Proof of Theorem bj-bary1lem1
| Step | Hyp | Ref
| Expression |
| 1 | | bj-bary1.s |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ ℂ) |
| 2 | | bj-bary1.t |
. . . . . . 7
⊢ (𝜑 → 𝑇 ∈ ℂ) |
| 3 | 1, 2 | pncand 11621 |
. . . . . 6
⊢ (𝜑 → ((𝑆 + 𝑇) − 𝑇) = 𝑆) |
| 4 | | oveq1 7438 |
. . . . . 6
⊢ ((𝑆 + 𝑇) = 1 → ((𝑆 + 𝑇) − 𝑇) = (1 − 𝑇)) |
| 5 | | pm5.31 831 |
. . . . . 6
⊢ ((((𝑆 + 𝑇) − 𝑇) = 𝑆 ∧ ((𝑆 + 𝑇) = 1 → ((𝑆 + 𝑇) − 𝑇) = (1 − 𝑇))) → ((𝑆 + 𝑇) = 1 → (((𝑆 + 𝑇) − 𝑇) = (1 − 𝑇) ∧ ((𝑆 + 𝑇) − 𝑇) = 𝑆))) |
| 6 | 3, 4, 5 | sylancl 586 |
. . . . 5
⊢ (𝜑 → ((𝑆 + 𝑇) = 1 → (((𝑆 + 𝑇) − 𝑇) = (1 − 𝑇) ∧ ((𝑆 + 𝑇) − 𝑇) = 𝑆))) |
| 7 | | eqtr2 2761 |
. . . . . 6
⊢ ((((𝑆 + 𝑇) − 𝑇) = (1 − 𝑇) ∧ ((𝑆 + 𝑇) − 𝑇) = 𝑆) → (1 − 𝑇) = 𝑆) |
| 8 | 7 | eqcomd 2743 |
. . . . 5
⊢ ((((𝑆 + 𝑇) − 𝑇) = (1 − 𝑇) ∧ ((𝑆 + 𝑇) − 𝑇) = 𝑆) → 𝑆 = (1 − 𝑇)) |
| 9 | 6, 8 | syl6 35 |
. . . 4
⊢ (𝜑 → ((𝑆 + 𝑇) = 1 → 𝑆 = (1 − 𝑇))) |
| 10 | | oveq1 7438 |
. . . . . . . 8
⊢ (𝑆 = (1 − 𝑇) → (𝑆 · 𝐴) = ((1 − 𝑇) · 𝐴)) |
| 11 | 10 | oveq1d 7446 |
. . . . . . 7
⊢ (𝑆 = (1 − 𝑇) → ((𝑆 · 𝐴) + (𝑇 · 𝐵)) = (((1 − 𝑇) · 𝐴) + (𝑇 · 𝐵))) |
| 12 | | eqtr 2760 |
. . . . . . 7
⊢ ((𝑋 = ((𝑆 · 𝐴) + (𝑇 · 𝐵)) ∧ ((𝑆 · 𝐴) + (𝑇 · 𝐵)) = (((1 − 𝑇) · 𝐴) + (𝑇 · 𝐵))) → 𝑋 = (((1 − 𝑇) · 𝐴) + (𝑇 · 𝐵))) |
| 13 | 11, 12 | sylan2 593 |
. . . . . 6
⊢ ((𝑋 = ((𝑆 · 𝐴) + (𝑇 · 𝐵)) ∧ 𝑆 = (1 − 𝑇)) → 𝑋 = (((1 − 𝑇) · 𝐴) + (𝑇 · 𝐵))) |
| 14 | | 1cnd 11256 |
. . . . . . . . 9
⊢ (𝜑 → 1 ∈
ℂ) |
| 15 | | bj-bary1.a |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 16 | 14, 2, 15 | subdird 11720 |
. . . . . . . 8
⊢ (𝜑 → ((1 − 𝑇) · 𝐴) = ((1 · 𝐴) − (𝑇 · 𝐴))) |
| 17 | 15 | mullidd 11279 |
. . . . . . . . 9
⊢ (𝜑 → (1 · 𝐴) = 𝐴) |
| 18 | 17 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝜑 → ((1 · 𝐴) − (𝑇 · 𝐴)) = (𝐴 − (𝑇 · 𝐴))) |
| 19 | 16, 18 | eqtrd 2777 |
. . . . . . 7
⊢ (𝜑 → ((1 − 𝑇) · 𝐴) = (𝐴 − (𝑇 · 𝐴))) |
| 20 | 19 | oveq1d 7446 |
. . . . . 6
⊢ (𝜑 → (((1 − 𝑇) · 𝐴) + (𝑇 · 𝐵)) = ((𝐴 − (𝑇 · 𝐴)) + (𝑇 · 𝐵))) |
| 21 | 13, 20 | sylan9eqr 2799 |
. . . . 5
⊢ ((𝜑 ∧ (𝑋 = ((𝑆 · 𝐴) + (𝑇 · 𝐵)) ∧ 𝑆 = (1 − 𝑇))) → 𝑋 = ((𝐴 − (𝑇 · 𝐴)) + (𝑇 · 𝐵))) |
| 22 | 21 | ex 412 |
. . . 4
⊢ (𝜑 → ((𝑋 = ((𝑆 · 𝐴) + (𝑇 · 𝐵)) ∧ 𝑆 = (1 − 𝑇)) → 𝑋 = ((𝐴 − (𝑇 · 𝐴)) + (𝑇 · 𝐵)))) |
| 23 | 9, 22 | sylan2d 605 |
. . 3
⊢ (𝜑 → ((𝑋 = ((𝑆 · 𝐴) + (𝑇 · 𝐵)) ∧ (𝑆 + 𝑇) = 1) → 𝑋 = ((𝐴 − (𝑇 · 𝐴)) + (𝑇 · 𝐵)))) |
| 24 | 2, 15 | mulcld 11281 |
. . . . . 6
⊢ (𝜑 → (𝑇 · 𝐴) ∈ ℂ) |
| 25 | | bj-bary1.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ ℂ) |
| 26 | 2, 25 | mulcld 11281 |
. . . . . 6
⊢ (𝜑 → (𝑇 · 𝐵) ∈ ℂ) |
| 27 | 15, 24, 26 | subadd23d 11642 |
. . . . 5
⊢ (𝜑 → ((𝐴 − (𝑇 · 𝐴)) + (𝑇 · 𝐵)) = (𝐴 + ((𝑇 · 𝐵) − (𝑇 · 𝐴)))) |
| 28 | 2, 25, 15 | subdid 11719 |
. . . . . . 7
⊢ (𝜑 → (𝑇 · (𝐵 − 𝐴)) = ((𝑇 · 𝐵) − (𝑇 · 𝐴))) |
| 29 | 28 | eqcomd 2743 |
. . . . . 6
⊢ (𝜑 → ((𝑇 · 𝐵) − (𝑇 · 𝐴)) = (𝑇 · (𝐵 − 𝐴))) |
| 30 | 29 | oveq2d 7447 |
. . . . 5
⊢ (𝜑 → (𝐴 + ((𝑇 · 𝐵) − (𝑇 · 𝐴))) = (𝐴 + (𝑇 · (𝐵 − 𝐴)))) |
| 31 | 27, 30 | eqtrd 2777 |
. . . 4
⊢ (𝜑 → ((𝐴 − (𝑇 · 𝐴)) + (𝑇 · 𝐵)) = (𝐴 + (𝑇 · (𝐵 − 𝐴)))) |
| 32 | 31 | eqeq2d 2748 |
. . 3
⊢ (𝜑 → (𝑋 = ((𝐴 − (𝑇 · 𝐴)) + (𝑇 · 𝐵)) ↔ 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))) |
| 33 | 23, 32 | sylibd 239 |
. 2
⊢ (𝜑 → ((𝑋 = ((𝑆 · 𝐴) + (𝑇 · 𝐵)) ∧ (𝑆 + 𝑇) = 1) → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))) |
| 34 | | oveq1 7438 |
. . 3
⊢ (𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))) → (𝑋 − 𝐴) = ((𝐴 + (𝑇 · (𝐵 − 𝐴))) − 𝐴)) |
| 35 | 25, 15 | subcld 11620 |
. . . . . 6
⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ) |
| 36 | 2, 35 | mulcld 11281 |
. . . . 5
⊢ (𝜑 → (𝑇 · (𝐵 − 𝐴)) ∈ ℂ) |
| 37 | 15, 36 | pncan2d 11622 |
. . . 4
⊢ (𝜑 → ((𝐴 + (𝑇 · (𝐵 − 𝐴))) − 𝐴) = (𝑇 · (𝐵 − 𝐴))) |
| 38 | 37 | eqeq2d 2748 |
. . 3
⊢ (𝜑 → ((𝑋 − 𝐴) = ((𝐴 + (𝑇 · (𝐵 − 𝐴))) − 𝐴) ↔ (𝑋 − 𝐴) = (𝑇 · (𝐵 − 𝐴)))) |
| 39 | 34, 38 | imbitrid 244 |
. 2
⊢ (𝜑 → (𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))) → (𝑋 − 𝐴) = (𝑇 · (𝐵 − 𝐴)))) |
| 40 | | eqcom 2744 |
. . 3
⊢ ((𝑋 − 𝐴) = (𝑇 · (𝐵 − 𝐴)) ↔ (𝑇 · (𝐵 − 𝐴)) = (𝑋 − 𝐴)) |
| 41 | 2, 35 | mulcomd 11282 |
. . . . 5
⊢ (𝜑 → (𝑇 · (𝐵 − 𝐴)) = ((𝐵 − 𝐴) · 𝑇)) |
| 42 | 41 | eqeq1d 2739 |
. . . 4
⊢ (𝜑 → ((𝑇 · (𝐵 − 𝐴)) = (𝑋 − 𝐴) ↔ ((𝐵 − 𝐴) · 𝑇) = (𝑋 − 𝐴))) |
| 43 | | bj-bary1.x |
. . . . . . 7
⊢ (𝜑 → 𝑋 ∈ ℂ) |
| 44 | 43, 15 | subcld 11620 |
. . . . . 6
⊢ (𝜑 → (𝑋 − 𝐴) ∈ ℂ) |
| 45 | | bj-bary1.neq |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ≠ 𝐵) |
| 46 | 45 | necomd 2996 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ≠ 𝐴) |
| 47 | 25, 15, 46 | subne0d 11629 |
. . . . . 6
⊢ (𝜑 → (𝐵 − 𝐴) ≠ 0) |
| 48 | 35, 2, 44, 47 | rdiv 12102 |
. . . . 5
⊢ (𝜑 → (((𝐵 − 𝐴) · 𝑇) = (𝑋 − 𝐴) ↔ 𝑇 = ((𝑋 − 𝐴) / (𝐵 − 𝐴)))) |
| 49 | 48 | biimpd 229 |
. . . 4
⊢ (𝜑 → (((𝐵 − 𝐴) · 𝑇) = (𝑋 − 𝐴) → 𝑇 = ((𝑋 − 𝐴) / (𝐵 − 𝐴)))) |
| 50 | 42, 49 | sylbid 240 |
. . 3
⊢ (𝜑 → ((𝑇 · (𝐵 − 𝐴)) = (𝑋 − 𝐴) → 𝑇 = ((𝑋 − 𝐴) / (𝐵 − 𝐴)))) |
| 51 | 40, 50 | biimtrid 242 |
. 2
⊢ (𝜑 → ((𝑋 − 𝐴) = (𝑇 · (𝐵 − 𝐴)) → 𝑇 = ((𝑋 − 𝐴) / (𝐵 − 𝐴)))) |
| 52 | 33, 39, 51 | 3syld 60 |
1
⊢ (𝜑 → ((𝑋 = ((𝑆 · 𝐴) + (𝑇 · 𝐵)) ∧ (𝑆 + 𝑇) = 1) → 𝑇 = ((𝑋 − 𝐴) / (𝐵 − 𝐴)))) |