| Step | Hyp | Ref
| Expression |
| 1 | | id 22 |
. . 3
⊢ (𝐶 ∈ ℂ → 𝐶 ∈
ℂ) |
| 2 | | oveq1 7438 |
. . . . . . 7
⊢ (𝑎 = 𝐶 → (𝑎 + (𝑏↑2)) = (𝐶 + (𝑏↑2))) |
| 3 | 2 | eqeq1d 2739 |
. . . . . 6
⊢ (𝑎 = 𝐶 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝐶 + (𝑏↑2)) = 𝐶)) |
| 4 | 3 | reubidv 3398 |
. . . . 5
⊢ (𝑎 = 𝐶 → (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶)) |
| 5 | | eqeq1 2741 |
. . . . . . 7
⊢ (𝑎 = 𝐶 → (𝑎 = 𝑐 ↔ 𝐶 = 𝑐)) |
| 6 | 5 | imbi2d 340 |
. . . . . 6
⊢ (𝑎 = 𝐶 → ((∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝑎 = 𝑐) ↔ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐))) |
| 7 | 6 | ralbidv 3178 |
. . . . 5
⊢ (𝑎 = 𝐶 → (∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝑎 = 𝑐) ↔ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐))) |
| 8 | 4, 7 | anbi12d 632 |
. . . 4
⊢ (𝑎 = 𝐶 → ((∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝑎 = 𝑐)) ↔ (∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐)))) |
| 9 | 8 | adantl 481 |
. . 3
⊢ ((𝐶 ∈ ℂ ∧ 𝑎 = 𝐶) → ((∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝑎 = 𝑐)) ↔ (∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐)))) |
| 10 | | 0cnd 11254 |
. . . . . 6
⊢ (𝐶 ∈ ℂ → 0 ∈
ℂ) |
| 11 | | reueq 3743 |
. . . . . 6
⊢ (0 ∈
ℂ ↔ ∃!𝑏
∈ ℂ 𝑏 =
0) |
| 12 | 10, 11 | sylib 218 |
. . . . 5
⊢ (𝐶 ∈ ℂ →
∃!𝑏 ∈ ℂ
𝑏 = 0) |
| 13 | | subid 11528 |
. . . . . . . . 9
⊢ (𝐶 ∈ ℂ → (𝐶 − 𝐶) = 0) |
| 14 | 13 | adantr 480 |
. . . . . . . 8
⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝐶 − 𝐶) = 0) |
| 15 | 14 | eqeq1d 2739 |
. . . . . . 7
⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶 − 𝐶) = (𝑏↑2) ↔ 0 = (𝑏↑2))) |
| 16 | | simpl 482 |
. . . . . . . 8
⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → 𝐶 ∈
ℂ) |
| 17 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → 𝑏 ∈
ℂ) |
| 18 | 17 | sqcld 14184 |
. . . . . . . 8
⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑏↑2) ∈
ℂ) |
| 19 | 16, 16, 18 | subaddd 11638 |
. . . . . . 7
⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶 − 𝐶) = (𝑏↑2) ↔ (𝐶 + (𝑏↑2)) = 𝐶)) |
| 20 | | eqcom 2744 |
. . . . . . . . 9
⊢ (0 =
(𝑏↑2) ↔ (𝑏↑2) = 0) |
| 21 | | sqeq0 14160 |
. . . . . . . . 9
⊢ (𝑏 ∈ ℂ → ((𝑏↑2) = 0 ↔ 𝑏 = 0)) |
| 22 | 20, 21 | bitrid 283 |
. . . . . . . 8
⊢ (𝑏 ∈ ℂ → (0 =
(𝑏↑2) ↔ 𝑏 = 0)) |
| 23 | 22 | adantl 481 |
. . . . . . 7
⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (0 =
(𝑏↑2) ↔ 𝑏 = 0)) |
| 24 | 15, 19, 23 | 3bitr3d 309 |
. . . . . 6
⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶 + (𝑏↑2)) = 𝐶 ↔ 𝑏 = 0)) |
| 25 | 24 | reubidva 3396 |
. . . . 5
⊢ (𝐶 ∈ ℂ →
(∃!𝑏 ∈ ℂ
(𝐶 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ 𝑏 = 0)) |
| 26 | 12, 25 | mpbird 257 |
. . . 4
⊢ (𝐶 ∈ ℂ →
∃!𝑏 ∈ ℂ
(𝐶 + (𝑏↑2)) = 𝐶) |
| 27 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → 𝑐 ∈
ℂ) |
| 28 | 27 | adantr 480 |
. . . . . . . 8
⊢ (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → 𝑐 ∈
ℂ) |
| 29 | | sqcl 14158 |
. . . . . . . . 9
⊢ (𝑏 ∈ ℂ → (𝑏↑2) ∈
ℂ) |
| 30 | 29 | adantl 481 |
. . . . . . . 8
⊢ (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → (𝑏↑2) ∈
ℂ) |
| 31 | | simpl 482 |
. . . . . . . . 9
⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → 𝐶 ∈
ℂ) |
| 32 | 31 | adantr 480 |
. . . . . . . 8
⊢ (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → 𝐶 ∈
ℂ) |
| 33 | 28, 30, 32 | addrsub 11680 |
. . . . . . 7
⊢ (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → ((𝑐 + (𝑏↑2)) = 𝐶 ↔ (𝑏↑2) = (𝐶 − 𝑐))) |
| 34 | 33 | reubidva 3396 |
. . . . . 6
⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) →
(∃!𝑏 ∈ ℂ
(𝑐 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶 − 𝑐))) |
| 35 | | subcl 11507 |
. . . . . . . 8
⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (𝐶 − 𝑐) ∈ ℂ) |
| 36 | | reusq0 15501 |
. . . . . . . 8
⊢ ((𝐶 − 𝑐) ∈ ℂ → (∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶 − 𝑐) ↔ (𝐶 − 𝑐) = 0)) |
| 37 | 35, 36 | syl 17 |
. . . . . . 7
⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) →
(∃!𝑏 ∈ ℂ
(𝑏↑2) = (𝐶 − 𝑐) ↔ (𝐶 − 𝑐) = 0)) |
| 38 | | subeq0 11535 |
. . . . . . . 8
⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ((𝐶 − 𝑐) = 0 ↔ 𝐶 = 𝑐)) |
| 39 | 38 | biimpd 229 |
. . . . . . 7
⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ((𝐶 − 𝑐) = 0 → 𝐶 = 𝑐)) |
| 40 | 37, 39 | sylbid 240 |
. . . . . 6
⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) →
(∃!𝑏 ∈ ℂ
(𝑏↑2) = (𝐶 − 𝑐) → 𝐶 = 𝑐)) |
| 41 | 34, 40 | sylbid 240 |
. . . . 5
⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) →
(∃!𝑏 ∈ ℂ
(𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐)) |
| 42 | 41 | ralrimiva 3146 |
. . . 4
⊢ (𝐶 ∈ ℂ →
∀𝑐 ∈ ℂ
(∃!𝑏 ∈ ℂ
(𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐)) |
| 43 | 26, 42 | jca 511 |
. . 3
⊢ (𝐶 ∈ ℂ →
(∃!𝑏 ∈ ℂ
(𝐶 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐))) |
| 44 | 1, 9, 43 | rspcedvd 3624 |
. 2
⊢ (𝐶 ∈ ℂ →
∃𝑎 ∈ ℂ
(∃!𝑏 ∈ ℂ
(𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝑎 = 𝑐))) |
| 45 | | oveq1 7438 |
. . . . 5
⊢ (𝑎 = 𝑐 → (𝑎 + (𝑏↑2)) = (𝑐 + (𝑏↑2))) |
| 46 | 45 | eqeq1d 2739 |
. . . 4
⊢ (𝑎 = 𝑐 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑐 + (𝑏↑2)) = 𝐶)) |
| 47 | 46 | reubidv 3398 |
. . 3
⊢ (𝑎 = 𝑐 → (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶)) |
| 48 | 47 | reu8 3739 |
. 2
⊢
(∃!𝑎 ∈
ℂ ∃!𝑏 ∈
ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃𝑎 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝑎 = 𝑐))) |
| 49 | 44, 48 | sylibr 234 |
1
⊢ (𝐶 ∈ ℂ →
∃!𝑎 ∈ ℂ
∃!𝑏 ∈ ℂ
(𝑎 + (𝑏↑2)) = 𝐶) |