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Theorem ovollb2lem 23703
 Description: Lemma for ovollb2 23704. (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 23684 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
42, 3syl 17 . . . 4 (𝜑 ran ([,] ∘ 𝐹) ⊆ ℝ)
51, 4sstrd 3831 . . 3 (𝜑𝐴 ⊆ ℝ)
6 ovolcl 23693 . . 3 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
75, 6syl 17 . 2 (𝜑 → (vol*‘𝐴) ∈ ℝ*)
8 ovolfcl 23681 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
92, 8sylan 575 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
109simp1d 1133 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
11 ovollb2.6 . . . . . . . . . . . . . . 15 (𝜑𝐵 ∈ ℝ+)
1211rphalfcld 12198 . . . . . . . . . . . . . 14 (𝜑 → (𝐵 / 2) ∈ ℝ+)
1312adantr 474 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐵 / 2) ∈ ℝ+)
14 2nn 11453 . . . . . . . . . . . . . . 15 2 ∈ ℕ
15 nnnn0 11655 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
1615adantl 475 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
17 nnexpcl 13196 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
1814, 16, 17sylancr 581 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℕ)
1918nnrpd 12184 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ+)
2013, 19rpdivcld 12203 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑛)) ∈ ℝ+)
2120rpred 12186 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑛)) ∈ ℝ)
2210, 21resubcld 10806 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) ∈ ℝ)
239simp2d 1134 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
2423, 21readdcld 10408 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))) ∈ ℝ)
2510, 20ltsubrpd 12218 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) < (1st ‘(𝐹𝑛)))
269simp3d 1135 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
2723, 20ltaddrpd 12219 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) < ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
2810, 23, 24, 26, 27lelttrd 10536 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) < ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
2922, 10, 24, 25, 28lttrd 10539 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) < ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
3022, 24, 29ltled 10526 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) ≤ ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
31 df-br 4889 . . . . . . . . 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 210 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ ∈ ≤ )
33 opelxpi 5394 . . . . . . . . 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 579 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ ∈ (ℝ × ℝ))
3532, 34elind 4021 . . . . . . 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 6650 . . . . . 6 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
38 eqid 2778 . . . . . . 7 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
39 ovollb2.3 . . . . . . 7 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
4038, 39ovolsf 23687 . . . . . 6 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
4137, 40syl 17 . . . . 5 (𝜑𝑇:ℕ⟶(0[,)+∞))
4241frnd 6300 . . . 4 (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
43 icossxr 12575 . . . 4 (0[,)+∞) ⊆ ℝ*
4442, 43syl6ss 3833 . . 3 (𝜑 → ran 𝑇 ⊆ ℝ*)
45 supxrcl 12462 . . 3 (ran 𝑇 ⊆ ℝ* → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
4644, 45syl 17 . 2 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
47 ovollb2.7 . . . 4 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
4811rpred 12186 . . . 4 (𝜑𝐵 ∈ ℝ)
4947, 48readdcld 10408 . . 3 (𝜑 → (sup(ran 𝑆, ℝ*, < ) + 𝐵) ∈ ℝ)
5049rexrd 10428 . 2 (𝜑 → (sup(ran 𝑆, ℝ*, < ) + 𝐵) ∈ ℝ*)
51 2fveq3 6453 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (1st ‘(𝐹𝑛)) = (1st ‘(𝐹𝑚)))
52 oveq2 6932 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (2↑𝑛) = (2↑𝑚))
5352oveq2d 6940 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → ((𝐵 / 2) / (2↑𝑛)) = ((𝐵 / 2) / (2↑𝑚)))
5451, 53oveq12d 6942 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) = ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))))
55 2fveq3 6453 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (2nd ‘(𝐹𝑛)) = (2nd ‘(𝐹𝑚)))
5655, 53oveq12d 6942 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
5754, 56opeq12d 4646 . . . . . . . . . . . . . . 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)))⟩)
58 opex 5166 . . . . . . . . . . . . . . 15 ⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩ ∈ V
5957, 36, 58fvmpt 6544 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → (𝐺𝑚) = ⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩)
6059adantl 475 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) = ⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩)
6160fveq2d 6452 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) = (1st ‘⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩))
62 ovex 6956 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))) ∈ V
63 ovex 6956 . . . . . . . . . . . . 13 ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))) ∈ V
6462, 63op1st 7455 . . . . . . . . . . . 12 (1st ‘⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩) = ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))
6561, 64syl6eq 2830 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) = ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))))
66 ovolfcl 23681 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) ∈ ℝ ∧ (2nd ‘(𝐹𝑚)) ∈ ℝ ∧ (1st ‘(𝐹𝑚)) ≤ (2nd ‘(𝐹𝑚))))
672, 66sylan 575 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) ∈ ℝ ∧ (2nd ‘(𝐹𝑚)) ∈ ℝ ∧ (1st ‘(𝐹𝑚)) ≤ (2nd ‘(𝐹𝑚))))
6867simp1d 1133 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐹𝑚)) ∈ ℝ)
6912adantr 474 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (𝐵 / 2) ∈ ℝ+)
70 nnnn0 11655 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
7170adantl 475 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℕ0)
72 nnexpcl 13196 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ0) → (2↑𝑚) ∈ ℕ)
7314, 71, 72sylancr 581 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (2↑𝑚) ∈ ℕ)
7473nnrpd 12184 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (2↑𝑚) ∈ ℝ+)
7569, 74rpdivcld 12203 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑚)) ∈ ℝ+)
7668, 75ltsubrpd 12218 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))) < (1st ‘(𝐹𝑚)))
7765, 76eqbrtrd 4910 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) < (1st ‘(𝐹𝑚)))
7877adantlr 705 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) < (1st ‘(𝐹𝑚)))
79 ovolfcl 23681 . . . . . . . . . . . . 13 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐺𝑚)) ∈ ℝ ∧ (2nd ‘(𝐺𝑚)) ∈ ℝ ∧ (1st ‘(𝐺𝑚)) ≤ (2nd ‘(𝐺𝑚))))
8037, 79sylan 575 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝐺𝑚)) ∈ ℝ ∧ (2nd ‘(𝐺𝑚)) ∈ ℝ ∧ (1st ‘(𝐺𝑚)) ≤ (2nd ‘(𝐺𝑚))))
8180simp1d 1133 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) ∈ ℝ)
8281adantlr 705 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) ∈ ℝ)
8368adantlr 705 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐹𝑚)) ∈ ℝ)
845sselda 3821 . . . . . . . . . . 11 ((𝜑𝑧𝐴) → 𝑧 ∈ ℝ)
8584adantr 474 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → 𝑧 ∈ ℝ)
86 ltletr 10470 . . . . . . . . . 10 (((1st ‘(𝐺𝑚)) ∈ ℝ ∧ (1st ‘(𝐹𝑚)) ∈ ℝ ∧ 𝑧 ∈ ℝ) → (((1st ‘(𝐺𝑚)) < (1st ‘(𝐹𝑚)) ∧ (1st ‘(𝐹𝑚)) ≤ 𝑧) → (1st ‘(𝐺𝑚)) < 𝑧))
8782, 83, 85, 86syl3anc 1439 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (((1st ‘(𝐺𝑚)) < (1st ‘(𝐹𝑚)) ∧ (1st ‘(𝐹𝑚)) ≤ 𝑧) → (1st ‘(𝐺𝑚)) < 𝑧))
8878, 87mpand 685 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) ≤ 𝑧 → (1st ‘(𝐺𝑚)) < 𝑧))
8967simp2d 1134 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) ∈ ℝ)
9089, 75ltaddrpd 12219 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) < ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
9160fveq2d 6452 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐺𝑚)) = (2nd ‘⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩))
9262, 63op2nd 7456 . . . . . . . . . . . 12 (2nd ‘⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))
9391, 92syl6eq 2830 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐺𝑚)) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
9490, 93breqtrrd 4916 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) < (2nd ‘(𝐺𝑚)))
9594adantlr 705 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) < (2nd ‘(𝐺𝑚)))
9689adantlr 705 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) ∈ ℝ)
9780simp2d 1134 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐺𝑚)) ∈ ℝ)
9897adantlr 705 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐺𝑚)) ∈ ℝ)
99 lelttr 10469 . . . . . . . . . 10 ((𝑧 ∈ ℝ ∧ (2nd ‘(𝐹𝑚)) ∈ ℝ ∧ (2nd ‘(𝐺𝑚)) ∈ ℝ) → ((𝑧 ≤ (2nd ‘(𝐹𝑚)) ∧ (2nd ‘(𝐹𝑚)) < (2nd ‘(𝐺𝑚))) → 𝑧 < (2nd ‘(𝐺𝑚))))
10085, 96, 98, 99syl3anc 1439 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → ((𝑧 ≤ (2nd ‘(𝐹𝑚)) ∧ (2nd ‘(𝐹𝑚)) < (2nd ‘(𝐺𝑚))) → 𝑧 < (2nd ‘(𝐺𝑚))))
10195, 100mpan2d 684 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (𝑧 ≤ (2nd ‘(𝐹𝑚)) → 𝑧 < (2nd ‘(𝐺𝑚))))
10288, 101anim12d 602 . . . . . . 7 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚))) → ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
103102reximdva 3198 . . . . . 6 ((𝜑𝑧𝐴) → (∃𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚))) → ∃𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
104103ralimdva 3144 . . . . 5 (𝜑 → (∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚))) → ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
105 ovolficc 23683 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ([,] ∘ 𝐹) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚)))))
1065, 2, 105syl2anc 579 . . . . 5 (𝜑 → (𝐴 ran ([,] ∘ 𝐹) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚)))))
107 ovolfioo 23682 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐺) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
1085, 37, 107syl2anc 579 . . . . 5 (𝜑 → (𝐴 ran ((,) ∘ 𝐺) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
109104, 106, 1083imtr4d 286 . . . 4 (𝜑 → (𝐴 ran ([,] ∘ 𝐹) → 𝐴 ran ((,) ∘ 𝐺)))
1101, 109mpd 15 . . 3 (𝜑𝐴 ran ((,) ∘ 𝐺))
11139ovollb 23694 . . 3 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ((,) ∘ 𝐺)) → (vol*‘𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
11237, 110, 111syl2anc 579 . 2 (𝜑 → (vol*‘𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
11339fveq1i 6449 . . . . . . 7 (𝑇𝑘) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘)
114 fzfid 13096 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (1...𝑘) ∈ Fin)
115 rge0ssre 12599 . . . . . . . . . . 11 (0[,)+∞) ⊆ ℝ
116 eqid 2778 . . . . . . . . . . . . . . 15 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
117116ovolfsf 23686 . . . . . . . . . . . . . 14 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
1182, 117syl 17 . . . . . . . . . . . . 13 (𝜑 → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
119118adantr 474 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
120 elfznn 12692 . . . . . . . . . . . 12 (𝑚 ∈ (1...𝑘) → 𝑚 ∈ ℕ)
121 ffvelrn 6623 . . . . . . . . . . . 12 ((((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ (0[,)+∞))
122119, 120, 121syl2an 589 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ (0[,)+∞))
123115, 122sseldi 3819 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ ℝ)
124123recnd 10407 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ ℂ)
12511adantr 474 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → 𝐵 ∈ ℝ+)
126125, 74rpdivcld 12203 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝐵 / (2↑𝑚)) ∈ ℝ+)
127126rpcnd 12188 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (𝐵 / (2↑𝑚)) ∈ ℂ)
128120, 127sylan2 586 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...𝑘)) → (𝐵 / (2↑𝑚)) ∈ ℂ)
129128adantlr 705 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (𝐵 / (2↑𝑚)) ∈ ℂ)
130114, 124, 129fsumadd 14886 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = (Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) + Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚))))
13138ovolfsval 23685 . . . . . . . . . . . . 13 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((2nd ‘(𝐺𝑚)) − (1st ‘(𝐺𝑚))))
13237, 131sylan 575 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((2nd ‘(𝐺𝑚)) − (1st ‘(𝐺𝑚))))
13389recnd 10407 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) ∈ ℂ)
13475rpcnd 12188 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑚)) ∈ ℂ)
13568recnd 10407 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐹𝑚)) ∈ ℂ)
136135, 134subcld 10736 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))) ∈ ℂ)
137133, 134, 136addsubassd 10756 . . . . . . . . . . . . 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))))))
13893, 65oveq12d 6942 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((2nd ‘(𝐺𝑚)) − (1st ‘(𝐺𝑚))) = (((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))))
139133, 135, 127subadd23d 10758 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (((2nd ‘(𝐹𝑚)) − (1st ‘(𝐹𝑚))) + (𝐵 / (2↑𝑚))) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹𝑚)))))
140116ovolfsval 23685 . . . . . . . . . . . . . . . 16 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) = ((2nd ‘(𝐹𝑚)) − (1st ‘(𝐹𝑚))))
1412, 140sylan 575 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) = ((2nd ‘(𝐹𝑚)) − (1st ‘(𝐹𝑚))))
142141oveq1d 6939 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = (((2nd ‘(𝐹𝑚)) − (1st ‘(𝐹𝑚))) + (𝐵 / (2↑𝑚))))
143134, 135, 134subsub3d 10766 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))) = ((((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − (1st ‘(𝐹𝑚))))
14469rpcnd 12188 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → (𝐵 / 2) ∈ ℂ)
14573nncnd 11397 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → (2↑𝑚) ∈ ℂ)
14673nnne0d 11430 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → (2↑𝑚) ≠ 0)
147144, 144, 145, 146divdird 11192 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) + (𝐵 / 2)) / (2↑𝑚)) = (((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
148125rpcnd 12188 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚 ∈ ℕ) → 𝐵 ∈ ℂ)
1491482halvesd 11633 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → ((𝐵 / 2) + (𝐵 / 2)) = 𝐵)
150149oveq1d 6939 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) + (𝐵 / 2)) / (2↑𝑚)) = (𝐵 / (2↑𝑚)))
151147, 150eqtr3d 2816 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) = (𝐵 / (2↑𝑚)))
152151oveq1d 6939 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → ((((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − (1st ‘(𝐹𝑚))) = ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹𝑚))))
153143, 152eqtrd 2814 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))) = ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹𝑚))))
154153oveq2d 6940 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((2nd ‘(𝐹𝑚)) + (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))))) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹𝑚)))))
155139, 142, 1543eqtr4d 2824 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = ((2nd ‘(𝐹𝑚)) + (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))))))
156137, 138, 1553eqtr4d 2824 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((2nd ‘(𝐺𝑚)) − (1st ‘(𝐺𝑚))) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
157132, 156eqtrd 2814 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
158120, 157sylan2 586 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
159158adantlr 705 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
160 simpr 479 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
161 nnuz 12034 . . . . . . . . . 10 ℕ = (ℤ‘1)
162160, 161syl6eleq 2869 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
163124, 129addcld 10398 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) ∈ ℂ)
164159, 162, 163fsumser 14877 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘))
165 eqidd 2779 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) = (((abs ∘ − ) ∘ 𝐹)‘𝑚))
166165, 162, 124fsumser 14877 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘))
167 ovollb2.1 . . . . . . . . . . 11 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
168167fveq1i 6449 . . . . . . . . . 10 (𝑆𝑘) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘)
169166, 168syl6eqr 2832 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) = (𝑆𝑘))
17011adantr 474 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → 𝐵 ∈ ℝ+)
171170rpcnd 12188 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝐵 ∈ ℂ)
172 geo2sum 15017 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ 𝐵 ∈ ℂ) → Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚)) = (𝐵 − (𝐵 / (2↑𝑘))))
173160, 171, 172syl2anc 579 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚)) = (𝐵 − (𝐵 / (2↑𝑘))))
174169, 173oveq12d 6942 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) + Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚))) = ((𝑆𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))))
175130, 164, 1743eqtr3d 2822 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) = ((𝑆𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))))
176113, 175syl5eq 2826 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝑇𝑘) = ((𝑆𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))))
177116, 167ovolsf 23687 . . . . . . . . . 10 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
1782, 177syl 17 . . . . . . . . 9 (𝜑𝑆:ℕ⟶(0[,)+∞))
179178ffvelrnda 6625 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ (0[,)+∞))
180115, 179sseldi 3819 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ℝ)
181170rpred 12186 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐵 ∈ ℝ)
182 nnnn0 11655 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
183182adantl 475 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ0)
184 nnexpcl 13196 . . . . . . . . . . . 12 ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ)
18514, 183, 184sylancr 581 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (2↑𝑘) ∈ ℕ)
186185nnrpd 12184 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (2↑𝑘) ∈ ℝ+)
187170, 186rpdivcld 12203 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐵 / (2↑𝑘)) ∈ ℝ+)
188187rpred 12186 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐵 / (2↑𝑘)) ∈ ℝ)
189181, 188resubcld 10806 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) ∈ ℝ)
19047adantr 474 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
191178frnd 6300 . . . . . . . . . 10 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
192191, 43syl6ss 3833 . . . . . . . . 9 (𝜑 → ran 𝑆 ⊆ ℝ*)
193192adantr 474 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → ran 𝑆 ⊆ ℝ*)
194178ffnd 6294 . . . . . . . . 9 (𝜑𝑆 Fn ℕ)
195 fnfvelrn 6622 . . . . . . . . 9 ((𝑆 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ran 𝑆)
196194, 195sylan 575 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ran 𝑆)
197 supxrub 12471 . . . . . . . 8 ((ran 𝑆 ⊆ ℝ* ∧ (𝑆𝑘) ∈ ran 𝑆) → (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
198193, 196, 197syl2anc 579 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
199181, 187ltsubrpd 12218 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) < 𝐵)
200189, 181, 199ltled 10526 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) ≤ 𝐵)
201180, 189, 190, 181, 198, 200le2addd 10997 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ((𝑆𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
202176, 201eqbrtrd 4910 . . . . 5 ((𝜑𝑘 ∈ ℕ) → (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
203202ralrimiva 3148 . . . 4 (𝜑 → ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
204 ffn 6293 . . . . 5 (𝑇:ℕ⟶(0[,)+∞) → 𝑇 Fn ℕ)
205 breq1 4891 . . . . . 6 (𝑦 = (𝑇𝑘) → (𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
206205ralrn 6628 . . . . 5 (𝑇 Fn ℕ → (∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
20741, 204, 2063syl 18 . . . 4 (𝜑 → (∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
208203, 207mpbird 249 . . 3 (𝜑 → ∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
209 supxrleub 12473 . . . 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)))
21044, 50, 209syl2anc 579 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ ∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
211208, 210mpbird 249 . 2 (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
2127, 46, 50, 112, 211xrletrd 12310 1 (𝜑 → (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ∀wral 3090  ∃wrex 3091   ∩ cin 3791   ⊆ wss 3792  ⟨cop 4404  ∪ cuni 4673   class class class wbr 4888   ↦ cmpt 4967   × cxp 5355  ran crn 5358   ∘ ccom 5361   Fn wfn 6132  ⟶wf 6133  ‘cfv 6137  (class class class)co 6924  1st c1st 7445  2nd c2nd 7446  supcsup 8636  ℂcc 10272  ℝcr 10273  0cc0 10274  1c1 10275   + caddc 10277  +∞cpnf 10410  ℝ*cxr 10412   < clt 10413   ≤ cle 10414   − cmin 10608   / cdiv 11035  ℕcn 11379  2c2 11435  ℕ0cn0 11647  ℤ≥cuz 11997  ℝ+crp 12142  (,)cioo 12492  [,)cico 12494  [,]cicc 12495  ...cfz 12648  seqcseq 13124  ↑cexp 13183  abscabs 14387  Σcsu 14833  vol*covol 23677 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-inf2 8837  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351  ax-pre-sup 10352 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-se 5317  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-isom 6146  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-er 8028  df-map 8144  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-sup 8638  df-inf 8639  df-oi 8706  df-card 9100  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-div 11036  df-nn 11380  df-2 11443  df-3 11444  df-n0 11648  df-z 11734  df-uz 11998  df-rp 12143  df-ioo 12496  df-ico 12498  df-icc 12499  df-fz 12649  df-fzo 12790  df-seq 13125  df-exp 13184  df-hash 13442  df-cj 14252  df-re 14253  df-im 14254  df-sqrt 14388  df-abs 14389  df-clim 14636  df-sum 14834  df-ovol 23679 This theorem is referenced by:  ovollb2  23704
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