Proof of Theorem sharhght
| Step | Hyp | Ref
| Expression |
| 1 | | sharhght.a |
. . . . . . . . 9
⊢ (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) |
| 2 | 1 | simp3d 1145 |
. . . . . . . 8
⊢ (𝜑 → 𝐶 ∈ ℂ) |
| 3 | 1 | simp1d 1143 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 4 | 2, 3 | subcld 11620 |
. . . . . . 7
⊢ (𝜑 → (𝐶 − 𝐴) ∈ ℂ) |
| 5 | 4 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → (𝐶 − 𝐴) ∈ ℂ) |
| 6 | | sharhght.b |
. . . . . . . . 9
⊢ (𝜑 → (𝐷 ∈ ℂ ∧ ((𝐴 − 𝐷)𝐺(𝐵 − 𝐷)) = 0)) |
| 7 | 6 | simpld 494 |
. . . . . . . 8
⊢ (𝜑 → 𝐷 ∈ ℂ) |
| 8 | 7, 3 | subcld 11620 |
. . . . . . 7
⊢ (𝜑 → (𝐷 − 𝐴) ∈ ℂ) |
| 9 | 8 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → (𝐷 − 𝐴) ∈ ℂ) |
| 10 | | sharhght.sigar |
. . . . . . 7
⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦
(ℑ‘((∗‘𝑥) · 𝑦))) |
| 11 | 10 | sigarim 46866 |
. . . . . 6
⊢ (((𝐶 − 𝐴) ∈ ℂ ∧ (𝐷 − 𝐴) ∈ ℂ) → ((𝐶 − 𝐴)𝐺(𝐷 − 𝐴)) ∈ ℝ) |
| 12 | 5, 9, 11 | syl2anc 584 |
. . . . 5
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → ((𝐶 − 𝐴)𝐺(𝐷 − 𝐴)) ∈ ℝ) |
| 13 | 12 | recnd 11289 |
. . . 4
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → ((𝐶 − 𝐴)𝐺(𝐷 − 𝐴)) ∈ ℂ) |
| 14 | 13 | mul01d 11460 |
. . 3
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → (((𝐶 − 𝐴)𝐺(𝐷 − 𝐴)) · 0) = 0) |
| 15 | 1 | simp2d 1144 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ ℂ) |
| 16 | 15 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → 𝐵 ∈ ℂ) |
| 17 | | simpr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → 𝐵 = 𝐷) |
| 18 | 16, 17 | subeq0bd 11689 |
. . . 4
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → (𝐵 − 𝐷) = 0) |
| 19 | 18 | oveq2d 7447 |
. . 3
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → (((𝐶 − 𝐴)𝐺(𝐷 − 𝐴)) · (𝐵 − 𝐷)) = (((𝐶 − 𝐴)𝐺(𝐷 − 𝐴)) · 0)) |
| 20 | 2, 15 | subcld 11620 |
. . . . . . . 8
⊢ (𝜑 → (𝐶 − 𝐵) ∈ ℂ) |
| 21 | 20 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → (𝐶 − 𝐵) ∈ ℂ) |
| 22 | 7, 15 | subcld 11620 |
. . . . . . . 8
⊢ (𝜑 → (𝐷 − 𝐵) ∈ ℂ) |
| 23 | 22 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → (𝐷 − 𝐵) ∈ ℂ) |
| 24 | 10 | sigarval 46865 |
. . . . . . 7
⊢ (((𝐶 − 𝐵) ∈ ℂ ∧ (𝐷 − 𝐵) ∈ ℂ) → ((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) = (ℑ‘((∗‘(𝐶 − 𝐵)) · (𝐷 − 𝐵)))) |
| 25 | 21, 23, 24 | syl2anc 584 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → ((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) = (ℑ‘((∗‘(𝐶 − 𝐵)) · (𝐷 − 𝐵)))) |
| 26 | 7 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → 𝐷 ∈ ℂ) |
| 27 | 17 | eqcomd 2743 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → 𝐷 = 𝐵) |
| 28 | 26, 27 | subeq0bd 11689 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → (𝐷 − 𝐵) = 0) |
| 29 | 28 | oveq2d 7447 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → ((∗‘(𝐶 − 𝐵)) · (𝐷 − 𝐵)) = ((∗‘(𝐶 − 𝐵)) · 0)) |
| 30 | 21 | cjcld 15235 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → (∗‘(𝐶 − 𝐵)) ∈ ℂ) |
| 31 | 30 | mul01d 11460 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → ((∗‘(𝐶 − 𝐵)) · 0) = 0) |
| 32 | 29, 31 | eqtrd 2777 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → ((∗‘(𝐶 − 𝐵)) · (𝐷 − 𝐵)) = 0) |
| 33 | 32 | fveq2d 6910 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐵 = 𝐷) →
(ℑ‘((∗‘(𝐶 − 𝐵)) · (𝐷 − 𝐵))) = (ℑ‘0)) |
| 34 | | 0red 11264 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → 0 ∈ ℝ) |
| 35 | 34 | reim0d 15264 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → (ℑ‘0) =
0) |
| 36 | 25, 33, 35 | 3eqtrd 2781 |
. . . . 5
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → ((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) = 0) |
| 37 | 36 | oveq1d 7446 |
. . . 4
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → (((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) · (𝐴 − 𝐷)) = (0 · (𝐴 − 𝐷))) |
| 38 | 3 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → 𝐴 ∈ ℂ) |
| 39 | 38, 26 | subcld 11620 |
. . . . 5
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → (𝐴 − 𝐷) ∈ ℂ) |
| 40 | 39 | mul02d 11459 |
. . . 4
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → (0 · (𝐴 − 𝐷)) = 0) |
| 41 | 37, 40 | eqtrd 2777 |
. . 3
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → (((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) · (𝐴 − 𝐷)) = 0) |
| 42 | 14, 19, 41 | 3eqtr4d 2787 |
. 2
⊢ ((𝜑 ∧ 𝐵 = 𝐷) → (((𝐶 − 𝐴)𝐺(𝐷 − 𝐴)) · (𝐵 − 𝐷)) = (((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) · (𝐴 − 𝐷))) |
| 43 | 2 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → 𝐶 ∈ ℂ) |
| 44 | 15 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → 𝐵 ∈ ℂ) |
| 45 | 3 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → 𝐴 ∈ ℂ) |
| 46 | 43, 44, 45 | npncand 11644 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶 − 𝐵) + (𝐵 − 𝐴)) = (𝐶 − 𝐴)) |
| 47 | 46 | oveq1d 7446 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶 − 𝐵) + (𝐵 − 𝐴))𝐺(𝐷 − 𝐴)) = ((𝐶 − 𝐴)𝐺(𝐷 − 𝐴))) |
| 48 | 43, 44 | subcld 11620 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐶 − 𝐵) ∈ ℂ) |
| 49 | 8 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐷 − 𝐴) ∈ ℂ) |
| 50 | 44, 45 | subcld 11620 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐵 − 𝐴) ∈ ℂ) |
| 51 | 10 | sigaraf 46868 |
. . . . . . . 8
⊢ (((𝐶 − 𝐵) ∈ ℂ ∧ (𝐷 − 𝐴) ∈ ℂ ∧ (𝐵 − 𝐴) ∈ ℂ) → (((𝐶 − 𝐵) + (𝐵 − 𝐴))𝐺(𝐷 − 𝐴)) = (((𝐶 − 𝐵)𝐺(𝐷 − 𝐴)) + ((𝐵 − 𝐴)𝐺(𝐷 − 𝐴)))) |
| 52 | 48, 49, 50, 51 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶 − 𝐵) + (𝐵 − 𝐴))𝐺(𝐷 − 𝐴)) = (((𝐶 − 𝐵)𝐺(𝐷 − 𝐴)) + ((𝐵 − 𝐴)𝐺(𝐷 − 𝐴)))) |
| 53 | 47, 52 | eqtr3d 2779 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶 − 𝐴)𝐺(𝐷 − 𝐴)) = (((𝐶 − 𝐵)𝐺(𝐷 − 𝐴)) + ((𝐵 − 𝐴)𝐺(𝐷 − 𝐴)))) |
| 54 | 6 | simprd 495 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐴 − 𝐷)𝐺(𝐵 − 𝐷)) = 0) |
| 55 | 54 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐴 − 𝐷)𝐺(𝐵 − 𝐷)) = 0) |
| 56 | 7 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → 𝐷 ∈ ℂ) |
| 57 | 10 | sigarperm 46875 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐷 ∈ ℂ) → ((𝐴 − 𝐷)𝐺(𝐵 − 𝐷)) = ((𝐵 − 𝐴)𝐺(𝐷 − 𝐴))) |
| 58 | 45, 44, 56, 57 | syl3anc 1373 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐴 − 𝐷)𝐺(𝐵 − 𝐷)) = ((𝐵 − 𝐴)𝐺(𝐷 − 𝐴))) |
| 59 | 55, 58 | eqtr3d 2779 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → 0 = ((𝐵 − 𝐴)𝐺(𝐷 − 𝐴))) |
| 60 | 59 | oveq2d 7447 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶 − 𝐵)𝐺(𝐷 − 𝐴)) + 0) = (((𝐶 − 𝐵)𝐺(𝐷 − 𝐴)) + ((𝐵 − 𝐴)𝐺(𝐷 − 𝐴)))) |
| 61 | 10 | sigarim 46866 |
. . . . . . . . 9
⊢ (((𝐶 − 𝐵) ∈ ℂ ∧ (𝐷 − 𝐴) ∈ ℂ) → ((𝐶 − 𝐵)𝐺(𝐷 − 𝐴)) ∈ ℝ) |
| 62 | 48, 49, 61 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶 − 𝐵)𝐺(𝐷 − 𝐴)) ∈ ℝ) |
| 63 | 62 | recnd 11289 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶 − 𝐵)𝐺(𝐷 − 𝐴)) ∈ ℂ) |
| 64 | 63 | addridd 11461 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶 − 𝐵)𝐺(𝐷 − 𝐴)) + 0) = ((𝐶 − 𝐵)𝐺(𝐷 − 𝐴))) |
| 65 | 53, 60, 64 | 3eqtr2d 2783 |
. . . . 5
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶 − 𝐴)𝐺(𝐷 − 𝐴)) = ((𝐶 − 𝐵)𝐺(𝐷 − 𝐴))) |
| 66 | 44, 56 | negsubdi2d 11636 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → -(𝐵 − 𝐷) = (𝐷 − 𝐵)) |
| 67 | 66 | eqcomd 2743 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐷 − 𝐵) = -(𝐵 − 𝐷)) |
| 68 | 67 | oveq1d 7446 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐷 − 𝐵) / (𝐵 − 𝐷)) = (-(𝐵 − 𝐷) / (𝐵 − 𝐷))) |
| 69 | 44, 56 | subcld 11620 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐵 − 𝐷) ∈ ℂ) |
| 70 | | simpr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ¬ 𝐵 = 𝐷) |
| 71 | 70 | neqned 2947 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → 𝐵 ≠ 𝐷) |
| 72 | 44, 56, 71 | subne0d 11629 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐵 − 𝐷) ≠ 0) |
| 73 | 69, 69, 72 | divnegd 12056 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → -((𝐵 − 𝐷) / (𝐵 − 𝐷)) = (-(𝐵 − 𝐷) / (𝐵 − 𝐷))) |
| 74 | 69, 72 | dividd 12041 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐵 − 𝐷) / (𝐵 − 𝐷)) = 1) |
| 75 | 74 | negeqd 11502 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → -((𝐵 − 𝐷) / (𝐵 − 𝐷)) = -1) |
| 76 | 68, 73, 75 | 3eqtr2d 2783 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐷 − 𝐵) / (𝐵 − 𝐷)) = -1) |
| 77 | 76 | oveq1d 7446 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐷 − 𝐵) / (𝐵 − 𝐷)) · (𝐴 − 𝐷)) = (-1 · (𝐴 − 𝐷))) |
| 78 | 45, 56 | subcld 11620 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐴 − 𝐷) ∈ ℂ) |
| 79 | 78 | mulm1d 11715 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (-1 · (𝐴 − 𝐷)) = -(𝐴 − 𝐷)) |
| 80 | 45, 56 | negsubdi2d 11636 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → -(𝐴 − 𝐷) = (𝐷 − 𝐴)) |
| 81 | 77, 79, 80 | 3eqtrd 2781 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐷 − 𝐵) / (𝐵 − 𝐷)) · (𝐴 − 𝐷)) = (𝐷 − 𝐴)) |
| 82 | 56, 44 | subcld 11620 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐷 − 𝐵) ∈ ℂ) |
| 83 | 82, 69, 78, 72 | div32d 12066 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐷 − 𝐵) / (𝐵 − 𝐷)) · (𝐴 − 𝐷)) = ((𝐷 − 𝐵) · ((𝐴 − 𝐷) / (𝐵 − 𝐷)))) |
| 84 | 81, 83 | eqtr3d 2779 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐷 − 𝐴) = ((𝐷 − 𝐵) · ((𝐴 − 𝐷) / (𝐵 − 𝐷)))) |
| 85 | 84 | oveq2d 7447 |
. . . . 5
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶 − 𝐵)𝐺(𝐷 − 𝐴)) = ((𝐶 − 𝐵)𝐺((𝐷 − 𝐵) · ((𝐴 − 𝐷) / (𝐵 − 𝐷))))) |
| 86 | 56, 45, 44 | 3jca 1129 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (𝐷 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) |
| 87 | 10, 86, 70, 55 | sigardiv 46876 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐴 − 𝐷) / (𝐵 − 𝐷)) ∈ ℝ) |
| 88 | 10 | sigarls 46872 |
. . . . . 6
⊢ (((𝐶 − 𝐵) ∈ ℂ ∧ (𝐷 − 𝐵) ∈ ℂ ∧ ((𝐴 − 𝐷) / (𝐵 − 𝐷)) ∈ ℝ) → ((𝐶 − 𝐵)𝐺((𝐷 − 𝐵) · ((𝐴 − 𝐷) / (𝐵 − 𝐷)))) = (((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) · ((𝐴 − 𝐷) / (𝐵 − 𝐷)))) |
| 89 | 48, 82, 87, 88 | syl3anc 1373 |
. . . . 5
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶 − 𝐵)𝐺((𝐷 − 𝐵) · ((𝐴 − 𝐷) / (𝐵 − 𝐷)))) = (((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) · ((𝐴 − 𝐷) / (𝐵 − 𝐷)))) |
| 90 | 65, 85, 89 | 3eqtrd 2781 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶 − 𝐴)𝐺(𝐷 − 𝐴)) = (((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) · ((𝐴 − 𝐷) / (𝐵 − 𝐷)))) |
| 91 | 90 | oveq1d 7446 |
. . 3
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶 − 𝐴)𝐺(𝐷 − 𝐴)) · (𝐵 − 𝐷)) = ((((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) · ((𝐴 − 𝐷) / (𝐵 − 𝐷))) · (𝐵 − 𝐷))) |
| 92 | 10 | sigarim 46866 |
. . . . . 6
⊢ (((𝐶 − 𝐵) ∈ ℂ ∧ (𝐷 − 𝐵) ∈ ℂ) → ((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) ∈ ℝ) |
| 93 | 92 | recnd 11289 |
. . . . 5
⊢ (((𝐶 − 𝐵) ∈ ℂ ∧ (𝐷 − 𝐵) ∈ ℂ) → ((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) ∈ ℂ) |
| 94 | 48, 82, 93 | syl2anc 584 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) ∈ ℂ) |
| 95 | 78, 69, 72 | divcld 12043 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((𝐴 − 𝐷) / (𝐵 − 𝐷)) ∈ ℂ) |
| 96 | 94, 95, 69 | mulassd 11284 |
. . 3
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → ((((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) · ((𝐴 − 𝐷) / (𝐵 − 𝐷))) · (𝐵 − 𝐷)) = (((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) · (((𝐴 − 𝐷) / (𝐵 − 𝐷)) · (𝐵 − 𝐷)))) |
| 97 | 78, 69, 72 | divcan1d 12044 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐴 − 𝐷) / (𝐵 − 𝐷)) · (𝐵 − 𝐷)) = (𝐴 − 𝐷)) |
| 98 | 97 | oveq2d 7447 |
. . 3
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) · (((𝐴 − 𝐷) / (𝐵 − 𝐷)) · (𝐵 − 𝐷))) = (((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) · (𝐴 − 𝐷))) |
| 99 | 91, 96, 98 | 3eqtrd 2781 |
. 2
⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐷) → (((𝐶 − 𝐴)𝐺(𝐷 − 𝐴)) · (𝐵 − 𝐷)) = (((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) · (𝐴 − 𝐷))) |
| 100 | 42, 99 | pm2.61dan 813 |
1
⊢ (𝜑 → (((𝐶 − 𝐴)𝐺(𝐷 − 𝐴)) · (𝐵 − 𝐷)) = (((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) · (𝐴 − 𝐷))) |