Proof of Theorem voliooico
| Step | Hyp | Ref
| Expression |
| 1 | | iftrue 4531 |
. . . . . 6
⊢ (𝐴 < 𝐵 → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = (𝐵 − 𝐴)) |
| 2 | 1 | adantl 481 |
. . . . 5
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝐴 < 𝐵) → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = (𝐵 − 𝐴)) |
| 3 | | voliooico.2 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈ ℝ) |
| 4 | 3 | recnd 11289 |
. . . . . . . . 9
⊢ (𝜑 → 𝐵 ∈ ℂ) |
| 5 | 4 | subidd 11608 |
. . . . . . . 8
⊢ (𝜑 → (𝐵 − 𝐵) = 0) |
| 6 | 5 | eqcomd 2743 |
. . . . . . 7
⊢ (𝜑 → 0 = (𝐵 − 𝐵)) |
| 7 | 6 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → 0 = (𝐵 − 𝐵)) |
| 8 | | iffalse 4534 |
. . . . . . 7
⊢ (¬
𝐴 < 𝐵 → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = 0) |
| 9 | 8 | adantl 481 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = 0) |
| 10 | | simpll 767 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → 𝜑) |
| 11 | | voliooico.1 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 12 | 10, 11 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → 𝐴 ∈ ℝ) |
| 13 | 10, 3 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → 𝐵 ∈ ℝ) |
| 14 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵) |
| 15 | 14 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → 𝐴 ≤ 𝐵) |
| 16 | | simpr 484 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → ¬ 𝐴 < 𝐵) |
| 17 | 12, 13, 15, 16 | lenlteq 45375 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → 𝐴 = 𝐵) |
| 18 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝐴 = 𝐵 → (𝐵 − 𝐴) = (𝐵 − 𝐵)) |
| 19 | 18 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐵 − 𝐴) = (𝐵 − 𝐵)) |
| 20 | 10, 17, 19 | syl2anc 584 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → (𝐵 − 𝐴) = (𝐵 − 𝐵)) |
| 21 | 7, 9, 20 | 3eqtr4d 2787 |
. . . . 5
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ ¬ 𝐴 < 𝐵) → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = (𝐵 − 𝐴)) |
| 22 | 2, 21 | pm2.61dan 813 |
. . . 4
⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = (𝐵 − 𝐴)) |
| 23 | 22 | eqcomd 2743 |
. . 3
⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐵 − 𝐴) = if(𝐴 < 𝐵, (𝐵 − 𝐴), 0)) |
| 24 | 11 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ) |
| 25 | 3 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ) |
| 26 | | volioo 25604 |
. . . 4
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴)) |
| 27 | 24, 25, 14, 26 | syl3anc 1373 |
. . 3
⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴)) |
| 28 | | volico 45998 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) →
(vol‘(𝐴[,)𝐵)) = if(𝐴 < 𝐵, (𝐵 − 𝐴), 0)) |
| 29 | 11, 3, 28 | syl2anc 584 |
. . . 4
⊢ (𝜑 → (vol‘(𝐴[,)𝐵)) = if(𝐴 < 𝐵, (𝐵 − 𝐴), 0)) |
| 30 | 29 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴[,)𝐵)) = if(𝐴 < 𝐵, (𝐵 − 𝐴), 0)) |
| 31 | 23, 27, 30 | 3eqtr4d 2787 |
. 2
⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (vol‘(𝐴[,)𝐵))) |
| 32 | | simpl 482 |
. . 3
⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐵) → 𝜑) |
| 33 | | simpr 484 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐵) → ¬ 𝐴 ≤ 𝐵) |
| 34 | 32, 3 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ) |
| 35 | 32, 11 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ) |
| 36 | 34, 35 | ltnled 11408 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐵) → (𝐵 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐵)) |
| 37 | 33, 36 | mpbird 257 |
. . 3
⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐵) → 𝐵 < 𝐴) |
| 38 | 3 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐵 ∈ ℝ) |
| 39 | 11 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐴 ∈ ℝ) |
| 40 | | simpr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐵 < 𝐴) |
| 41 | 38, 39, 40 | ltled 11409 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐵 ≤ 𝐴) |
| 42 | 39 | rexrd 11311 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐴 ∈
ℝ*) |
| 43 | 38 | rexrd 11311 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐵 < 𝐴) → 𝐵 ∈
ℝ*) |
| 44 | | ioo0 13412 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴)) |
| 45 | 42, 43, 44 | syl2anc 584 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴)) |
| 46 | 41, 45 | mpbird 257 |
. . . . 5
⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴(,)𝐵) = ∅) |
| 47 | | ico0 13433 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴)) |
| 48 | 42, 43, 47 | syl2anc 584 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴)) |
| 49 | 41, 48 | mpbird 257 |
. . . . 5
⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴[,)𝐵) = ∅) |
| 50 | 46, 49 | eqtr4d 2780 |
. . . 4
⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴(,)𝐵) = (𝐴[,)𝐵)) |
| 51 | 50 | fveq2d 6910 |
. . 3
⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (vol‘(𝐴(,)𝐵)) = (vol‘(𝐴[,)𝐵))) |
| 52 | 32, 37, 51 | syl2anc 584 |
. 2
⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (vol‘(𝐴[,)𝐵))) |
| 53 | 31, 52 | pm2.61dan 813 |
1
⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) = (vol‘(𝐴[,)𝐵))) |