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Theorem ovolscalem1 23088
 Description: Lemma for ovolsca 23090. (Contributed by Mario Carneiro, 6-Apr-2015.)
Hypotheses
Ref Expression
ovolsca.1 (𝜑𝐴 ⊆ ℝ)
ovolsca.2 (𝜑𝐶 ∈ ℝ+)
ovolsca.3 (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴})
ovolsca.4 (𝜑 → (vol*‘𝐴) ∈ ℝ)
ovolsca.5 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
ovolsca.6 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩)
ovolsca.7 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
ovolsca.8 (𝜑𝐴 ran ((,) ∘ 𝐹))
ovolsca.9 (𝜑𝑅 ∈ ℝ+)
ovolsca.10 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
Assertion
Ref Expression
ovolscalem1 (𝜑 → (vol*‘𝐵) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
Distinct variable groups:   𝑥,𝑛,𝐴   𝐵,𝑛   𝑛,𝐹,𝑥   𝑛,𝐺   𝑥,𝑅   𝐶,𝑛,𝑥   𝜑,𝑛   𝑥,𝑆
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝑅(𝑛)   𝑆(𝑛)   𝐺(𝑥)

Proof of Theorem ovolscalem1
Dummy variables 𝑘 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovolsca.3 . . . 4 (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴})
2 ssrab2 3650 . . . 4 {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴} ⊆ ℝ
31, 2syl6eqss 3618 . . 3 (𝜑𝐵 ⊆ ℝ)
4 ovolcl 23053 . . 3 (𝐵 ⊆ ℝ → (vol*‘𝐵) ∈ ℝ*)
53, 4syl 17 . 2 (𝜑 → (vol*‘𝐵) ∈ ℝ*)
6 ovolsca.7 . . . . . . . . . . . 12 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
7 ovolfcl 23042 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
86, 7sylan 487 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
98simp3d 1068 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
108simp1d 1066 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
118simp2d 1067 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
12 ovolsca.2 . . . . . . . . . . . . 13 (𝜑𝐶 ∈ ℝ+)
1312rpregt0d 11754 . . . . . . . . . . . 12 (𝜑 → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
1413adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
15 lediv1 10767 . . . . . . . . . . 11 (((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)) ↔ ((1st ‘(𝐹𝑛)) / 𝐶) ≤ ((2nd ‘(𝐹𝑛)) / 𝐶)))
1610, 11, 14, 15syl3anc 1318 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)) ↔ ((1st ‘(𝐹𝑛)) / 𝐶) ≤ ((2nd ‘(𝐹𝑛)) / 𝐶)))
179, 16mpbid 221 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) / 𝐶) ≤ ((2nd ‘(𝐹𝑛)) / 𝐶))
18 df-br 4584 . . . . . . . . 9 (((1st ‘(𝐹𝑛)) / 𝐶) ≤ ((2nd ‘(𝐹𝑛)) / 𝐶) ↔ ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩ ∈ ≤ )
1917, 18sylib 207 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩ ∈ ≤ )
2012adantr 480 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝐶 ∈ ℝ+)
2110, 20rerpdivcld 11779 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) / 𝐶) ∈ ℝ)
2211, 20rerpdivcld 11779 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) / 𝐶) ∈ ℝ)
23 opelxpi 5072 . . . . . . . . 9 ((((1st ‘(𝐹𝑛)) / 𝐶) ∈ ℝ ∧ ((2nd ‘(𝐹𝑛)) / 𝐶) ∈ ℝ) → ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩ ∈ (ℝ × ℝ))
2421, 22, 23syl2anc 691 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩ ∈ (ℝ × ℝ))
2519, 24elind 3760 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
26 ovolsca.6 . . . . . . 7 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩)
2725, 26fmptd 6292 . . . . . 6 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
28 eqid 2610 . . . . . . 7 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
29 eqid 2610 . . . . . . 7 seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
3028, 29ovolsf 23048 . . . . . 6 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
3127, 30syl 17 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
32 frn 5966 . . . . 5 (seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ (0[,)+∞))
3331, 32syl 17 . . . 4 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ (0[,)+∞))
34 icossxr 12129 . . . 4 (0[,)+∞) ⊆ ℝ*
3533, 34syl6ss 3580 . . 3 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ*)
36 supxrcl 12017 . . 3 (ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ* → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ ℝ*)
3735, 36syl 17 . 2 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ ℝ*)
38 ovolsca.4 . . . . 5 (𝜑 → (vol*‘𝐴) ∈ ℝ)
3938, 12rerpdivcld 11779 . . . 4 (𝜑 → ((vol*‘𝐴) / 𝐶) ∈ ℝ)
40 ovolsca.9 . . . . 5 (𝜑𝑅 ∈ ℝ+)
4140rpred 11748 . . . 4 (𝜑𝑅 ∈ ℝ)
4239, 41readdcld 9948 . . 3 (𝜑 → (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ)
4342rexrd 9968 . 2 (𝜑 → (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ*)
441eleq2d 2673 . . . . . . 7 (𝜑 → (𝑦𝐵𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴}))
45 oveq2 6557 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐶 · 𝑥) = (𝐶 · 𝑦))
4645eleq1d 2672 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐶 · 𝑥) ∈ 𝐴 ↔ (𝐶 · 𝑦) ∈ 𝐴))
4746elrab 3331 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴} ↔ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴))
4844, 47syl6bb 275 . . . . . 6 (𝜑 → (𝑦𝐵 ↔ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)))
49 simprr 792 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → (𝐶 · 𝑦) ∈ 𝐴)
50 ovolsca.8 . . . . . . . . . . 11 (𝜑𝐴 ran ((,) ∘ 𝐹))
51 ovolsca.1 . . . . . . . . . . . 12 (𝜑𝐴 ⊆ ℝ)
52 ovolfioo 23043 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛)))))
5351, 6, 52syl2anc 691 . . . . . . . . . . 11 (𝜑 → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛)))))
5450, 53mpbid 221 . . . . . . . . . 10 (𝜑 → ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))))
5554adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))))
56 breq2 4587 . . . . . . . . . . . 12 (𝑥 = (𝐶 · 𝑦) → ((1st ‘(𝐹𝑛)) < 𝑥 ↔ (1st ‘(𝐹𝑛)) < (𝐶 · 𝑦)))
57 breq1 4586 . . . . . . . . . . . 12 (𝑥 = (𝐶 · 𝑦) → (𝑥 < (2nd ‘(𝐹𝑛)) ↔ (𝐶 · 𝑦) < (2nd ‘(𝐹𝑛))))
5856, 57anbi12d 743 . . . . . . . . . . 11 (𝑥 = (𝐶 · 𝑦) → (((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) ↔ ((1st ‘(𝐹𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹𝑛)))))
5958rexbidv 3034 . . . . . . . . . 10 (𝑥 = (𝐶 · 𝑦) → (∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹𝑛)))))
6059rspcv 3278 . . . . . . . . 9 ((𝐶 · 𝑦) ∈ 𝐴 → (∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹𝑛)))))
6149, 55, 60sylc 63 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹𝑛))))
62 opex 4859 . . . . . . . . . . . . . . . 16 ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩ ∈ V
6326fvmpt2 6200 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ ∧ ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩ ∈ V) → (𝐺𝑛) = ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩)
6462, 63mpan2 703 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (𝐺𝑛) = ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩)
6564fveq2d 6107 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (1st ‘(𝐺𝑛)) = (1st ‘⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩))
66 ovex 6577 . . . . . . . . . . . . . . 15 ((1st ‘(𝐹𝑛)) / 𝐶) ∈ V
67 ovex 6577 . . . . . . . . . . . . . . 15 ((2nd ‘(𝐹𝑛)) / 𝐶) ∈ V
6866, 67op1st 7067 . . . . . . . . . . . . . 14 (1st ‘⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩) = ((1st ‘(𝐹𝑛)) / 𝐶)
6965, 68syl6eq 2660 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (1st ‘(𝐺𝑛)) = ((1st ‘(𝐹𝑛)) / 𝐶))
7069adantl 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = ((1st ‘(𝐹𝑛)) / 𝐶))
7170breq1d 4593 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺𝑛)) < 𝑦 ↔ ((1st ‘(𝐹𝑛)) / 𝐶) < 𝑦))
7210adantlr 747 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
73 simplrl 796 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑦 ∈ ℝ)
7414adantlr 747 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
75 ltdivmul 10777 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑛)) ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (((1st ‘(𝐹𝑛)) / 𝐶) < 𝑦 ↔ (1st ‘(𝐹𝑛)) < (𝐶 · 𝑦)))
7672, 73, 74, 75syl3anc 1318 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹𝑛)) / 𝐶) < 𝑦 ↔ (1st ‘(𝐹𝑛)) < (𝐶 · 𝑦)))
7771, 76bitr2d 268 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) < (𝐶 · 𝑦) ↔ (1st ‘(𝐺𝑛)) < 𝑦))
7811adantlr 747 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
79 ltmuldiv2 10776 . . . . . . . . . . . 12 ((𝑦 ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐶 · 𝑦) < (2nd ‘(𝐹𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹𝑛)) / 𝐶)))
8073, 78, 74, 79syl3anc 1318 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝐶 · 𝑦) < (2nd ‘(𝐹𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹𝑛)) / 𝐶)))
8164fveq2d 6107 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (2nd ‘(𝐺𝑛)) = (2nd ‘⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩))
8266, 67op2nd 7068 . . . . . . . . . . . . . 14 (2nd ‘⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩) = ((2nd ‘(𝐹𝑛)) / 𝐶)
8381, 82syl6eq 2660 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (2nd ‘(𝐺𝑛)) = ((2nd ‘(𝐹𝑛)) / 𝐶))
8483adantl 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = ((2nd ‘(𝐹𝑛)) / 𝐶))
8584breq2d 4595 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑦 < (2nd ‘(𝐺𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹𝑛)) / 𝐶)))
8680, 85bitr4d 270 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝐶 · 𝑦) < (2nd ‘(𝐹𝑛)) ↔ 𝑦 < (2nd ‘(𝐺𝑛))))
8777, 86anbi12d 743 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹𝑛))) ↔ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛)))))
8887rexbidva 3031 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → (∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹𝑛))) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛)))))
8961, 88mpbid 221 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))))
9089ex 449 . . . . . 6 (𝜑 → ((𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛)))))
9148, 90sylbid 229 . . . . 5 (𝜑 → (𝑦𝐵 → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛)))))
9291ralrimiv 2948 . . . 4 (𝜑 → ∀𝑦𝐵𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))))
93 ovolfioo 23043 . . . . 5 ((𝐵 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ran ((,) ∘ 𝐺) ↔ ∀𝑦𝐵𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛)))))
943, 27, 93syl2anc 691 . . . 4 (𝜑 → (𝐵 ran ((,) ∘ 𝐺) ↔ ∀𝑦𝐵𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛)))))
9592, 94mpbird 246 . . 3 (𝜑𝐵 ran ((,) ∘ 𝐺))
9629ovollb 23054 . . 3 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐵 ran ((,) ∘ 𝐺)) → (vol*‘𝐵) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ))
9727, 95, 96syl2anc 691 . 2 (𝜑 → (vol*‘𝐵) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ))
98 fzfid 12634 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (1...𝑘) ∈ Fin)
9912rpcnd 11750 . . . . . . . . 9 (𝜑𝐶 ∈ ℂ)
10099adantr 480 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐶 ∈ ℂ)
101 simpl 472 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝜑)
102 elfznn 12241 . . . . . . . . . 10 (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
10311, 10resubcld 10337 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ∈ ℝ)
104101, 102, 103syl2an 493 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ∈ ℝ)
105104recnd 9947 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ∈ ℂ)
10612rpne0d 11753 . . . . . . . . 9 (𝜑𝐶 ≠ 0)
107106adantr 480 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐶 ≠ 0)
10898, 100, 105, 107fsumdivc 14360 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) = Σ𝑛 ∈ (1...𝑘)(((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶))
10983, 69oveq12d 6567 . . . . . . . . . . 11 (𝑛 ∈ ℕ → ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))) = (((2nd ‘(𝐹𝑛)) / 𝐶) − ((1st ‘(𝐹𝑛)) / 𝐶)))
110109adantl 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))) = (((2nd ‘(𝐹𝑛)) / 𝐶) − ((1st ‘(𝐹𝑛)) / 𝐶)))
11128ovolfsval 23046 . . . . . . . . . . 11 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))))
11227, 111sylan 487 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))))
11311recnd 9947 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℂ)
11410recnd 9947 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℂ)
11512rpcnne0d 11757 . . . . . . . . . . . 12 (𝜑 → (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0))
116115adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0))
117 divsubdir 10600 . . . . . . . . . . 11 (((2nd ‘(𝐹𝑛)) ∈ ℂ ∧ (1st ‘(𝐹𝑛)) ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) = (((2nd ‘(𝐹𝑛)) / 𝐶) − ((1st ‘(𝐹𝑛)) / 𝐶)))
118113, 114, 116, 117syl3anc 1318 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) = (((2nd ‘(𝐹𝑛)) / 𝐶) − ((1st ‘(𝐹𝑛)) / 𝐶)))
119110, 112, 1183eqtr4d 2654 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶))
120101, 102, 119syl2an 493 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶))
121 simpr 476 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
122 nnuz 11599 . . . . . . . . 9 ℕ = (ℤ‘1)
123121, 122syl6eleq 2698 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
124103, 20rerpdivcld 11779 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) ∈ ℝ)
125124recnd 9947 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) ∈ ℂ)
126101, 102, 125syl2an 493 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) ∈ ℂ)
127120, 123, 126fsumser 14308 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)(((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘))
128108, 127eqtrd 2644 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘))
129 ovolsca.10 . . . . . . . . . . 11 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
130 eqid 2610 . . . . . . . . . . . . . . . 16 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
131 ovolsca.5 . . . . . . . . . . . . . . . 16 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
132130, 131ovolsf 23048 . . . . . . . . . . . . . . 15 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
1336, 132syl 17 . . . . . . . . . . . . . 14 (𝜑𝑆:ℕ⟶(0[,)+∞))
134 frn 5966 . . . . . . . . . . . . . 14 (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
135133, 134syl 17 . . . . . . . . . . . . 13 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
136135, 34syl6ss 3580 . . . . . . . . . . . 12 (𝜑 → ran 𝑆 ⊆ ℝ*)
13712, 40rpmulcld 11764 . . . . . . . . . . . . . . 15 (𝜑 → (𝐶 · 𝑅) ∈ ℝ+)
138137rpred 11748 . . . . . . . . . . . . . 14 (𝜑 → (𝐶 · 𝑅) ∈ ℝ)
13938, 138readdcld 9948 . . . . . . . . . . . . 13 (𝜑 → ((vol*‘𝐴) + (𝐶 · 𝑅)) ∈ ℝ)
140139rexrd 9968 . . . . . . . . . . . 12 (𝜑 → ((vol*‘𝐴) + (𝐶 · 𝑅)) ∈ ℝ*)
141 supxrleub 12028 . . . . . . . . . . . 12 ((ran 𝑆 ⊆ ℝ* ∧ ((vol*‘𝐴) + (𝐶 · 𝑅)) ∈ ℝ*) → (sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ ∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
142136, 140, 141syl2anc 691 . . . . . . . . . . 11 (𝜑 → (sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ ∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
143129, 142mpbid 221 . . . . . . . . . 10 (𝜑 → ∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
144 ffn 5958 . . . . . . . . . . . 12 (𝑆:ℕ⟶(0[,)+∞) → 𝑆 Fn ℕ)
145133, 144syl 17 . . . . . . . . . . 11 (𝜑𝑆 Fn ℕ)
146 breq1 4586 . . . . . . . . . . . 12 (𝑥 = (𝑆𝑘) → (𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ (𝑆𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
147146ralrn 6270 . . . . . . . . . . 11 (𝑆 Fn ℕ → (∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
148145, 147syl 17 . . . . . . . . . 10 (𝜑 → (∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
149143, 148mpbid 221 . . . . . . . . 9 (𝜑 → ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
150149r19.21bi 2916 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
1516adantr 480 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
152130ovolfsval 23046 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
153151, 102, 152syl2an 493 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
154153, 123, 105fsumser 14308 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘))
155131fveq1i 6104 . . . . . . . . 9 (𝑆𝑘) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘)
156154, 155syl6eqr 2662 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) = (𝑆𝑘))
15739recnd 9947 . . . . . . . . . . 11 (𝜑 → ((vol*‘𝐴) / 𝐶) ∈ ℂ)
15840rpcnd 11750 . . . . . . . . . . 11 (𝜑𝑅 ∈ ℂ)
15999, 157, 158adddid 9943 . . . . . . . . . 10 (𝜑 → (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)) = ((𝐶 · ((vol*‘𝐴) / 𝐶)) + (𝐶 · 𝑅)))
16038recnd 9947 . . . . . . . . . . . 12 (𝜑 → (vol*‘𝐴) ∈ ℂ)
161160, 99, 106divcan2d 10682 . . . . . . . . . . 11 (𝜑 → (𝐶 · ((vol*‘𝐴) / 𝐶)) = (vol*‘𝐴))
162161oveq1d 6564 . . . . . . . . . 10 (𝜑 → ((𝐶 · ((vol*‘𝐴) / 𝐶)) + (𝐶 · 𝑅)) = ((vol*‘𝐴) + (𝐶 · 𝑅)))
163159, 162eqtrd 2644 . . . . . . . . 9 (𝜑 → (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)) = ((vol*‘𝐴) + (𝐶 · 𝑅)))
164163adantr 480 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)) = ((vol*‘𝐴) + (𝐶 · 𝑅)))
165150, 156, 1643brtr4d 4615 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ≤ (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)))
16698, 104fsumrecl 14312 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ∈ ℝ)
16742adantr 480 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ)
16813adantr 480 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
169 ledivmul 10778 . . . . . . . 8 ((Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ∈ ℝ ∧ (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ≤ (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅))))
170166, 167, 168, 169syl3anc 1318 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → ((Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ≤ (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅))))
171165, 170mpbird 246 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
172128, 171eqbrtrrd 4607 . . . . 5 ((𝜑𝑘 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
173172ralrimiva 2949 . . . 4 (𝜑 → ∀𝑘 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
174 ffn 5958 . . . . . 6 (seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞) → seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ)
17531, 174syl 17 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ)
176 breq1 4586 . . . . . 6 (𝑦 = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) → (𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅)))
177176ralrn 6270 . . . . 5 (seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ → (∀𝑦 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ ∀𝑘 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅)))
178175, 177syl 17 . . . 4 (𝜑 → (∀𝑦 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ ∀𝑘 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅)))
179173, 178mpbird 246 . . 3 (𝜑 → ∀𝑦 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
180 supxrleub 12028 . . . 4 ((ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ* ∧ (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ*) → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ ∀𝑦 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅)))
18135, 43, 180syl2anc 691 . . 3 (𝜑 → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ ∀𝑦 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅)))
182179, 181mpbird 246 . 2 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
1835, 37, 43, 97, 182xrletrd 11869 1 (𝜑 → (vol*‘𝐵) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ⟨cop 4131  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ran crn 5039   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  supcsup 8229  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  +∞cpnf 9950  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145   / cdiv 10563  ℕcn 10897  ℤ≥cuz 11563  ℝ+crp 11708  (,)cioo 12046  [,)cico 12048  ...cfz 12197  seqcseq 12663  abscabs 13822  Σcsu 14264  vol*covol 23038 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-ioo 12050  df-ico 12052  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-ovol 23040 This theorem is referenced by:  ovolscalem2  23089
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