Proof of Theorem lediv2
| Step | Hyp | Ref
| Expression |
| 1 | | gt0ne0 11728 |
. . . . 5
⊢ ((𝐵 ∈ ℝ ∧ 0 <
𝐵) → 𝐵 ≠ 0) |
| 2 | | rereccl 11985 |
. . . . 5
⊢ ((𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (1 / 𝐵) ∈
ℝ) |
| 3 | 1, 2 | syldan 591 |
. . . 4
⊢ ((𝐵 ∈ ℝ ∧ 0 <
𝐵) → (1 / 𝐵) ∈
ℝ) |
| 4 | 3 | 3ad2ant2 1135 |
. . 3
⊢ (((𝐴 ∈ ℝ ∧ 0 <
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (1 / 𝐵) ∈ ℝ) |
| 5 | | gt0ne0 11728 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 0 <
𝐴) → 𝐴 ≠ 0) |
| 6 | | rereccl 11985 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈
ℝ) |
| 7 | 5, 6 | syldan 591 |
. . . 4
⊢ ((𝐴 ∈ ℝ ∧ 0 <
𝐴) → (1 / 𝐴) ∈
ℝ) |
| 8 | 7 | 3ad2ant1 1134 |
. . 3
⊢ (((𝐴 ∈ ℝ ∧ 0 <
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (1 / 𝐴) ∈ ℝ) |
| 9 | | simp3l 1202 |
. . 3
⊢ (((𝐴 ∈ ℝ ∧ 0 <
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → 𝐶 ∈ ℝ) |
| 10 | | simp3r 1203 |
. . 3
⊢ (((𝐴 ∈ ℝ ∧ 0 <
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → 0 < 𝐶) |
| 11 | | lemul2 12120 |
. . 3
⊢ (((1 /
𝐵) ∈ ℝ ∧ (1
/ 𝐴) ∈ ℝ ∧
(𝐶 ∈ ℝ ∧ 0
< 𝐶)) → ((1 / 𝐵) ≤ (1 / 𝐴) ↔ (𝐶 · (1 / 𝐵)) ≤ (𝐶 · (1 / 𝐴)))) |
| 12 | 4, 8, 9, 10, 11 | syl112anc 1376 |
. 2
⊢ (((𝐴 ∈ ℝ ∧ 0 <
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((1 / 𝐵) ≤ (1 / 𝐴) ↔ (𝐶 · (1 / 𝐵)) ≤ (𝐶 · (1 / 𝐴)))) |
| 13 | | lerec 12151 |
. . 3
⊢ (((𝐴 ∈ ℝ ∧ 0 <
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 ≤ 𝐵 ↔ (1 / 𝐵) ≤ (1 / 𝐴))) |
| 14 | 13 | 3adant3 1133 |
. 2
⊢ (((𝐴 ∈ ℝ ∧ 0 <
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ (1 / 𝐵) ≤ (1 / 𝐴))) |
| 15 | | recn 11245 |
. . . . . . 7
⊢ (𝐶 ∈ ℝ → 𝐶 ∈
ℂ) |
| 16 | | recn 11245 |
. . . . . . . . 9
⊢ (𝐵 ∈ ℝ → 𝐵 ∈
ℂ) |
| 17 | 16 | adantr 480 |
. . . . . . . 8
⊢ ((𝐵 ∈ ℝ ∧ 0 <
𝐵) → 𝐵 ∈ ℂ) |
| 18 | 17, 1 | jca 511 |
. . . . . . 7
⊢ ((𝐵 ∈ ℝ ∧ 0 <
𝐵) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) |
| 19 | | divrec 11938 |
. . . . . . . 8
⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐶 / 𝐵) = (𝐶 · (1 / 𝐵))) |
| 20 | 19 | 3expb 1121 |
. . . . . . 7
⊢ ((𝐶 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐶 / 𝐵) = (𝐶 · (1 / 𝐵))) |
| 21 | 15, 18, 20 | syl2an 596 |
. . . . . 6
⊢ ((𝐶 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 <
𝐵)) → (𝐶 / 𝐵) = (𝐶 · (1 / 𝐵))) |
| 22 | 21 | 3adant2 1132 |
. . . . 5
⊢ ((𝐶 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 <
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐶 / 𝐵) = (𝐶 · (1 / 𝐵))) |
| 23 | | recn 11245 |
. . . . . . . . 9
⊢ (𝐴 ∈ ℝ → 𝐴 ∈
ℂ) |
| 24 | 23 | adantr 480 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 0 <
𝐴) → 𝐴 ∈ ℂ) |
| 25 | 24, 5 | jca 511 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 0 <
𝐴) → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) |
| 26 | | divrec 11938 |
. . . . . . . 8
⊢ ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐶 / 𝐴) = (𝐶 · (1 / 𝐴))) |
| 27 | 26 | 3expb 1121 |
. . . . . . 7
⊢ ((𝐶 ∈ ℂ ∧ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) → (𝐶 / 𝐴) = (𝐶 · (1 / 𝐴))) |
| 28 | 15, 25, 27 | syl2an 596 |
. . . . . 6
⊢ ((𝐶 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 <
𝐴)) → (𝐶 / 𝐴) = (𝐶 · (1 / 𝐴))) |
| 29 | 28 | 3adant3 1133 |
. . . . 5
⊢ ((𝐶 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 <
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐶 / 𝐴) = (𝐶 · (1 / 𝐴))) |
| 30 | 22, 29 | breq12d 5156 |
. . . 4
⊢ ((𝐶 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 <
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐶 / 𝐵) ≤ (𝐶 / 𝐴) ↔ (𝐶 · (1 / 𝐵)) ≤ (𝐶 · (1 / 𝐴)))) |
| 31 | 30 | 3coml 1128 |
. . 3
⊢ (((𝐴 ∈ ℝ ∧ 0 <
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ 𝐶 ∈ ℝ) → ((𝐶 / 𝐵) ≤ (𝐶 / 𝐴) ↔ (𝐶 · (1 / 𝐵)) ≤ (𝐶 · (1 / 𝐴)))) |
| 32 | 31 | 3adant3r 1182 |
. 2
⊢ (((𝐴 ∈ ℝ ∧ 0 <
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐶 / 𝐵) ≤ (𝐶 / 𝐴) ↔ (𝐶 · (1 / 𝐵)) ≤ (𝐶 · (1 / 𝐴)))) |
| 33 | 12, 14, 32 | 3bitr4d 311 |
1
⊢ (((𝐴 ∈ ℝ ∧ 0 <
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ (𝐶 / 𝐵) ≤ (𝐶 / 𝐴))) |