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Theorem ovolscalem1 25562
Description: Lemma for ovolsca 25564. (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 4090 . . . 4 {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴} ⊆ ℝ
31, 2eqsstrdi 4050 . . 3 (𝜑𝐵 ⊆ ℝ)
4 ovolcl 25527 . . 3 (𝐵 ⊆ ℝ → (vol*‘𝐵) ∈ ℝ*)
53, 4syl 17 . 2 (𝜑 → (vol*‘𝐵) ∈ ℝ*)
6 ovolsca.7 . . . . . . . . . . . 12 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
7 ovolfcl 25515 . . . . . . . . . . . 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 1143 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
108simp1d 1141 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
118simp2d 1142 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
12 ovolsca.2 . . . . . . . . . . . . 13 (𝜑𝐶 ∈ ℝ+)
1312rpregt0d 13081 . . . . . . . . . . . 12 (𝜑 → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
1413adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
15 lediv1 12131 . . . . . . . . . . 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 1370 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)) ↔ ((1st ‘(𝐹𝑛)) / 𝐶) ≤ ((2nd ‘(𝐹𝑛)) / 𝐶)))
179, 16mpbid 232 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) / 𝐶) ≤ ((2nd ‘(𝐹𝑛)) / 𝐶))
18 df-br 5149 . . . . . . . . 9 (((1st ‘(𝐹𝑛)) / 𝐶) ≤ ((2nd ‘(𝐹𝑛)) / 𝐶) ↔ ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩ ∈ ≤ )
1917, 18sylib 218 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩ ∈ ≤ )
2012adantr 480 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝐶 ∈ ℝ+)
2110, 20rerpdivcld 13106 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) / 𝐶) ∈ ℝ)
2211, 20rerpdivcld 13106 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) / 𝐶) ∈ ℝ)
2321, 22opelxpd 5728 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩ ∈ (ℝ × ℝ))
2419, 23elind 4210 . . . . . . 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 7134 . . . . . 6 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
27 eqid 2735 . . . . . . 7 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
28 eqid 2735 . . . . . . 7 seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
2927, 28ovolsf 25521 . . . . . 6 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
3026, 29syl 17 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
3130frnd 6745 . . . 4 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ (0[,)+∞))
32 icossxr 13469 . . . 4 (0[,)+∞) ⊆ ℝ*
3331, 32sstrdi 4008 . . 3 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ*)
34 supxrcl 13354 . . 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 13106 . . . 4 (𝜑 → ((vol*‘𝐴) / 𝐶) ∈ ℝ)
38 ovolsca.9 . . . . 5 (𝜑𝑅 ∈ ℝ+)
3938rpred 13075 . . . 4 (𝜑𝑅 ∈ ℝ)
4037, 39readdcld 11288 . . 3 (𝜑 → (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ)
4140rexrd 11309 . 2 (𝜑 → (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ*)
421eleq2d 2825 . . . . . . 7 (𝜑 → (𝑦𝐵𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴}))
43 oveq2 7439 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐶 · 𝑥) = (𝐶 · 𝑦))
4443eleq1d 2824 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐶 · 𝑥) ∈ 𝐴 ↔ (𝐶 · 𝑦) ∈ 𝐴))
4544elrab 3695 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴} ↔ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴))
4642, 45bitrdi 287 . . . . . 6 (𝜑 → (𝑦𝐵 ↔ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)))
47 breq2 5152 . . . . . . . . . . 11 (𝑥 = (𝐶 · 𝑦) → ((1st ‘(𝐹𝑛)) < 𝑥 ↔ (1st ‘(𝐹𝑛)) < (𝐶 · 𝑦)))
48 breq1 5151 . . . . . . . . . . 11 (𝑥 = (𝐶 · 𝑦) → (𝑥 < (2nd ‘(𝐹𝑛)) ↔ (𝐶 · 𝑦) < (2nd ‘(𝐹𝑛))))
4947, 48anbi12d 632 . . . . . . . . . 10 (𝑥 = (𝐶 · 𝑦) → (((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) ↔ ((1st ‘(𝐹𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹𝑛)))))
5049rexbidv 3177 . . . . . . . . 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 25516 . . . . . . . . . . . 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 232 . . . . . . . . . 10 (𝜑 → ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))))
5655adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))))
57 simprr 773 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → (𝐶 · 𝑦) ∈ 𝐴)
5850, 56, 57rspcdva 3623 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹𝑛))))
59 opex 5475 . . . . . . . . . . . . . . . 16 ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩ ∈ V
6025fvmpt2 7027 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ ∧ ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩ ∈ V) → (𝐺𝑛) = ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩)
6159, 60mpan2 691 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (𝐺𝑛) = ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩)
6261fveq2d 6911 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (1st ‘(𝐺𝑛)) = (1st ‘⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩))
63 ovex 7464 . . . . . . . . . . . . . . 15 ((1st ‘(𝐹𝑛)) / 𝐶) ∈ V
64 ovex 7464 . . . . . . . . . . . . . . 15 ((2nd ‘(𝐹𝑛)) / 𝐶) ∈ V
6563, 64op1st 8021 . . . . . . . . . . . . . 14 (1st ‘⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩) = ((1st ‘(𝐹𝑛)) / 𝐶)
6662, 65eqtrdi 2791 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (1st ‘(𝐺𝑛)) = ((1st ‘(𝐹𝑛)) / 𝐶))
6766adantl 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = ((1st ‘(𝐹𝑛)) / 𝐶))
6867breq1d 5158 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺𝑛)) < 𝑦 ↔ ((1st ‘(𝐹𝑛)) / 𝐶) < 𝑦))
6910adantlr 715 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
70 simplrl 777 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑦 ∈ ℝ)
7114adantlr 715 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
72 ltdivmul 12141 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑛)) ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (((1st ‘(𝐹𝑛)) / 𝐶) < 𝑦 ↔ (1st ‘(𝐹𝑛)) < (𝐶 · 𝑦)))
7369, 70, 71, 72syl3anc 1370 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹𝑛)) / 𝐶) < 𝑦 ↔ (1st ‘(𝐹𝑛)) < (𝐶 · 𝑦)))
7468, 73bitr2d 280 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) < (𝐶 · 𝑦) ↔ (1st ‘(𝐺𝑛)) < 𝑦))
7511adantlr 715 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
76 ltmuldiv2 12140 . . . . . . . . . . . 12 ((𝑦 ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐶 · 𝑦) < (2nd ‘(𝐹𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹𝑛)) / 𝐶)))
7770, 75, 71, 76syl3anc 1370 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝐶 · 𝑦) < (2nd ‘(𝐹𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹𝑛)) / 𝐶)))
7861fveq2d 6911 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (2nd ‘(𝐺𝑛)) = (2nd ‘⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩))
7963, 64op2nd 8022 . . . . . . . . . . . . . 14 (2nd ‘⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩) = ((2nd ‘(𝐹𝑛)) / 𝐶)
8078, 79eqtrdi 2791 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (2nd ‘(𝐺𝑛)) = ((2nd ‘(𝐹𝑛)) / 𝐶))
8180adantl 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = ((2nd ‘(𝐹𝑛)) / 𝐶))
8281breq2d 5160 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑦 < (2nd ‘(𝐺𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹𝑛)) / 𝐶)))
8377, 82bitr4d 282 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝐶 · 𝑦) < (2nd ‘(𝐹𝑛)) ↔ 𝑦 < (2nd ‘(𝐺𝑛))))
8474, 83anbi12d 632 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹𝑛))) ↔ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛)))))
8584rexbidva 3175 . . . . . . . 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 232 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))))
8786ex 412 . . . . . 6 (𝜑 → ((𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛)))))
8846, 87sylbid 240 . . . . 5 (𝜑 → (𝑦𝐵 → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛)))))
8988ralrimiv 3143 . . . 4 (𝜑 → ∀𝑦𝐵𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))))
90 ovolfioo 25516 . . . . 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 257 . . 3 (𝜑𝐵 ran ((,) ∘ 𝐺))
9328ovollb 25528 . . 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 14011 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (1...𝑘) ∈ Fin)
9612rpcnd 13077 . . . . . . . . 9 (𝜑𝐶 ∈ ℂ)
9796adantr 480 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐶 ∈ ℂ)
98 simpl 482 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝜑)
99 elfznn 13590 . . . . . . . . . 10 (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
10011, 10resubcld 11689 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ∈ ℝ)
10198, 99, 100syl2an 596 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ∈ ℝ)
102101recnd 11287 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ∈ ℂ)
10312rpne0d 13080 . . . . . . . . 9 (𝜑𝐶 ≠ 0)
104103adantr 480 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐶 ≠ 0)
10595, 97, 102, 104fsumdivc 15819 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) = Σ𝑛 ∈ (1...𝑘)(((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶))
10680, 66oveq12d 7449 . . . . . . . . . . 11 (𝑛 ∈ ℕ → ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))) = (((2nd ‘(𝐹𝑛)) / 𝐶) − ((1st ‘(𝐹𝑛)) / 𝐶)))
107106adantl 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))) = (((2nd ‘(𝐹𝑛)) / 𝐶) − ((1st ‘(𝐹𝑛)) / 𝐶)))
10827ovolfsval 25519 . . . . . . . . . . 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 11287 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℂ)
11110recnd 11287 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℂ)
11212rpcnne0d 13084 . . . . . . . . . . . 12 (𝜑 → (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0))
113112adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0))
114 divsubdir 11959 . . . . . . . . . . 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 1370 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) = (((2nd ‘(𝐹𝑛)) / 𝐶) − ((1st ‘(𝐹𝑛)) / 𝐶)))
116107, 109, 1153eqtr4d 2785 . . . . . . . . 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 484 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
119 nnuz 12919 . . . . . . . . 9 ℕ = (ℤ‘1)
120118, 119eleqtrdi 2849 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
121100, 20rerpdivcld 13106 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) ∈ ℝ)
122121recnd 11287 . . . . . . . . 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 15763 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)(((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘))
125105, 124eqtrd 2775 . . . . . 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 2735 . . . . . . . . . . . . . . . 16 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
128 ovolsca.5 . . . . . . . . . . . . . . . 16 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
129127, 128ovolsf 25521 . . . . . . . . . . . . . . 15 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
1306, 129syl 17 . . . . . . . . . . . . . 14 (𝜑𝑆:ℕ⟶(0[,)+∞))
131130frnd 6745 . . . . . . . . . . . . 13 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
132131, 32sstrdi 4008 . . . . . . . . . . . 12 (𝜑 → ran 𝑆 ⊆ ℝ*)
13312, 38rpmulcld 13091 . . . . . . . . . . . . . . 15 (𝜑 → (𝐶 · 𝑅) ∈ ℝ+)
134133rpred 13075 . . . . . . . . . . . . . 14 (𝜑 → (𝐶 · 𝑅) ∈ ℝ)
13536, 134readdcld 11288 . . . . . . . . . . . . 13 (𝜑 → ((vol*‘𝐴) + (𝐶 · 𝑅)) ∈ ℝ)
136135rexrd 11309 . . . . . . . . . . . 12 (𝜑 → ((vol*‘𝐴) + (𝐶 · 𝑅)) ∈ ℝ*)
137 supxrleub 13365 . . . . . . . . . . . 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 232 . . . . . . . . . 10 (𝜑 → ∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
140130ffnd 6738 . . . . . . . . . . 11 (𝜑𝑆 Fn ℕ)
141 breq1 5151 . . . . . . . . . . . 12 (𝑥 = (𝑆𝑘) → (𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ (𝑆𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
142141ralrn 7108 . . . . . . . . . . 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 232 . . . . . . . . 9 (𝜑 → ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
145144r19.21bi 3249 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
1466adantr 480 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
147127ovolfsval 25519 . . . . . . . . . . 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 15763 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘))
150128fveq1i 6908 . . . . . . . . 9 (𝑆𝑘) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘)
151149, 150eqtr4di 2793 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) = (𝑆𝑘))
15237recnd 11287 . . . . . . . . . . 11 (𝜑 → ((vol*‘𝐴) / 𝐶) ∈ ℂ)
15338rpcnd 13077 . . . . . . . . . . 11 (𝜑𝑅 ∈ ℂ)
15496, 152, 153adddid 11283 . . . . . . . . . 10 (𝜑 → (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)) = ((𝐶 · ((vol*‘𝐴) / 𝐶)) + (𝐶 · 𝑅)))
15536recnd 11287 . . . . . . . . . . . 12 (𝜑 → (vol*‘𝐴) ∈ ℂ)
156155, 96, 103divcan2d 12043 . . . . . . . . . . 11 (𝜑 → (𝐶 · ((vol*‘𝐴) / 𝐶)) = (vol*‘𝐴))
157156oveq1d 7446 . . . . . . . . . 10 (𝜑 → ((𝐶 · ((vol*‘𝐴) / 𝐶)) + (𝐶 · 𝑅)) = ((vol*‘𝐴) + (𝐶 · 𝑅)))
158154, 157eqtrd 2775 . . . . . . . . 9 (𝜑 → (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)) = ((vol*‘𝐴) + (𝐶 · 𝑅)))
159158adantr 480 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)) = ((vol*‘𝐴) + (𝐶 · 𝑅)))
160145, 151, 1593brtr4d 5180 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ≤ (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)))
16195, 101fsumrecl 15767 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ∈ ℝ)
16240adantr 480 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ)
16313adantr 480 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
164 ledivmul 12142 . . . . . . . 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 1370 . . . . . . 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 257 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
167125, 166eqbrtrrd 5172 . . . . 5 ((𝜑𝑘 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
168167ralrimiva 3144 . . . 4 (𝜑 → ∀𝑘 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
16930ffnd 6738 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ)
170 breq1 5151 . . . . . 6 (𝑦 = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) → (𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅)))
171170ralrn 7108 . . . . 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 257 . . 3 (𝜑 → ∀𝑦 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
174 supxrleub 13365 . . . 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 257 . 2 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
1775, 35, 41, 94, 176xrletrd 13201 1 (𝜑 → (vol*‘𝐵) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cin 3962  wss 3963  cop 4637   cuni 4912   class class class wbr 5148  cmpt 5231   × cxp 5687  ran crn 5690  ccom 5693   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  supcsup 9478  cc 11151  cr 11152  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158  +∞cpnf 11290  *cxr 11292   < clt 11293  cle 11294  cmin 11490   / cdiv 11918  cn 12264  cuz 12876  +crp 13032  (,)cioo 13384  [,)cico 13386  ...cfz 13544  seqcseq 14039  abscabs 15270  Σcsu 15719  vol*covol 25511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-ioo 13388  df-ico 13390  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-sum 15720  df-ovol 25513
This theorem is referenced by:  ovolscalem2  25563
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