ ℝ+)
1211rphalfcld 11718 . . . . . . . . . . . . . 14 (𝜑 → (𝐵 / 2) ∈ ℝ+)
1312adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐵 / 2) ∈ ℝ+)
14 2nn 11034 . . . . . . . . . . . . . . 15 2 ∈ ℕ
15 nnnn0 11148 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
1615adantl 480 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
17 nnexpcl 12692 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
1814, 16, 17sylancr 693 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℕ)
1918nnrpd 11704 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ+)
2013, 19rpdivcld 11723 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑛)) ∈ ℝ+)
2120rpred 11706 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑛)) ∈ ℝ)
2210, 21resubcld 10309 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) ∈ ℝ)
239simp2d 1066 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
2423, 21readdcld 9925 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))) ∈ ℝ)
2510, 20ltsubrpd 11738 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) < (1st ‘(𝐹𝑛)))
269simp3d 1067 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
2723, 20ltaddrpd 11739 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) < ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
2810, 23, 24, 26, 27lelttrd 10046 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) < ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
2922, 10, 24, 25, 28lttrd 10049 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) < ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
3022, 24, 29ltled 10036 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) ≤ ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
31 df-br 4578 . . . . . . . . 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 206 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ ∈ ≤ )
33 opelxpi 5061 . . . . . . . . 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 690 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ ∈ (ℝ × ℝ))
3532, 34elind 3759 . . . . . . 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 6276 . . . . . 6 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
38 eqid 2609 . . . . . . 7 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
39 ovollb2.3 . . . . . . 7 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
4038, 39ovolsf 22992 . . . . . 6 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
4137, 40syl 17 . . . . 5 (𝜑𝑇:ℕ⟶(0[,)+∞))
42 frn 5951 . . . . 5 (𝑇:ℕ⟶(0[,)+∞) → ran 𝑇 ⊆ (0[,)+∞))
4341, 42syl 17 . . . 4 (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
44 icossxr 12087 . . . 4 (0[,)+∞) ⊆ ℝ*
4543, 44syl6ss 3579 . . 3 (𝜑 → ran 𝑇 ⊆ ℝ*)
46 supxrcl 11975 . . 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 11706 . . . 4 (𝜑𝐵 ∈ ℝ)
5048, 49readdcld 9925 . . 3 (𝜑 → (sup(ran 𝑆, ℝ*, < ) + 𝐵) ∈ ℝ)
5150rexrd 9945 . 2 (𝜑 → (sup(ran 𝑆, ℝ*, < ) + 𝐵) ∈ ℝ*)
52 fveq2 6087 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
5352fveq2d 6091 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (1st ‘(𝐹𝑛)) = (1st ‘(𝐹𝑚)))
54 oveq2 6534 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (2↑𝑛) = (2↑𝑚))
5554oveq2d 6542 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → ((𝐵 / 2) / (2↑𝑛)) = ((𝐵 / 2) / (2↑𝑚)))
5653, 55oveq12d 6544 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) = ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))))
5752fveq2d 6091 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (2nd ‘(𝐹𝑛)) = (2nd ‘(𝐹𝑚)))
5857, 55oveq12d 6544 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
5956, 58opeq12d 4342 . . . . . . . . . . . . . . 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 4852 . . . . . . . . . . . . . . 15 ⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩ ∈ V
6159, 36, 60fvmpt 6175 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → (𝐺𝑚) = ⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩)
6261adantl 480 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) = ⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩)
6362fveq2d 6091 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) = (1st ‘⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩))
64 ovex 6554 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))) ∈ V
65 ovex 6554 . . . . . . . . . . . . 13 ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))) ∈ V
6664, 65op1st 7044 . . . . . . . . . . . 12 (1st ‘⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩) = ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))
6763, 66syl6eq 2659 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) = ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))))
68 ovolfcl 22986 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) ∈ ℝ ∧ (2nd ‘(𝐹𝑚)) ∈ ℝ ∧ (1st ‘(𝐹𝑚)) ≤ (2nd ‘(𝐹𝑚))))
692, 68sylan 486 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) ∈ ℝ ∧ (2nd ‘(𝐹𝑚)) ∈ ℝ ∧ (1st ‘(𝐹𝑚)) ≤ (2nd ‘(𝐹𝑚))))
7069simp1d 1065 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐹𝑚)) ∈ ℝ)
7112adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (𝐵 / 2) ∈ ℝ+)
72 nnnn0 11148 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
7372adantl 480 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℕ0)
74 nnexpcl 12692 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ0) → (2↑𝑚) ∈ ℕ)
7514, 73, 74sylancr 693 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (2↑𝑚) ∈ ℕ)
7675nnrpd 11704 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (2↑𝑚) ∈ ℝ+)
7771, 76rpdivcld 11723 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑚)) ∈ ℝ+)
7870, 77ltsubrpd 11738 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))) < (1st ‘(𝐹𝑚)))
7967, 78eqbrtrd 4599 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) < (1st ‘(𝐹𝑚)))
8079adantlr 746 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) < (1st ‘(𝐹𝑚)))
81 ovolfcl 22986 . . . . . . . . . . . . 13 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐺𝑚)) ∈ ℝ ∧ (2nd ‘(𝐺𝑚)) ∈ ℝ ∧ (1st ‘(𝐺𝑚)) ≤ (2nd ‘(𝐺𝑚))))
8237, 81sylan 486 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝐺𝑚)) ∈ ℝ ∧ (2nd ‘(𝐺𝑚)) ∈ ℝ ∧ (1st ‘(𝐺𝑚)) ≤ (2nd ‘(𝐺𝑚))))
8382simp1d 1065 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) ∈ ℝ)
8483adantlr 746 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) ∈ ℝ)
8570adantlr 746 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐹𝑚)) ∈ ℝ)
865sselda 3567 . . . . . . . . . . 11 ((𝜑𝑧𝐴) → 𝑧 ∈ ℝ)
8786adantr 479 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → 𝑧 ∈ ℝ)
88 ltletr 9980 . . . . . . . . . 10 (((1st ‘(𝐺𝑚)) ∈ ℝ ∧ (1st ‘(𝐹𝑚)) ∈ ℝ ∧ 𝑧 ∈ ℝ) → (((1st ‘(𝐺𝑚)) < (1st ‘(𝐹𝑚)) ∧ (1st ‘(𝐹𝑚)) ≤ 𝑧) → (1st ‘(𝐺𝑚)) < 𝑧))
8984, 85, 87, 88syl3anc 1317 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (((1st ‘(𝐺𝑚)) < (1st ‘(𝐹𝑚)) ∧ (1st ‘(𝐹𝑚)) ≤ 𝑧) → (1st ‘(𝐺𝑚)) < 𝑧))
9080, 89mpand 706 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) ≤ 𝑧 → (1st ‘(𝐺𝑚)) < 𝑧))
9169simp2d 1066 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) ∈ ℝ)
9291, 77ltaddrpd 11739 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) < ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
9362fveq2d 6091 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐺𝑚)) = (2nd ‘⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩))
9464, 65op2nd 7045 . . . . . . . . . . . 12 (2nd ‘⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))
9593, 94syl6eq 2659 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐺𝑚)) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
9692, 95breqtrrd 4605 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) < (2nd ‘(𝐺𝑚)))
9796adantlr 746 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) < (2nd ‘(𝐺𝑚)))
9891adantlr 746 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) ∈ ℝ)
9982simp2d 1066 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐺𝑚)) ∈ ℝ)
10099adantlr 746 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐺𝑚)) ∈ ℝ)
101 lelttr 9979 . . . . . . . . . 10 ((𝑧 ∈ ℝ ∧ (2nd ‘(𝐹𝑚)) ∈ ℝ ∧ (2nd ‘(𝐺𝑚)) ∈ ℝ) → ((𝑧 ≤ (2nd ‘(𝐹𝑚)) ∧ (2nd ‘(𝐹𝑚)) < (2nd ‘(𝐺𝑚))) → 𝑧 < (2nd ‘(𝐺𝑚))))
10287, 98, 100, 101syl3anc 1317 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → ((𝑧 ≤ (2nd ‘(𝐹𝑚)) ∧ (2nd ‘(𝐹𝑚)) < (2nd ‘(𝐺𝑚))) → 𝑧 < (2nd ‘(𝐺𝑚))))
10397, 102mpan2d 705 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (𝑧 ≤ (2nd ‘(𝐹𝑚)) → 𝑧 < (2nd ‘(𝐺𝑚))))
10490, 103anim12d 583 . . . . . . 7 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚))) → ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
105104reximdva 2999 . . . . . 6 ((𝜑𝑧𝐴) → (∃𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚))) → ∃𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
106105ralimdva 2944 . . . . 5 (𝜑 → (∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚))) → ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
107 ovolficc 22988 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ([,] ∘ 𝐹) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚)))))
1085, 2, 107syl2anc 690 . . . . 5 (𝜑 → (𝐴 ran ([,] ∘ 𝐹) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚)))))
109 ovolfioo 22987 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐺) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
1105, 37, 109syl2anc 690 . . . . 5 (𝜑 → (𝐴 ran ((,) ∘ 𝐺) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
111106, 108, 1103imtr4d 281 . . . 4 (𝜑 → (𝐴 ran ([,] ∘ 𝐹) → 𝐴 ran ((,) ∘ 𝐺)))
1121, 111mpd 15 . . 3 (𝜑𝐴 ran ((,) ∘ 𝐺))
11339ovollb 22998 . . 3 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ((,) ∘ 𝐺)) → (vol*‘𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
11437, 112, 113syl2anc 690 . 2 (𝜑 → (vol*‘𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
11539fveq1i 6088 . . . . . . 7 (𝑇𝑘) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘)
116 fzfid 12591 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (1...𝑘) ∈ Fin)
117 rge0ssre 12109 . . . . . . . . . . 11 (0[,)+∞) ⊆ ℝ
118 eqid 2609 . . . . . . . . . . . . . . 15 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
119118ovolfsf 22991 . . . . . . . . . . . . . 14 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
1202, 119syl 17 . . . . . . . . . . . . 13 (𝜑 → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
121120adantr 479 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
122 elfznn 12198 . . . . . . . . . . . 12 (𝑚 ∈ (1...𝑘) → 𝑚 ∈ ℕ)
123 ffvelrn 6249 . . . . . . . . . . . 12 ((((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ (0[,)+∞))
124121, 122, 123syl2an 492 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ (0[,)+∞))
125117, 124sseldi 3565 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ ℝ)
126125recnd 9924 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ ℂ)
12711adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → 𝐵 ∈ ℝ+)
128127, 76rpdivcld 11723 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝐵 / (2↑𝑚)) ∈ ℝ+)
129128rpcnd 11708 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (𝐵 / (2↑𝑚)) ∈ ℂ)
130122, 129sylan2 489 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...𝑘)) → (𝐵 / (2↑𝑚)) ∈ ℂ)
131130adantlr 746 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (𝐵 / (2↑𝑚)) ∈ ℂ)
132116, 126, 131fsumadd 14265 . . . . . . . 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 22990 . . . . . . . . . . . . 13 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((2nd ‘(𝐺𝑚)) − (1st ‘(𝐺𝑚))))
13437, 133sylan 486 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((2nd ‘(𝐺𝑚)) − (1st ‘(𝐺𝑚))))
13591recnd 9924 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) ∈ ℂ)
13677rpcnd 11708 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑚)) ∈ ℂ)
13770recnd 9924 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐹𝑚)) ∈ ℂ)
138137, 136subcld 10243 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))) ∈ ℂ)
139135, 136, 138addsubassd 10263 . . . . . . . . . . . . 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 6544 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((2nd ‘(𝐺𝑚)) − (1st ‘(𝐺𝑚))) = (((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))))
141135, 137, 129subadd23d 10265 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (((2nd ‘(𝐹𝑚)) − (1st ‘(𝐹𝑚))) + (𝐵 / (2↑𝑚))) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹𝑚)))))
142118ovolfsval 22990 . . . . . . . . . . . . . . . 16 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) = ((2nd ‘(𝐹𝑚)) − (1st ‘(𝐹𝑚))))
1432, 142sylan 486 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) = ((2nd ‘(𝐹𝑚)) − (1st ‘(𝐹𝑚))))
144143oveq1d 6541 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = (((2nd ‘(𝐹𝑚)) − (1st ‘(𝐹𝑚))) + (𝐵 / (2↑𝑚))))
145136, 137, 136subsub3d 10273 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))) = ((((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − (1st ‘(𝐹𝑚))))
14671rpcnd 11708 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → (𝐵 / 2) ∈ ℂ)
14775nncnd 10885 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → (2↑𝑚) ∈ ℂ)
14875nnne0d 10914 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → (2↑𝑚) ≠ 0)
149146, 146, 147, 148divdird 10690 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) + (𝐵 / 2)) / (2↑𝑚)) = (((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
150127rpcnd 11708 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚 ∈ ℕ) → 𝐵 ∈ ℂ)
1511502halvesd 11127 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → ((𝐵 / 2) + (𝐵 / 2)) = 𝐵)
152151oveq1d 6541 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) + (𝐵 / 2)) / (2↑𝑚)) = (𝐵 / (2↑𝑚)))
153149, 152eqtr3d 2645 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) = (𝐵 / (2↑𝑚)))
154153oveq1d 6541 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → ((((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − (1st ‘(𝐹𝑚))) = ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹𝑚))))
155145, 154eqtrd 2643 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))) = ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹𝑚))))
156155oveq2d 6542 . . . . . . . . . . . . . 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 2653 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = ((2nd ‘(𝐹𝑚)) + (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))))))
158139, 140, 1573eqtr4d 2653 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((2nd ‘(𝐺𝑚)) − (1st ‘(𝐺𝑚))) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
159134, 158eqtrd 2643 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
160122, 159sylan2 489 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
161160adantlr 746 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
162 simpr 475 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
163 nnuz 11557 . . . . . . . . . 10 ℕ = (ℤ‘1)
164162, 163syl6eleq 2697 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
165126, 131addcld 9915 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) ∈ ℂ)
166161, 164, 165fsumser 14256 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘))
167 eqidd 2610 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) = (((abs ∘ − ) ∘ 𝐹)‘𝑚))
168167, 164, 126fsumser 14256 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘))
169 ovollb2.1 . . . . . . . . . . 11 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
170169fveq1i 6088 . . . . . . . . . 10 (𝑆𝑘) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘)
171168, 170syl6eqr 2661 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) = (𝑆𝑘))
17211adantr 479 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → 𝐵 ∈ ℝ+)
173172rpcnd 11708 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝐵 ∈ ℂ)
174 geo2sum 14391 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ 𝐵 ∈ ℂ) → Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚)) = (𝐵 − (𝐵 / (2↑𝑘))))
175162, 173, 174syl2anc 690 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚)) = (𝐵 − (𝐵 / (2↑𝑘))))
176171, 175oveq12d 6544 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) + Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚))) = ((𝑆𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))))
177132, 166, 1763eqtr3d 2651 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) = ((𝑆𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))))
178115, 177syl5eq 2655 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝑇𝑘) = ((𝑆𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))))
179118, 169ovolsf 22992 . . . . . . . . . 10 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
1802, 179syl 17 . . . . . . . . 9 (𝜑𝑆:ℕ⟶(0[,)+∞))
181180ffvelrnda 6251 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ (0[,)+∞))
182117, 181sseldi 3565 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ℝ)
183172rpred 11706 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐵 ∈ ℝ)
184 nnnn0 11148 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
185184adantl 480 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ0)
186 nnexpcl 12692 . . . . . . . . . . . 12 ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ)
18714, 185, 186sylancr 693 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (2↑𝑘) ∈ ℕ)
188187nnrpd 11704 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (2↑𝑘) ∈ ℝ+)
189172, 188rpdivcld 11723 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐵 / (2↑𝑘)) ∈ ℝ+)
190189rpred 11706 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐵 / (2↑𝑘)) ∈ ℝ)
191183, 190resubcld 10309 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) ∈ ℝ)
19248adantr 479 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
193 frn 5951 . . . . . . . . . . 11 (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
194180, 193syl 17 . . . . . . . . . 10 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
195194, 44syl6ss 3579 . . . . . . . . 9 (𝜑 → ran 𝑆 ⊆ ℝ*)
196195adantr 479 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → ran 𝑆 ⊆ ℝ*)
197 ffn 5943 . . . . . . . . . 10 (𝑆:ℕ⟶(0[,)+∞) → 𝑆 Fn ℕ)
198180, 197syl 17 . . . . . . . . 9 (𝜑𝑆 Fn ℕ)
199 fnfvelrn 6248 . . . . . . . . 9 ((𝑆 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ran 𝑆)
200198, 199sylan 486 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ran 𝑆)
201 supxrub 11984 . . . . . . . 8 ((ran 𝑆 ⊆ ℝ* ∧ (𝑆𝑘) ∈ ran 𝑆) → (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
202196, 200, 201syl2anc 690 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
203183, 189ltsubrpd 11738 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) < 𝐵)
204191, 183, 203ltled 10036 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) ≤ 𝐵)
205182, 191, 192, 183, 202, 204le2addd 10497 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ((𝑆𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
206178, 205eqbrtrd 4599 . . . . 5 ((𝜑𝑘 ∈ ℕ) → (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
207206ralrimiva 2948 . . . 4 (𝜑 → ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
208 ffn 5943 . . . . 5 (𝑇:ℕ⟶(0[,)+∞) → 𝑇 Fn ℕ)
209 breq1 4580 . . . . . 6 (𝑦 = (𝑇𝑘) → (𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
210209ralrn 6254 . . . . 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 245 . . 3 (𝜑 → ∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
213 supxrleub 11986 . . . 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 690 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ ∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
215212, 214mpbird 245 . 2 (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
2167, 47, 51, 114, 215xrletrd 11830 1 (𝜑 → (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  wrex 2896  cin 3538  wss 3539  cop 4130   cuni 4366   class class class wbr 4577  cmpt 4637   × cxp 5025  ran crn 5028  ccom 5031   Fn wfn 5784  wf 5785  cfv 5789  (class class class)co 6526  1st c1st 7034  2nd c2nd 7035  supcsup 8206  cc 9790  cr 9791  0cc0 9792  1c1 9793   + caddc 9795  +∞cpnf 9927  *cxr 9929   < clt 9930  cle 9931  cmin 10117   / cdiv 10535  cn 10869  2c2 10919  0cn0 11141  cuz 11521  +crp 11666  (,)cioo 12004  [,)cico 12006  [,]cicc 12007  ...cfz 12154  seqcseq 12620  cexp 12679  abscabs 13770  Σcsu 14212  vol*covol 22982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-se 4987  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-isom 5798  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-sup 8208  df-inf 8209  df-oi 8275  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10536  df-nn 10870  df-2 10928  df-3 10929  df-n0 11142  df-z 11213  df-uz 11522  df-rp 11667  df-ioo 12008  df-ico 12010  df-icc 12011  df-fz 12155  df-fzo 12292  df-seq 12621  df-exp 12680  df-hash 12937  df-cj 13635  df-re 13636  df-im 13637  df-sqrt 13771  df-abs 13772  df-clim 14015  df-sum 14213  df-ovol 22984
This theorem is referenced by:  ovollb2  23008
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