| Step | Hyp | Ref
| Expression |
| 1 | | hoidmv1lelem3.b |
. . 3
⊢ (𝜑 → 𝐵 ∈ ℝ) |
| 2 | | hoidmv1lelem3.a |
. . 3
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 3 | 1, 2 | resubcld 11691 |
. 2
⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
| 4 | | nnex 12272 |
. . . . . . 7
⊢ ℕ
∈ V |
| 5 | 4 | a1i 11 |
. . . . . 6
⊢ (𝜑 → ℕ ∈
V) |
| 6 | | icossicc 13476 |
. . . . . . . 8
⊢
(0[,)+∞) ⊆ (0[,]+∞) |
| 7 | | 0xr 11308 |
. . . . . . . . . 10
⊢ 0 ∈
ℝ* |
| 8 | 7 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 0 ∈
ℝ*) |
| 9 | | pnfxr 11315 |
. . . . . . . . . 10
⊢ +∞
∈ ℝ* |
| 10 | 9 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → +∞ ∈
ℝ*) |
| 11 | | hoidmv1lelem3.c |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐶:ℕ⟶ℝ) |
| 12 | 11 | ffvelcdmda 7104 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ ℝ) |
| 13 | | hoidmv1lelem3.d |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐷:ℕ⟶ℝ) |
| 14 | 13 | ffvelcdmda 7104 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ ℝ) |
| 15 | 1 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐵 ∈ ℝ) |
| 16 | 14, 15 | ifcld 4572 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ∈ ℝ) |
| 17 | | volicore 46596 |
. . . . . . . . . . 11
⊢ (((𝐶‘𝑗) ∈ ℝ ∧ if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ∈ ℝ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈ ℝ) |
| 18 | 12, 16, 17 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈ ℝ) |
| 19 | 18 | rexrd 11311 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈
ℝ*) |
| 20 | 16 | rexrd 11311 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ∈
ℝ*) |
| 21 | | icombl 25599 |
. . . . . . . . . . 11
⊢ (((𝐶‘𝑗) ∈ ℝ ∧ if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ∈ ℝ*) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)) ∈ dom vol) |
| 22 | 12, 20, 21 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)) ∈ dom vol) |
| 23 | | volge0 45976 |
. . . . . . . . . 10
⊢ (((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)) ∈ dom vol → 0 ≤
(vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))) |
| 24 | 22, 23 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 0 ≤
(vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))) |
| 25 | 18 | ltpnfd 13163 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) < +∞) |
| 26 | 8, 10, 19, 24, 25 | elicod 13437 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈ (0[,)+∞)) |
| 27 | 6, 26 | sselid 3981 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈ (0[,]+∞)) |
| 28 | | eqid 2737 |
. . . . . . 7
⊢ (𝑗 ∈ ℕ ↦
(vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))) |
| 29 | 27, 28 | fmptd 7134 |
. . . . . 6
⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))):ℕ⟶(0[,]+∞)) |
| 30 | 5, 29 | sge0xrcl 46400 |
. . . . 5
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) ∈
ℝ*) |
| 31 | 9 | a1i 11 |
. . . . 5
⊢ (𝜑 → +∞ ∈
ℝ*) |
| 32 | | hoidmv1lelem3.r |
. . . . . . 7
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈ ℝ) |
| 33 | 32 | rexrd 11311 |
. . . . . 6
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈
ℝ*) |
| 34 | | nfv 1914 |
. . . . . . 7
⊢
Ⅎ𝑗𝜑 |
| 35 | | volf 25564 |
. . . . . . . . 9
⊢ vol:dom
vol⟶(0[,]+∞) |
| 36 | 35 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → vol:dom
vol⟶(0[,]+∞)) |
| 37 | 14 | rexrd 11311 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈
ℝ*) |
| 38 | | icombl 25599 |
. . . . . . . . 9
⊢ (((𝐶‘𝑗) ∈ ℝ ∧ (𝐷‘𝑗) ∈ ℝ*) → ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol) |
| 39 | 12, 37, 38 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol) |
| 40 | 36, 39 | ffvelcdmd 7105 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∈ (0[,]+∞)) |
| 41 | 12 | rexrd 11311 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈
ℝ*) |
| 42 | 12 | leidd 11829 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ≤ (𝐶‘𝑗)) |
| 43 | | min1 13231 |
. . . . . . . . . 10
⊢ (((𝐷‘𝑗) ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ≤ (𝐷‘𝑗)) |
| 44 | 14, 15, 43 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ≤ (𝐷‘𝑗)) |
| 45 | | icossico 13457 |
. . . . . . . . 9
⊢ ((((𝐶‘𝑗) ∈ ℝ* ∧ (𝐷‘𝑗) ∈ ℝ*) ∧ ((𝐶‘𝑗) ≤ (𝐶‘𝑗) ∧ if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ≤ (𝐷‘𝑗))) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) |
| 46 | 41, 37, 42, 44, 45 | syl22anc 839 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) |
| 47 | | volss 25568 |
. . . . . . . 8
⊢ ((((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)) ∈ dom vol ∧ ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol ∧ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ≤ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗)))) |
| 48 | 22, 39, 46, 47 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ≤ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗)))) |
| 49 | 34, 5, 27, 40, 48 | sge0lempt 46425 |
. . . . . 6
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗)))))) |
| 50 | 32 | ltpnfd 13163 |
. . . . . 6
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) < +∞) |
| 51 | 30, 33, 31, 49, 50 | xrlelttrd 13202 |
. . . . 5
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) < +∞) |
| 52 | 30, 31, 51 | xrltned 45368 |
. . . 4
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) ≠ +∞) |
| 53 | 52 | neneqd 2945 |
. . 3
⊢ (𝜑 → ¬
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) = +∞) |
| 54 | 5, 29 | sge0repnf 46401 |
. . 3
⊢ (𝜑 →
((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) ∈ ℝ ↔ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) = +∞)) |
| 55 | 53, 54 | mpbird 257 |
. 2
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) ∈ ℝ) |
| 56 | 1 | rexrd 11311 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
| 57 | 2, 1 | iccssred 13474 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
| 58 | | hoidmv1lelem3.u |
. . . . . . . . . . 11
⊢ 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))} |
| 59 | | ssrab2 4080 |
. . . . . . . . . . 11
⊢ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))} ⊆ (𝐴[,]𝐵) |
| 60 | 58, 59 | eqsstri 4030 |
. . . . . . . . . 10
⊢ 𝑈 ⊆ (𝐴[,]𝐵) |
| 61 | | hoidmv1lelem3.l |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 < 𝐵) |
| 62 | | hoidmv1lelem3.s |
. . . . . . . . . . . 12
⊢ 𝑆 = sup(𝑈, ℝ, < ) |
| 63 | 2, 1, 61, 11, 13, 32, 58, 62 | hoidmv1lelem1 46606 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑆 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥)) |
| 64 | 63 | simp1d 1143 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑆 ∈ 𝑈) |
| 65 | 60, 64 | sselid 3981 |
. . . . . . . . 9
⊢ (𝜑 → 𝑆 ∈ (𝐴[,]𝐵)) |
| 66 | 57, 65 | sseldd 3984 |
. . . . . . . 8
⊢ (𝜑 → 𝑆 ∈ ℝ) |
| 67 | 66 | rexrd 11311 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈
ℝ*) |
| 68 | | simpl 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → 𝜑) |
| 69 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → ¬ 𝐵 ≤ 𝑆) |
| 70 | 68, 66 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → 𝑆 ∈ ℝ) |
| 71 | 68, 1 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → 𝐵 ∈ ℝ) |
| 72 | 70, 71 | ltnled 11408 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → (𝑆 < 𝐵 ↔ ¬ 𝐵 ≤ 𝑆)) |
| 73 | 69, 72 | mpbird 257 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → 𝑆 < 𝐵) |
| 74 | | hoidmv1lelem3.x |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐴[,)𝐵) ⊆ ∪ 𝑗 ∈ ℕ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) |
| 75 | 74 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑆 < 𝐵) → (𝐴[,)𝐵) ⊆ ∪ 𝑗 ∈ ℕ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) |
| 76 | 2 | rexrd 11311 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
| 77 | 76 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐴 ∈
ℝ*) |
| 78 | 56 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐵 ∈
ℝ*) |
| 79 | 67 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 ∈
ℝ*) |
| 80 | 60, 57 | sstrid 3995 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑈 ⊆ ℝ) |
| 81 | 64 | ne0d 4342 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑈 ≠ ∅) |
| 82 | 63 | simp3d 1145 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥) |
| 83 | 63 | simp2d 1144 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐴 ∈ 𝑈) |
| 84 | | suprub 12229 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥) ∧ 𝐴 ∈ 𝑈) → 𝐴 ≤ sup(𝑈, ℝ, < )) |
| 85 | 80, 81, 82, 83, 84 | syl31anc 1375 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐴 ≤ sup(𝑈, ℝ, < )) |
| 86 | 85, 62 | breqtrrdi 5185 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐴 ≤ 𝑆) |
| 87 | 86 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐴 ≤ 𝑆) |
| 88 | | simpr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 < 𝐵) |
| 89 | 77, 78, 79, 87, 88 | elicod 13437 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 ∈ (𝐴[,)𝐵)) |
| 90 | 75, 89 | sseldd 3984 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 ∈ ∪
𝑗 ∈ ℕ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) |
| 91 | | eliun 4995 |
. . . . . . . . . . 11
⊢ (𝑆 ∈ ∪ 𝑗 ∈ ℕ ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ↔ ∃𝑗 ∈ ℕ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) |
| 92 | 90, 91 | sylib 218 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑆 < 𝐵) → ∃𝑗 ∈ ℕ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) |
| 93 | 2 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐴 ∈ ℝ) |
| 94 | 93 | 3ad2ant1 1134 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝐴 ∈ ℝ) |
| 95 | 1 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐵 ∈ ℝ) |
| 96 | 95 | 3ad2ant1 1134 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝐵 ∈ ℝ) |
| 97 | 11 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐶:ℕ⟶ℝ) |
| 98 | 97 | 3ad2ant1 1134 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝐶:ℕ⟶ℝ) |
| 99 | 13 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐷:ℕ⟶ℝ) |
| 100 | 99 | 3ad2ant1 1134 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝐷:ℕ⟶ℝ) |
| 101 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 = 𝑗 → (𝐶‘𝑖) = (𝐶‘𝑗)) |
| 102 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗)) |
| 103 | 101, 102 | oveq12d 7449 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)[,)(𝐷‘𝑖)) = ((𝐶‘𝑗)[,)(𝐷‘𝑗))) |
| 104 | 103 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = 𝑗 → (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))) = (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗)))) |
| 105 | 104 | cbvmptv 5255 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 ∈ ℕ ↦
(vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗)))) |
| 106 | 105 | fveq2i 6909 |
. . . . . . . . . . . . . . . 16
⊢
(Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))))) =
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) |
| 107 | 106, 32 | eqeltrid 2845 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 →
(Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))))) ∈ ℝ) |
| 108 | 107 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑆 < 𝐵) →
(Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))))) ∈ ℝ) |
| 109 | 108 | 3ad2ant1 1134 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) →
(Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))))) ∈ ℝ) |
| 110 | 102 | breq1d 5153 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 = 𝑗 → ((𝐷‘𝑖) ≤ 𝑧 ↔ (𝐷‘𝑗) ≤ 𝑧)) |
| 111 | 110, 102 | ifbieq1d 4550 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 = 𝑗 → if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧) = if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) |
| 112 | 101, 111 | oveq12d 7449 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧)) = ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))) |
| 113 | 112 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 = 𝑗 → (vol‘((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧))) = (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))) |
| 114 | 113 | cbvmptv 5255 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ ℕ ↦
(vol‘((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))) |
| 115 | 114 | eqcomi 2746 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ ℕ ↦
(vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))) = (𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧)))) |
| 116 | 115 | fveq2i 6909 |
. . . . . . . . . . . . . . . 16
⊢
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) =
(Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧))))) |
| 117 | 116 | breq2i 5151 |
. . . . . . . . . . . . . . 15
⊢ ((𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ↔ (𝑧 − 𝐴) ≤
(Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧)))))) |
| 118 | 117 | rabbii 3442 |
. . . . . . . . . . . . . 14
⊢ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))} = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤
(Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧)))))} |
| 119 | 58, 118 | eqtri 2765 |
. . . . . . . . . . . . 13
⊢ 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤
(Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧)))))} |
| 120 | 64 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 ∈ 𝑈) |
| 121 | 120 | 3ad2ant1 1134 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑆 ∈ 𝑈) |
| 122 | 87 | 3ad2ant1 1134 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝐴 ≤ 𝑆) |
| 123 | 88 | 3ad2ant1 1134 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑆 < 𝐵) |
| 124 | | simp2 1138 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑗 ∈ ℕ) |
| 125 | | simp3 1139 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) |
| 126 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) = if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) |
| 127 | 94, 96, 98, 100, 109, 119, 121, 122, 123, 124, 125, 126 | hoidmv1lelem2 46607 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢) |
| 128 | 127 | 3exp 1120 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑆 < 𝐵) → (𝑗 ∈ ℕ → (𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢))) |
| 129 | 128 | rexlimdv 3153 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑆 < 𝐵) → (∃𝑗 ∈ ℕ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)) |
| 130 | 92, 129 | mpd 15 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑆 < 𝐵) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢) |
| 131 | 68, 73, 130 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢) |
| 132 | 57 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴[,]𝐵) ⊆ ℝ) |
| 133 | 60, 132 | sstrid 3995 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ⊆ ℝ) |
| 134 | 81 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ≠ ∅) |
| 135 | 2, 1 | jca 511 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) |
| 136 | 135 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) |
| 137 | 60 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ⊆ (𝐴[,]𝐵)) |
| 138 | 64 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑆 ∈ 𝑈) |
| 139 | | iccsupr 13482 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑈 ⊆ (𝐴[,]𝐵) ∧ 𝑆 ∈ 𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥)) |
| 140 | 136, 137,
138, 139 | syl3anc 1373 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥)) |
| 141 | 140 | simp3d 1145 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥) |
| 142 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ 𝑈) |
| 143 | | suprub 12229 |
. . . . . . . . . . . . . 14
⊢ (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥) ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < )) |
| 144 | 133, 134,
141, 142, 143 | syl31anc 1375 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < )) |
| 145 | 144, 62 | breqtrrdi 5185 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ 𝑆) |
| 146 | 145 | ralrimiva 3146 |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑆) |
| 147 | 60 | sseli 3979 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 ∈ 𝑈 → 𝑢 ∈ (𝐴[,]𝐵)) |
| 148 | 147 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ (𝐴[,]𝐵)) |
| 149 | 132, 148 | sseldd 3984 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ ℝ) |
| 150 | 66 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑆 ∈ ℝ) |
| 151 | 149, 150 | lenltd 11407 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 ≤ 𝑆 ↔ ¬ 𝑆 < 𝑢)) |
| 152 | 151 | ralbidva 3176 |
. . . . . . . . . . 11
⊢ (𝜑 → (∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑆 ↔ ∀𝑢 ∈ 𝑈 ¬ 𝑆 < 𝑢)) |
| 153 | 146, 152 | mpbid 232 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑢 ∈ 𝑈 ¬ 𝑆 < 𝑢) |
| 154 | | ralnex 3072 |
. . . . . . . . . 10
⊢
(∀𝑢 ∈
𝑈 ¬ 𝑆 < 𝑢 ↔ ¬ ∃𝑢 ∈ 𝑈 𝑆 < 𝑢) |
| 155 | 153, 154 | sylib 218 |
. . . . . . . . 9
⊢ (𝜑 → ¬ ∃𝑢 ∈ 𝑈 𝑆 < 𝑢) |
| 156 | 155 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → ¬ ∃𝑢 ∈ 𝑈 𝑆 < 𝑢) |
| 157 | 131, 156 | condan 818 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ≤ 𝑆) |
| 158 | | iccleub 13442 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝑆
∈ (𝐴[,]𝐵)) → 𝑆 ≤ 𝐵) |
| 159 | 76, 56, 65, 158 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ≤ 𝐵) |
| 160 | 56, 67, 157, 159 | xrletrid 13197 |
. . . . . 6
⊢ (𝜑 → 𝐵 = 𝑆) |
| 161 | 160, 64 | eqeltrd 2841 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ 𝑈) |
| 162 | 161, 58 | eleqtrdi 2851 |
. . . 4
⊢ (𝜑 → 𝐵 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))}) |
| 163 | | oveq1 7438 |
. . . . . 6
⊢ (𝑧 = 𝐵 → (𝑧 − 𝐴) = (𝐵 − 𝐴)) |
| 164 | | breq2 5147 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝐵 → ((𝐷‘𝑗) ≤ 𝑧 ↔ (𝐷‘𝑗) ≤ 𝐵)) |
| 165 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝐵 → 𝑧 = 𝐵) |
| 166 | 164, 165 | ifbieq2d 4552 |
. . . . . . . . . 10
⊢ (𝑧 = 𝐵 → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)) |
| 167 | 166 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝑧 = 𝐵 → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) = ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) |
| 168 | 167 | fveq2d 6910 |
. . . . . . . 8
⊢ (𝑧 = 𝐵 → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))) = (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))) |
| 169 | 168 | mpteq2dv 5244 |
. . . . . . 7
⊢ (𝑧 = 𝐵 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) |
| 170 | 169 | fveq2d 6910 |
. . . . . 6
⊢ (𝑧 = 𝐵 →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) =
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))))) |
| 171 | 163, 170 | breq12d 5156 |
. . . . 5
⊢ (𝑧 = 𝐵 → ((𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ↔ (𝐵 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))))) |
| 172 | 171 | elrab 3692 |
. . . 4
⊢ (𝐵 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))} ↔ (𝐵 ∈ (𝐴[,]𝐵) ∧ (𝐵 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))))) |
| 173 | 162, 172 | sylib 218 |
. . 3
⊢ (𝜑 → (𝐵 ∈ (𝐴[,]𝐵) ∧ (𝐵 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))))) |
| 174 | 173 | simprd 495 |
. 2
⊢ (𝜑 → (𝐵 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))))) |
| 175 | 3, 55, 32, 174, 49 | letrd 11418 |
1
⊢ (𝜑 → (𝐵 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗)))))) |