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Theorem ovolscalem1 25548
Description: Lemma for ovolsca 25550. (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 4080 . . . 4 {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴} ⊆ ℝ
31, 2eqsstrdi 4028 . . 3 (𝜑𝐵 ⊆ ℝ)
4 ovolcl 25513 . . 3 (𝐵 ⊆ ℝ → (vol*‘𝐵) ∈ ℝ*)
53, 4syl 17 . 2 (𝜑 → (vol*‘𝐵) ∈ ℝ*)
6 ovolsca.7 . . . . . . . . . . . 12 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
7 ovolfcl 25501 . . . . . . . . . . . 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 1145 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
108simp1d 1143 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
118simp2d 1144 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
12 ovolsca.2 . . . . . . . . . . . . 13 (𝜑𝐶 ∈ ℝ+)
1312rpregt0d 13083 . . . . . . . . . . . 12 (𝜑 → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
1413adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
15 lediv1 12133 . . . . . . . . . . 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 1373 . . . . . . . . . 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 5144 . . . . . . . . 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 13108 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) / 𝐶) ∈ ℝ)
2211, 20rerpdivcld 13108 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) / 𝐶) ∈ ℝ)
2321, 22opelxpd 5724 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩ ∈ (ℝ × ℝ))
2419, 23elind 4200 . . . . . . 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 2737 . . . . . . 7 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
28 eqid 2737 . . . . . . 7 seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
2927, 28ovolsf 25507 . . . . . 6 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
3026, 29syl 17 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
3130frnd 6744 . . . 4 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ (0[,)+∞))
32 icossxr 13472 . . . 4 (0[,)+∞) ⊆ ℝ*
3331, 32sstrdi 3996 . . 3 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ*)
34 supxrcl 13357 . . 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 13108 . . . 4 (𝜑 → ((vol*‘𝐴) / 𝐶) ∈ ℝ)
38 ovolsca.9 . . . . 5 (𝜑𝑅 ∈ ℝ+)
3938rpred 13077 . . . 4 (𝜑𝑅 ∈ ℝ)
4037, 39readdcld 11290 . . 3 (𝜑 → (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ)
4140rexrd 11311 . 2 (𝜑 → (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ*)
421eleq2d 2827 . . . . . . 7 (𝜑 → (𝑦𝐵𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴}))
43 oveq2 7439 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐶 · 𝑥) = (𝐶 · 𝑦))
4443eleq1d 2826 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐶 · 𝑥) ∈ 𝐴 ↔ (𝐶 · 𝑦) ∈ 𝐴))
4544elrab 3692 . . . . . . 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 5147 . . . . . . . . . . 11 (𝑥 = (𝐶 · 𝑦) → ((1st ‘(𝐹𝑛)) < 𝑥 ↔ (1st ‘(𝐹𝑛)) < (𝐶 · 𝑦)))
48 breq1 5146 . . . . . . . . . . 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 3179 . . . . . . . . 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 25502 . . . . . . . . . . . 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 5469 . . . . . . . . . . . . . . . 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 6910 . . . . . . . . . . . . . 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 8022 . . . . . . . . . . . . . 14 (1st ‘⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩) = ((1st ‘(𝐹𝑛)) / 𝐶)
6662, 65eqtrdi 2793 . . . . . . . . . . . . 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 5153 . . . . . . . . . . 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 12143 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑛)) ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (((1st ‘(𝐹𝑛)) / 𝐶) < 𝑦 ↔ (1st ‘(𝐹𝑛)) < (𝐶 · 𝑦)))
7369, 70, 71, 72syl3anc 1373 . . . . . . . . . . 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 12142 . . . . . . . . . . . 12 ((𝑦 ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐶 · 𝑦) < (2nd ‘(𝐹𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹𝑛)) / 𝐶)))
7770, 75, 71, 76syl3anc 1373 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝐶 · 𝑦) < (2nd ‘(𝐹𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹𝑛)) / 𝐶)))
7861fveq2d 6910 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (2nd ‘(𝐺𝑛)) = (2nd ‘⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩))
7963, 64op2nd 8023 . . . . . . . . . . . . . 14 (2nd ‘⟨((1st ‘(𝐹𝑛)) / 𝐶), ((2nd ‘(𝐹𝑛)) / 𝐶)⟩) = ((2nd ‘(𝐹𝑛)) / 𝐶)
8078, 79eqtrdi 2793 . . . . . . . . . . . . 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 5155 . . . . . . . . . . 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 3177 . . . . . . . 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 3145 . . . 4 (𝜑 → ∀𝑦𝐵𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))))
90 ovolfioo 25502 . . . . 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 25514 . . 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 14014 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (1...𝑘) ∈ Fin)
9612rpcnd 13079 . . . . . . . . 9 (𝜑𝐶 ∈ ℂ)
9796adantr 480 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐶 ∈ ℂ)
98 simpl 482 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝜑)
99 elfznn 13593 . . . . . . . . . 10 (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
10011, 10resubcld 11691 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ∈ ℝ)
10198, 99, 100syl2an 596 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ∈ ℝ)
102101recnd 11289 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ∈ ℂ)
10312rpne0d 13082 . . . . . . . . 9 (𝜑𝐶 ≠ 0)
104103adantr 480 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐶 ≠ 0)
10595, 97, 102, 104fsumdivc 15822 . . . . . . 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 25505 . . . . . . . . . . 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 11289 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℂ)
11110recnd 11289 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℂ)
11212rpcnne0d 13086 . . . . . . . . . . . 12 (𝜑 → (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0))
113112adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0))
114 divsubdir 11961 . . . . . . . . . . 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 1373 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) = (((2nd ‘(𝐹𝑛)) / 𝐶) − ((1st ‘(𝐹𝑛)) / 𝐶)))
116107, 109, 1153eqtr4d 2787 . . . . . . . . 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 12921 . . . . . . . . 9 ℕ = (ℤ‘1)
120118, 119eleqtrdi 2851 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
121100, 20rerpdivcld 13108 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) ∈ ℝ)
122121recnd 11289 . . . . . . . . 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 15766 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)(((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) / 𝐶) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘))
125105, 124eqtrd 2777 . . . . . 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 2737 . . . . . . . . . . . . . . . 16 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
128 ovolsca.5 . . . . . . . . . . . . . . . 16 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
129127, 128ovolsf 25507 . . . . . . . . . . . . . . 15 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
1306, 129syl 17 . . . . . . . . . . . . . 14 (𝜑𝑆:ℕ⟶(0[,)+∞))
131130frnd 6744 . . . . . . . . . . . . 13 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
132131, 32sstrdi 3996 . . . . . . . . . . . 12 (𝜑 → ran 𝑆 ⊆ ℝ*)
13312, 38rpmulcld 13093 . . . . . . . . . . . . . . 15 (𝜑 → (𝐶 · 𝑅) ∈ ℝ+)
134133rpred 13077 . . . . . . . . . . . . . 14 (𝜑 → (𝐶 · 𝑅) ∈ ℝ)
13536, 134readdcld 11290 . . . . . . . . . . . . 13 (𝜑 → ((vol*‘𝐴) + (𝐶 · 𝑅)) ∈ ℝ)
136135rexrd 11311 . . . . . . . . . . . 12 (𝜑 → ((vol*‘𝐴) + (𝐶 · 𝑅)) ∈ ℝ*)
137 supxrleub 13368 . . . . . . . . . . . 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 6737 . . . . . . . . . . 11 (𝜑𝑆 Fn ℕ)
141 breq1 5146 . . . . . . . . . . . 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 3251 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
1466adantr 480 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
147127ovolfsval 25505 . . . . . . . . . . 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 15766 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘))
150128fveq1i 6907 . . . . . . . . 9 (𝑆𝑘) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘)
151149, 150eqtr4di 2795 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) = (𝑆𝑘))
15237recnd 11289 . . . . . . . . . . 11 (𝜑 → ((vol*‘𝐴) / 𝐶) ∈ ℂ)
15338rpcnd 13079 . . . . . . . . . . 11 (𝜑𝑅 ∈ ℂ)
15496, 152, 153adddid 11285 . . . . . . . . . 10 (𝜑 → (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)) = ((𝐶 · ((vol*‘𝐴) / 𝐶)) + (𝐶 · 𝑅)))
15536recnd 11289 . . . . . . . . . . . 12 (𝜑 → (vol*‘𝐴) ∈ ℂ)
156155, 96, 103divcan2d 12045 . . . . . . . . . . 11 (𝜑 → (𝐶 · ((vol*‘𝐴) / 𝐶)) = (vol*‘𝐴))
157156oveq1d 7446 . . . . . . . . . 10 (𝜑 → ((𝐶 · ((vol*‘𝐴) / 𝐶)) + (𝐶 · 𝑅)) = ((vol*‘𝐴) + (𝐶 · 𝑅)))
158154, 157eqtrd 2777 . . . . . . . . 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 5175 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))) ≤ (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)))
16195, 101fsumrecl 15770 . . . . . . . 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 12144 . . . . . . . 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 1373 . . . . . . 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 5167 . . . . 5 ((𝜑𝑘 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
168167ralrimiva 3146 . . . 4 (𝜑 → ∀𝑘 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
16930ffnd 6737 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ)
170 breq1 5146 . . . . . 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 13368 . . . 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 13204 1 (𝜑 → (vol*‘𝐵) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cin 3950  wss 3951  cop 4632   cuni 4907   class class class wbr 5143  cmpt 5225   × cxp 5683  ran crn 5686  ccom 5689   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  supcsup 9480  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  +∞cpnf 11292  *cxr 11294   < clt 11295  cle 11296  cmin 11492   / cdiv 11920  cn 12266  cuz 12878  +crp 13034  (,)cioo 13387  [,)cico 13389  ...cfz 13547  seqcseq 14042  abscabs 15273  Σcsu 15722  vol*covol 25497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-ioo 13391  df-ico 13393  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-ovol 25499
This theorem is referenced by:  ovolscalem2  25549
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