Proof of Theorem dcubic1
| Step | Hyp | Ref
| Expression |
| 1 | | dcubic.3 |
. . . . . . 7
⊢ (𝜑 → (𝑇↑3) = (𝐺 − 𝑁)) |
| 2 | 1 | oveq1d 7446 |
. . . . . 6
⊢ (𝜑 → ((𝑇↑3)↑2) = ((𝐺 − 𝑁)↑2)) |
| 3 | | dcubic.g |
. . . . . . 7
⊢ (𝜑 → 𝐺 ∈ ℂ) |
| 4 | | dcubic.n |
. . . . . . . 8
⊢ (𝜑 → 𝑁 = (𝑄 / 2)) |
| 5 | | dcubic.d |
. . . . . . . . 9
⊢ (𝜑 → 𝑄 ∈ ℂ) |
| 6 | 5 | halfcld 12511 |
. . . . . . . 8
⊢ (𝜑 → (𝑄 / 2) ∈ ℂ) |
| 7 | 4, 6 | eqeltrd 2841 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 8 | | binom2sub 14259 |
. . . . . . 7
⊢ ((𝐺 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝐺 − 𝑁)↑2) = (((𝐺↑2) − (2 · (𝐺 · 𝑁))) + (𝑁↑2))) |
| 9 | 3, 7, 8 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → ((𝐺 − 𝑁)↑2) = (((𝐺↑2) − (2 · (𝐺 · 𝑁))) + (𝑁↑2))) |
| 10 | | dcubic.2 |
. . . . . . . 8
⊢ (𝜑 → (𝐺↑2) = ((𝑁↑2) + (𝑀↑3))) |
| 11 | | 2cnd 12344 |
. . . . . . . . . 10
⊢ (𝜑 → 2 ∈
ℂ) |
| 12 | 11, 3, 7 | mul12d 11470 |
. . . . . . . . 9
⊢ (𝜑 → (2 · (𝐺 · 𝑁)) = (𝐺 · (2 · 𝑁))) |
| 13 | 4 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (𝜑 → (2 · 𝑁) = (2 · (𝑄 / 2))) |
| 14 | | 2ne0 12370 |
. . . . . . . . . . . . 13
⊢ 2 ≠
0 |
| 15 | 14 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → 2 ≠ 0) |
| 16 | 5, 11, 15 | divcan2d 12045 |
. . . . . . . . . . 11
⊢ (𝜑 → (2 · (𝑄 / 2)) = 𝑄) |
| 17 | 13, 16 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (𝜑 → (2 · 𝑁) = 𝑄) |
| 18 | 17 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺 · (2 · 𝑁)) = (𝐺 · 𝑄)) |
| 19 | 3, 5 | mulcomd 11282 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺 · 𝑄) = (𝑄 · 𝐺)) |
| 20 | 12, 18, 19 | 3eqtrd 2781 |
. . . . . . . 8
⊢ (𝜑 → (2 · (𝐺 · 𝑁)) = (𝑄 · 𝐺)) |
| 21 | 10, 20 | oveq12d 7449 |
. . . . . . 7
⊢ (𝜑 → ((𝐺↑2) − (2 · (𝐺 · 𝑁))) = (((𝑁↑2) + (𝑀↑3)) − (𝑄 · 𝐺))) |
| 22 | 21 | oveq1d 7446 |
. . . . . 6
⊢ (𝜑 → (((𝐺↑2) − (2 · (𝐺 · 𝑁))) + (𝑁↑2)) = ((((𝑁↑2) + (𝑀↑3)) − (𝑄 · 𝐺)) + (𝑁↑2))) |
| 23 | 2, 9, 22 | 3eqtrd 2781 |
. . . . 5
⊢ (𝜑 → ((𝑇↑3)↑2) = ((((𝑁↑2) + (𝑀↑3)) − (𝑄 · 𝐺)) + (𝑁↑2))) |
| 24 | 7 | sqcld 14184 |
. . . . . . 7
⊢ (𝜑 → (𝑁↑2) ∈ ℂ) |
| 25 | | dcubic.m |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 = (𝑃 / 3)) |
| 26 | | dcubic.c |
. . . . . . . . . 10
⊢ (𝜑 → 𝑃 ∈ ℂ) |
| 27 | | 3cn 12347 |
. . . . . . . . . . 11
⊢ 3 ∈
ℂ |
| 28 | 27 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → 3 ∈
ℂ) |
| 29 | | 3ne0 12372 |
. . . . . . . . . . 11
⊢ 3 ≠
0 |
| 30 | 29 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → 3 ≠ 0) |
| 31 | 26, 28, 30 | divcld 12043 |
. . . . . . . . 9
⊢ (𝜑 → (𝑃 / 3) ∈ ℂ) |
| 32 | 25, 31 | eqeltrd 2841 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ ℂ) |
| 33 | | 3nn0 12544 |
. . . . . . . 8
⊢ 3 ∈
ℕ0 |
| 34 | | expcl 14120 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℂ ∧ 3 ∈
ℕ0) → (𝑀↑3) ∈ ℂ) |
| 35 | 32, 33, 34 | sylancl 586 |
. . . . . . 7
⊢ (𝜑 → (𝑀↑3) ∈ ℂ) |
| 36 | 24, 35 | addcld 11280 |
. . . . . 6
⊢ (𝜑 → ((𝑁↑2) + (𝑀↑3)) ∈ ℂ) |
| 37 | 5, 3 | mulcld 11281 |
. . . . . 6
⊢ (𝜑 → (𝑄 · 𝐺) ∈ ℂ) |
| 38 | 36, 24, 37 | addsubd 11641 |
. . . . 5
⊢ (𝜑 → ((((𝑁↑2) + (𝑀↑3)) + (𝑁↑2)) − (𝑄 · 𝐺)) = ((((𝑁↑2) + (𝑀↑3)) − (𝑄 · 𝐺)) + (𝑁↑2))) |
| 39 | 24, 35, 24 | add32d 11489 |
. . . . . . 7
⊢ (𝜑 → (((𝑁↑2) + (𝑀↑3)) + (𝑁↑2)) = (((𝑁↑2) + (𝑁↑2)) + (𝑀↑3))) |
| 40 | 24 | 2timesd 12509 |
. . . . . . . 8
⊢ (𝜑 → (2 · (𝑁↑2)) = ((𝑁↑2) + (𝑁↑2))) |
| 41 | 40 | oveq1d 7446 |
. . . . . . 7
⊢ (𝜑 → ((2 · (𝑁↑2)) + (𝑀↑3)) = (((𝑁↑2) + (𝑁↑2)) + (𝑀↑3))) |
| 42 | 39, 41 | eqtr4d 2780 |
. . . . . 6
⊢ (𝜑 → (((𝑁↑2) + (𝑀↑3)) + (𝑁↑2)) = ((2 · (𝑁↑2)) + (𝑀↑3))) |
| 43 | 42 | oveq1d 7446 |
. . . . 5
⊢ (𝜑 → ((((𝑁↑2) + (𝑀↑3)) + (𝑁↑2)) − (𝑄 · 𝐺)) = (((2 · (𝑁↑2)) + (𝑀↑3)) − (𝑄 · 𝐺))) |
| 44 | 23, 38, 43 | 3eqtr2d 2783 |
. . . 4
⊢ (𝜑 → ((𝑇↑3)↑2) = (((2 · (𝑁↑2)) + (𝑀↑3)) − (𝑄 · 𝐺))) |
| 45 | 5, 3, 7 | subdid 11719 |
. . . . . . 7
⊢ (𝜑 → (𝑄 · (𝐺 − 𝑁)) = ((𝑄 · 𝐺) − (𝑄 · 𝑁))) |
| 46 | 1 | oveq2d 7447 |
. . . . . . 7
⊢ (𝜑 → (𝑄 · (𝑇↑3)) = (𝑄 · (𝐺 − 𝑁))) |
| 47 | 7 | sqvald 14183 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁↑2) = (𝑁 · 𝑁)) |
| 48 | 47 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝜑 → (2 · (𝑁↑2)) = (2 · (𝑁 · 𝑁))) |
| 49 | 11, 7, 7 | mulassd 11284 |
. . . . . . . . 9
⊢ (𝜑 → ((2 · 𝑁) · 𝑁) = (2 · (𝑁 · 𝑁))) |
| 50 | 17 | oveq1d 7446 |
. . . . . . . . 9
⊢ (𝜑 → ((2 · 𝑁) · 𝑁) = (𝑄 · 𝑁)) |
| 51 | 48, 49, 50 | 3eqtr2d 2783 |
. . . . . . . 8
⊢ (𝜑 → (2 · (𝑁↑2)) = (𝑄 · 𝑁)) |
| 52 | 51 | oveq2d 7447 |
. . . . . . 7
⊢ (𝜑 → ((𝑄 · 𝐺) − (2 · (𝑁↑2))) = ((𝑄 · 𝐺) − (𝑄 · 𝑁))) |
| 53 | 45, 46, 52 | 3eqtr4d 2787 |
. . . . . 6
⊢ (𝜑 → (𝑄 · (𝑇↑3)) = ((𝑄 · 𝐺) − (2 · (𝑁↑2)))) |
| 54 | 53 | oveq1d 7446 |
. . . . 5
⊢ (𝜑 → ((𝑄 · (𝑇↑3)) − (𝑀↑3)) = (((𝑄 · 𝐺) − (2 · (𝑁↑2))) − (𝑀↑3))) |
| 55 | | 2cn 12341 |
. . . . . . 7
⊢ 2 ∈
ℂ |
| 56 | | mulcl 11239 |
. . . . . . 7
⊢ ((2
∈ ℂ ∧ (𝑁↑2) ∈ ℂ) → (2 ·
(𝑁↑2)) ∈
ℂ) |
| 57 | 55, 24, 56 | sylancr 587 |
. . . . . 6
⊢ (𝜑 → (2 · (𝑁↑2)) ∈
ℂ) |
| 58 | 37, 57, 35 | subsub4d 11651 |
. . . . 5
⊢ (𝜑 → (((𝑄 · 𝐺) − (2 · (𝑁↑2))) − (𝑀↑3)) = ((𝑄 · 𝐺) − ((2 · (𝑁↑2)) + (𝑀↑3)))) |
| 59 | 54, 58 | eqtrd 2777 |
. . . 4
⊢ (𝜑 → ((𝑄 · (𝑇↑3)) − (𝑀↑3)) = ((𝑄 · 𝐺) − ((2 · (𝑁↑2)) + (𝑀↑3)))) |
| 60 | 44, 59 | oveq12d 7449 |
. . 3
⊢ (𝜑 → (((𝑇↑3)↑2) + ((𝑄 · (𝑇↑3)) − (𝑀↑3))) = ((((2 · (𝑁↑2)) + (𝑀↑3)) − (𝑄 · 𝐺)) + ((𝑄 · 𝐺) − ((2 · (𝑁↑2)) + (𝑀↑3))))) |
| 61 | 57, 35 | addcld 11280 |
. . . 4
⊢ (𝜑 → ((2 · (𝑁↑2)) + (𝑀↑3)) ∈ ℂ) |
| 62 | | npncan2 11536 |
. . . 4
⊢ ((((2
· (𝑁↑2)) +
(𝑀↑3)) ∈ ℂ
∧ (𝑄 · 𝐺) ∈ ℂ) → ((((2
· (𝑁↑2)) +
(𝑀↑3)) − (𝑄 · 𝐺)) + ((𝑄 · 𝐺) − ((2 · (𝑁↑2)) + (𝑀↑3)))) = 0) |
| 63 | 61, 37, 62 | syl2anc 584 |
. . 3
⊢ (𝜑 → ((((2 · (𝑁↑2)) + (𝑀↑3)) − (𝑄 · 𝐺)) + ((𝑄 · 𝐺) − ((2 · (𝑁↑2)) + (𝑀↑3)))) = 0) |
| 64 | 60, 63 | eqtrd 2777 |
. 2
⊢ (𝜑 → (((𝑇↑3)↑2) + ((𝑄 · (𝑇↑3)) − (𝑀↑3))) = 0) |
| 65 | | dcubic.x |
. . 3
⊢ (𝜑 → 𝑋 ∈ ℂ) |
| 66 | | dcubic.t |
. . 3
⊢ (𝜑 → 𝑇 ∈ ℂ) |
| 67 | | dcubic.0 |
. . 3
⊢ (𝜑 → 𝑇 ≠ 0) |
| 68 | | dcubic1.x |
. . 3
⊢ (𝜑 → 𝑋 = (𝑇 − (𝑀 / 𝑇))) |
| 69 | 26, 5, 65, 66, 1, 3, 10, 25, 4, 67, 66, 67, 68 | dcubic1lem 26886 |
. 2
⊢ (𝜑 → (((𝑋↑3) + ((𝑃 · 𝑋) + 𝑄)) = 0 ↔ (((𝑇↑3)↑2) + ((𝑄 · (𝑇↑3)) − (𝑀↑3))) = 0)) |
| 70 | 64, 69 | mpbird 257 |
1
⊢ (𝜑 → ((𝑋↑3) + ((𝑃 · 𝑋) + 𝑄)) = 0) |