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Theorem ovolscalem1 24877
Description: Lemma for ovolsca 24879. (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 4037 . . . 4 {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴} ⊆ ℝ
31, 2eqsstrdi 3998 . . 3 (𝜑𝐵 ⊆ ℝ)
4 ovolcl 24842 . . 3 (𝐵 ⊆ ℝ → (vol*‘𝐵) ∈ ℝ*)
53, 4syl 17 . 2 (𝜑 → (vol*‘𝐵) ∈ ℝ*)
6 ovolsca.7 . . . . . . . . . . . 12 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
7 ovolfcl 24830 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
86, 7sylan 580 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
98simp3d 1144 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
108simp1d 1142 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
118simp2d 1143 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
12 ovolsca.2 . . . . . . . . . . . . 13 (𝜑𝐶 ∈ ℝ+)
1312rpregt0d 12963 . . . . . . . . . . . 12 (𝜑 → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
1413adantr 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
15 lediv1 12020 . . . . . . . . . . 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 1371 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)) ↔ ((1st ‘(𝐹𝑛)) / 𝐶) ≤ ((2nd ‘(𝐹𝑛)) / 𝐶)))
179, 16mpbid 231 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) / 𝐶) ≤ ((2nd ‘(𝐹𝑛)) / 𝐶))
18 df-br 5106 . . . . . . . . 9 (((1st ‘(𝐹𝑛)) / 𝐶) ≤ ((2nd ‘(𝐹𝑛)) / 𝐶) ↔ ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩ ∈ ≤ )
1917, 18sylib 217 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩ ∈ ≤ )
2012adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝐶 ∈ ℝ+)
2110, 20rerpdivcld 12988 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) / 𝐶) ∈ ℝ)
2211, 20rerpdivcld 12988 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) / 𝐶) ∈ ℝ)
2321, 22opelxpd 5671 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩ ∈ (ℝ × ℝ))
2419, 23elind 4154 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
25 ovolsca.6 . . . . . . 7 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩)
2624, 25fmptd 7062 . . . . . 6 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
27 eqid 2736 . . . . . . 7 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
28 eqid 2736 . . . . . . 7 seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
2927, 28ovolsf 24836 . . . . . 6 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
3026, 29syl 17 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
3130frnd 6676 . . . 4 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ (0[,)+∞))
32 icossxr 13349 . . . 4 (0[,)+∞) ⊆ ℝ*
3331, 32sstrdi 3956 . . 3 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ*)
34 supxrcl 13234 . . 3 (ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ* → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ ℝ*)
3533, 34syl 17 . 2 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ ℝ*)
36 ovolsca.4 . . . . 5 (𝜑 → (vol*‘𝐴) ∈ ℝ)
3736, 12rerpdivcld 12988 . . . 4 (𝜑 → ((vol*‘𝐴) / 𝐶) ∈ ℝ)
38 ovolsca.9 . . . . 5 (𝜑𝑅 ∈ ℝ+)
3938rpred 12957 . . . 4 (𝜑𝑅 ∈ ℝ)
4037, 39readdcld 11184 . . 3 (𝜑 → (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ)
4140rexrd 11205 . 2 (𝜑 → (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ*)
421eleq2d 2823 . . . . . . 7 (𝜑 → (𝑦𝐵𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴}))
43 oveq2 7365 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐶 · 𝑥) = (𝐶 · 𝑦))
4443eleq1d 2822 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐶 · 𝑥) ∈ 𝐴 ↔ (𝐶 · 𝑦) ∈ 𝐴))
4544elrab 3645 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴} ↔ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴))
4642, 45bitrdi 286 . . . . . 6 (𝜑 → (𝑦𝐵 ↔ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)))
47 breq2 5109 . . . . . . . . . . 11 (𝑥 = (𝐶 · 𝑦) → ((1st ‘(𝐹𝑛)) < 𝑥 ↔ (1st ‘(𝐹𝑛)) < (𝐶 · 𝑦)))
48 breq1 5108 . . . . . . . . . . 11 (𝑥 = (𝐶 · 𝑦) → (𝑥 < (2nd ‘(𝐹𝑛)) ↔ (𝐶 · 𝑦) < (2nd ‘(𝐹𝑛))))
4947, 48anbi12d 631 . . . . . . . . . 10 (𝑥 = (𝐶 · 𝑦) → (((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) ↔ ((1st ‘(𝐹𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹𝑛)))))
5049rexbidv 3175 . . . . . . . . 9 (𝑥 = (𝐶 · 𝑦) → (∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹𝑛)))))
51 ovolsca.8 . . . . . . . . . . 11 (𝜑𝐴 ran ((,) ∘ 𝐹))
52 ovolsca.1 . . . . . . . . . . . 12 (𝜑𝐴 ⊆ ℝ)
53 ovolfioo 24831 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛)))))
5452, 6, 53syl2anc 584 . . . . . . . . . . 11 (𝜑 → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛)))))
5551, 54mpbid 231 . . . . . . . . . 10 (𝜑 → ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))))
5655adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))))
57 simprr 771 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → (𝐶 · 𝑦) ∈ 𝐴)
5850, 56, 57rspcdva 3582 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹𝑛))))
59 opex 5421 . . . . . . . . . . . . . . . 16 ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩ ∈ V
6025fvmpt2 6959 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ ∧ ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩ ∈ V) → (𝐺𝑛) = ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩)
6159, 60mpan2 689 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (𝐺𝑛) = ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩)
6261fveq2d 6846 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (1st ‘(𝐺𝑛)) = (1st ‘⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩))
63 ovex 7390 . . . . . . . . . . . . . . 15 ((1st ‘(𝐹𝑛)) / 𝐶) ∈ V
64 ovex 7390 . . . . . . . . . . . . . . 15 ((2nd ‘(𝐹𝑛)) / 𝐶) ∈ V
6563, 64op1st 7929 . . . . . . . . . . . . . 14 (1st ‘⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩) = ((1st ‘(𝐹𝑛)) / 𝐶)
6662, 65eqtrdi 2792 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (1st ‘(𝐺𝑛)) = ((1st ‘(𝐹𝑛)) / 𝐶))
6766adantl 482 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = ((1st ‘(𝐹𝑛)) / 𝐶))
6867breq1d 5115 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺𝑛)) < 𝑦 ↔ ((1st ‘(𝐹𝑛)) / 𝐶) < 𝑦))
6910adantlr 713 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
70 simplrl 775 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑦 ∈ ℝ)
7114adantlr 713 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
72 ltdivmul 12030 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑛)) ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (((1st ‘(𝐹𝑛)) / 𝐶) < 𝑦 ↔ (1st ‘(𝐹𝑛)) < (𝐶 · 𝑦)))
7369, 70, 71, 72syl3anc 1371 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹𝑛)) / 𝐶) < 𝑦 ↔ (1st ‘(𝐹𝑛)) < (𝐶 · 𝑦)))
7468, 73bitr2d 279 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) < (𝐶 · 𝑦) ↔ (1st ‘(𝐺𝑛)) < 𝑦))
7511adantlr 713 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
76 ltmuldiv2 12029 . . . . . . . . . . . 12 ((𝑦 ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐶 · 𝑦) < (2nd ‘(𝐹𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹𝑛)) / 𝐶)))
7770, 75, 71, 76syl3anc 1371 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝐶 · 𝑦) < (2nd ‘(𝐹𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹𝑛)) / 𝐶)))
7861fveq2d 6846 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (2nd ‘(𝐺𝑛)) = (2nd ‘⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩))
7963, 64op2nd 7930 . . . . . . . . . . . . . 14 (2nd ‘⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩) = ((2nd ‘(𝐹𝑛)) / 𝐶)
8078, 79eqtrdi 2792 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (2nd ‘(𝐺𝑛)) = ((2nd ‘(𝐹𝑛)) / 𝐶))
8180adantl 482 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = ((2nd ‘(𝐹𝑛)) / 𝐶))
8281breq2d 5117 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑦 < (2nd ‘(𝐺𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹𝑛)) / 𝐶)))
8377, 82bitr4d 281 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝐶 · 𝑦) < (2nd ‘(𝐹𝑛)) ↔ 𝑦 < (2nd ‘(𝐺𝑛))))
8474, 83anbi12d 631 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹𝑛))) ↔ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛)))))
8584rexbidva 3173 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → (∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹𝑛))) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛)))))
8658, 85mpbid 231 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))))
8786ex 413 . . . . . 6 (𝜑 → ((𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛)))))
8846, 87sylbid 239 . . . . 5 (𝜑 → (𝑦𝐵 → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛)))))
8988ralrimiv 3142 . . . 4 (𝜑 → ∀𝑦𝐵𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))))
90 ovolfioo 24831 . . . . 5 ((𝐵 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ran ((,) ∘ 𝐺) ↔ ∀𝑦𝐵𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛)))))
913, 26, 90syl2anc 584 . . . 4 (𝜑 → (𝐵 ran ((,) ∘ 𝐺) ↔ ∀𝑦𝐵𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛)))))
9289, 91mpbird 256 . . 3 (𝜑𝐵 ran ((,) ∘ 𝐺))
9328ovollb 24843 . . 3 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐵 ran ((,) ∘ 𝐺)) → (vol*‘𝐵) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ))
9426, 92, 93syl2anc 584 . 2 (𝜑 → (vol*‘𝐵) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ))
95 fzfid 13878 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (1...𝑘) ∈ Fin)
9612rpcnd 12959 . . . . . . . . 9 (𝜑𝐶 ∈ ℂ)
9796adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐶 ∈ ℂ)
98 simpl 483 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝜑)
99 elfznn 13470 . . . . . . . . . 10 (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
10011, 10resubcld 11583 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ∈ ℝ)
10198, 99, 100syl2an 596 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ∈ ℝ)
102101recnd 11183 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ∈ ℂ)
10312rpne0d 12962 . . . . . . . . 9 (𝜑𝐶 ≠ 0)
104103adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐶 ≠ 0)
10595, 97, 102, 104fsumdivc 15671 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) = Σ𝑛 ∈ (1...𝑘)(((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶))
10680, 66oveq12d 7375 . . . . . . . . . . 11 (𝑛 ∈ ℕ → ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))) = (((2nd ‘(𝐹𝑛)) / 𝐶) − ((1st ‘(𝐹𝑛)) / 𝐶)))
107106adantl 482 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))) = (((2nd ‘(𝐹𝑛)) / 𝐶) − ((1st ‘(𝐹𝑛)) / 𝐶)))
10827ovolfsval 24834 . . . . . . . . . . 11 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))))
10926, 108sylan 580 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))))
11011recnd 11183 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℂ)
11110recnd 11183 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℂ)
11212rpcnne0d 12966 . . . . . . . . . . . 12 (𝜑 → (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0))
113112adantr 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0))
114 divsubdir 11849 . . . . . . . . . . 11 (((2nd ‘(𝐹𝑛)) ∈ ℂ ∧ (1st ‘(𝐹𝑛)) ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) = (((2nd ‘(𝐹𝑛)) / 𝐶) − ((1st ‘(𝐹𝑛)) / 𝐶)))
115110, 111, 113, 114syl3anc 1371 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) = (((2nd ‘(𝐹𝑛)) / 𝐶) − ((1st ‘(𝐹𝑛)) / 𝐶)))
116107, 109, 1153eqtr4d 2786 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶))
11798, 99, 116syl2an 596 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶))
118 simpr 485 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
119 nnuz 12806 . . . . . . . . 9 ℕ = (ℤ‘1)
120118, 119eleqtrdi 2848 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
121100, 20rerpdivcld 12988 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) ∈ ℝ)
122121recnd 11183 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) ∈ ℂ)
12398, 99, 122syl2an 596 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) ∈ ℂ)
124117, 120, 123fsumser 15615 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)(((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘))
125105, 124eqtrd 2776 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘))
126 ovolsca.10 . . . . . . . . . . 11 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
127 eqid 2736 . . . . . . . . . . . . . . . 16 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
128 ovolsca.5 . . . . . . . . . . . . . . . 16 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
129127, 128ovolsf 24836 . . . . . . . . . . . . . . 15 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
1306, 129syl 17 . . . . . . . . . . . . . 14 (𝜑𝑆:ℕ⟶(0[,)+∞))
131130frnd 6676 . . . . . . . . . . . . 13 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
132131, 32sstrdi 3956 . . . . . . . . . . . 12 (𝜑 → ran 𝑆 ⊆ ℝ*)
13312, 38rpmulcld 12973 . . . . . . . . . . . . . . 15 (𝜑 → (𝐶 · 𝑅) ∈ ℝ+)
134133rpred 12957 . . . . . . . . . . . . . 14 (𝜑 → (𝐶 · 𝑅) ∈ ℝ)
13536, 134readdcld 11184 . . . . . . . . . . . . 13 (𝜑 → ((vol*‘𝐴) + (𝐶 · 𝑅)) ∈ ℝ)
136135rexrd 11205 . . . . . . . . . . . 12 (𝜑 → ((vol*‘𝐴) + (𝐶 · 𝑅)) ∈ ℝ*)
137 supxrleub 13245 . . . . . . . . . . . 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))))
138132, 136, 137syl2anc 584 . . . . . . . . . . 11 (𝜑 → (sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ ∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
139126, 138mpbid 231 . . . . . . . . . 10 (𝜑 → ∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
140130ffnd 6669 . . . . . . . . . . 11 (𝜑𝑆 Fn ℕ)
141 breq1 5108 . . . . . . . . . . . 12 (𝑥 = (𝑆𝑘) → (𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ (𝑆𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
142141ralrn 7038 . . . . . . . . . . 11 (𝑆 Fn ℕ → (∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
143140, 142syl 17 . . . . . . . . . 10 (𝜑 → (∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
144139, 143mpbid 231 . . . . . . . . 9 (𝜑 → ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
145144r19.21bi 3234 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
1466adantr 481 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
147127ovolfsval 24834 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
148146, 99, 147syl2an 596 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
149148, 120, 102fsumser 15615 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘))
150128fveq1i 6843 . . . . . . . . 9 (𝑆𝑘) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘)
151149, 150eqtr4di 2794 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) = (𝑆𝑘))
15237recnd 11183 . . . . . . . . . . 11 (𝜑 → ((vol*‘𝐴) / 𝐶) ∈ ℂ)
15338rpcnd 12959 . . . . . . . . . . 11 (𝜑𝑅 ∈ ℂ)
15496, 152, 153adddid 11179 . . . . . . . . . 10 (𝜑 → (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)) = ((𝐶 · ((vol*‘𝐴) / 𝐶)) + (𝐶 · 𝑅)))
15536recnd 11183 . . . . . . . . . . . 12 (𝜑 → (vol*‘𝐴) ∈ ℂ)
156155, 96, 103divcan2d 11933 . . . . . . . . . . 11 (𝜑 → (𝐶 · ((vol*‘𝐴) / 𝐶)) = (vol*‘𝐴))
157156oveq1d 7372 . . . . . . . . . 10 (𝜑 → ((𝐶 · ((vol*‘𝐴) / 𝐶)) + (𝐶 · 𝑅)) = ((vol*‘𝐴) + (𝐶 · 𝑅)))
158154, 157eqtrd 2776 . . . . . . . . 9 (𝜑 → (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)) = ((vol*‘𝐴) + (𝐶 · 𝑅)))
159158adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)) = ((vol*‘𝐴) + (𝐶 · 𝑅)))
160145, 151, 1593brtr4d 5137 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ≤ (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)))
16195, 101fsumrecl 15619 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ∈ ℝ)
16240adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ)
16313adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
164 ledivmul 12031 . . . . . . . 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))))
165161, 162, 163, 164syl3anc 1371 . . . . . . 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))))
166160, 165mpbird 256 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
167125, 166eqbrtrrd 5129 . . . . 5 ((𝜑𝑘 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
168167ralrimiva 3143 . . . 4 (𝜑 → ∀𝑘 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
16930ffnd 6669 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ)
170 breq1 5108 . . . . . 6 (𝑦 = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) → (𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅)))
171170ralrn 7038 . . . . 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)))
172169, 171syl 17 . . . 4 (𝜑 → (∀𝑦 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ ∀𝑘 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅)))
173168, 172mpbird 256 . . 3 (𝜑 → ∀𝑦 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
174 supxrleub 13245 . . . 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)))
17533, 41, 174syl2anc 584 . . 3 (𝜑 → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ ∀𝑦 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅)))
176173, 175mpbird 256 . 2 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
1775, 35, 41, 94, 176xrletrd 13081 1 (𝜑 → (vol*‘𝐵) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  cin 3909  wss 3910  cop 4592   cuni 4865   class class class wbr 5105  cmpt 5188   × cxp 5631  ran crn 5634  ccom 5637   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  supcsup 9376  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  +∞cpnf 11186  *cxr 11188   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  cn 12153  cuz 12763  +crp 12915  (,)cioo 13264  [,)cico 13266  ...cfz 13424  seqcseq 13906  abscabs 15119  Σcsu 15570  vol*covol 24826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-ioo 13268  df-ico 13270  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-ovol 24828
This theorem is referenced by:  ovolscalem2  24878
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