Step |
Hyp |
Ref |
Expression |
1 |
|
peano2cnm |
⊢ ( 𝐶 ∈ ℂ → ( 𝐶 − 1 ) ∈ ℂ ) |
2 |
|
id |
⊢ ( 𝐶 ∈ ℂ → 𝐶 ∈ ℂ ) |
3 |
|
4cn |
⊢ 4 ∈ ℂ |
4 |
3
|
a1i |
⊢ ( 𝐶 ∈ ℂ → 4 ∈ ℂ ) |
5 |
2 4
|
subcld |
⊢ ( 𝐶 ∈ ℂ → ( 𝐶 − 4 ) ∈ ℂ ) |
6 |
|
1cnd |
⊢ ( 𝐶 ∈ ℂ → 1 ∈ ℂ ) |
7 |
|
1re |
⊢ 1 ∈ ℝ |
8 |
|
1lt4 |
⊢ 1 < 4 |
9 |
7 8
|
ltneii |
⊢ 1 ≠ 4 |
10 |
9
|
a1i |
⊢ ( 𝐶 ∈ ℂ → 1 ≠ 4 ) |
11 |
2 6 4 10
|
subneintrd |
⊢ ( 𝐶 ∈ ℂ → ( 𝐶 − 1 ) ≠ ( 𝐶 − 4 ) ) |
12 |
|
oveq1 |
⊢ ( 𝑏 = 1 → ( 𝑏 ↑ 2 ) = ( 1 ↑ 2 ) ) |
13 |
12
|
oveq2d |
⊢ ( 𝑏 = 1 → ( ( 𝐶 − 1 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝐶 − 1 ) + ( 1 ↑ 2 ) ) ) |
14 |
13
|
eqeq1d |
⊢ ( 𝑏 = 1 → ( ( ( 𝐶 − 1 ) + ( 𝑏 ↑ 2 ) ) = 𝐶 ↔ ( ( 𝐶 − 1 ) + ( 1 ↑ 2 ) ) = 𝐶 ) ) |
15 |
14
|
adantl |
⊢ ( ( 𝐶 ∈ ℂ ∧ 𝑏 = 1 ) → ( ( ( 𝐶 − 1 ) + ( 𝑏 ↑ 2 ) ) = 𝐶 ↔ ( ( 𝐶 − 1 ) + ( 1 ↑ 2 ) ) = 𝐶 ) ) |
16 |
|
sq1 |
⊢ ( 1 ↑ 2 ) = 1 |
17 |
16
|
oveq2i |
⊢ ( ( 𝐶 − 1 ) + ( 1 ↑ 2 ) ) = ( ( 𝐶 − 1 ) + 1 ) |
18 |
|
npcan1 |
⊢ ( 𝐶 ∈ ℂ → ( ( 𝐶 − 1 ) + 1 ) = 𝐶 ) |
19 |
17 18
|
syl5eq |
⊢ ( 𝐶 ∈ ℂ → ( ( 𝐶 − 1 ) + ( 1 ↑ 2 ) ) = 𝐶 ) |
20 |
6 15 19
|
rspcedvd |
⊢ ( 𝐶 ∈ ℂ → ∃ 𝑏 ∈ ℂ ( ( 𝐶 − 1 ) + ( 𝑏 ↑ 2 ) ) = 𝐶 ) |
21 |
|
2cnd |
⊢ ( 𝐶 ∈ ℂ → 2 ∈ ℂ ) |
22 |
|
oveq1 |
⊢ ( 𝑏 = 2 → ( 𝑏 ↑ 2 ) = ( 2 ↑ 2 ) ) |
23 |
22
|
oveq2d |
⊢ ( 𝑏 = 2 → ( ( 𝐶 − 4 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝐶 − 4 ) + ( 2 ↑ 2 ) ) ) |
24 |
23
|
eqeq1d |
⊢ ( 𝑏 = 2 → ( ( ( 𝐶 − 4 ) + ( 𝑏 ↑ 2 ) ) = 𝐶 ↔ ( ( 𝐶 − 4 ) + ( 2 ↑ 2 ) ) = 𝐶 ) ) |
25 |
24
|
adantl |
⊢ ( ( 𝐶 ∈ ℂ ∧ 𝑏 = 2 ) → ( ( ( 𝐶 − 4 ) + ( 𝑏 ↑ 2 ) ) = 𝐶 ↔ ( ( 𝐶 − 4 ) + ( 2 ↑ 2 ) ) = 𝐶 ) ) |
26 |
|
2cn |
⊢ 2 ∈ ℂ |
27 |
26
|
sqcli |
⊢ ( 2 ↑ 2 ) ∈ ℂ |
28 |
27
|
a1i |
⊢ ( 𝐶 ∈ ℂ → ( 2 ↑ 2 ) ∈ ℂ ) |
29 |
2 4 28
|
subadd23d |
⊢ ( 𝐶 ∈ ℂ → ( ( 𝐶 − 4 ) + ( 2 ↑ 2 ) ) = ( 𝐶 + ( ( 2 ↑ 2 ) − 4 ) ) ) |
30 |
|
sq2 |
⊢ ( 2 ↑ 2 ) = 4 |
31 |
30
|
a1i |
⊢ ( 𝐶 ∈ ℂ → ( 2 ↑ 2 ) = 4 ) |
32 |
28 31
|
subeq0bd |
⊢ ( 𝐶 ∈ ℂ → ( ( 2 ↑ 2 ) − 4 ) = 0 ) |
33 |
27 3
|
subcli |
⊢ ( ( 2 ↑ 2 ) − 4 ) ∈ ℂ |
34 |
|
addid0 |
⊢ ( ( 𝐶 ∈ ℂ ∧ ( ( 2 ↑ 2 ) − 4 ) ∈ ℂ ) → ( ( 𝐶 + ( ( 2 ↑ 2 ) − 4 ) ) = 𝐶 ↔ ( ( 2 ↑ 2 ) − 4 ) = 0 ) ) |
35 |
33 34
|
mpan2 |
⊢ ( 𝐶 ∈ ℂ → ( ( 𝐶 + ( ( 2 ↑ 2 ) − 4 ) ) = 𝐶 ↔ ( ( 2 ↑ 2 ) − 4 ) = 0 ) ) |
36 |
32 35
|
mpbird |
⊢ ( 𝐶 ∈ ℂ → ( 𝐶 + ( ( 2 ↑ 2 ) − 4 ) ) = 𝐶 ) |
37 |
29 36
|
eqtrd |
⊢ ( 𝐶 ∈ ℂ → ( ( 𝐶 − 4 ) + ( 2 ↑ 2 ) ) = 𝐶 ) |
38 |
21 25 37
|
rspcedvd |
⊢ ( 𝐶 ∈ ℂ → ∃ 𝑏 ∈ ℂ ( ( 𝐶 − 4 ) + ( 𝑏 ↑ 2 ) ) = 𝐶 ) |
39 |
|
oveq1 |
⊢ ( 𝑎 = ( 𝐶 − 1 ) → ( 𝑎 + ( 𝑏 ↑ 2 ) ) = ( ( 𝐶 − 1 ) + ( 𝑏 ↑ 2 ) ) ) |
40 |
39
|
eqeq1d |
⊢ ( 𝑎 = ( 𝐶 − 1 ) → ( ( 𝑎 + ( 𝑏 ↑ 2 ) ) = 𝐶 ↔ ( ( 𝐶 − 1 ) + ( 𝑏 ↑ 2 ) ) = 𝐶 ) ) |
41 |
40
|
rexbidv |
⊢ ( 𝑎 = ( 𝐶 − 1 ) → ( ∃ 𝑏 ∈ ℂ ( 𝑎 + ( 𝑏 ↑ 2 ) ) = 𝐶 ↔ ∃ 𝑏 ∈ ℂ ( ( 𝐶 − 1 ) + ( 𝑏 ↑ 2 ) ) = 𝐶 ) ) |
42 |
|
oveq1 |
⊢ ( 𝑎 = ( 𝐶 − 4 ) → ( 𝑎 + ( 𝑏 ↑ 2 ) ) = ( ( 𝐶 − 4 ) + ( 𝑏 ↑ 2 ) ) ) |
43 |
42
|
eqeq1d |
⊢ ( 𝑎 = ( 𝐶 − 4 ) → ( ( 𝑎 + ( 𝑏 ↑ 2 ) ) = 𝐶 ↔ ( ( 𝐶 − 4 ) + ( 𝑏 ↑ 2 ) ) = 𝐶 ) ) |
44 |
43
|
rexbidv |
⊢ ( 𝑎 = ( 𝐶 − 4 ) → ( ∃ 𝑏 ∈ ℂ ( 𝑎 + ( 𝑏 ↑ 2 ) ) = 𝐶 ↔ ∃ 𝑏 ∈ ℂ ( ( 𝐶 − 4 ) + ( 𝑏 ↑ 2 ) ) = 𝐶 ) ) |
45 |
41 44
|
2nreu |
⊢ ( ( ( 𝐶 − 1 ) ∈ ℂ ∧ ( 𝐶 − 4 ) ∈ ℂ ∧ ( 𝐶 − 1 ) ≠ ( 𝐶 − 4 ) ) → ( ( ∃ 𝑏 ∈ ℂ ( ( 𝐶 − 1 ) + ( 𝑏 ↑ 2 ) ) = 𝐶 ∧ ∃ 𝑏 ∈ ℂ ( ( 𝐶 − 4 ) + ( 𝑏 ↑ 2 ) ) = 𝐶 ) → ¬ ∃! 𝑎 ∈ ℂ ∃ 𝑏 ∈ ℂ ( 𝑎 + ( 𝑏 ↑ 2 ) ) = 𝐶 ) ) |
46 |
45
|
imp |
⊢ ( ( ( ( 𝐶 − 1 ) ∈ ℂ ∧ ( 𝐶 − 4 ) ∈ ℂ ∧ ( 𝐶 − 1 ) ≠ ( 𝐶 − 4 ) ) ∧ ( ∃ 𝑏 ∈ ℂ ( ( 𝐶 − 1 ) + ( 𝑏 ↑ 2 ) ) = 𝐶 ∧ ∃ 𝑏 ∈ ℂ ( ( 𝐶 − 4 ) + ( 𝑏 ↑ 2 ) ) = 𝐶 ) ) → ¬ ∃! 𝑎 ∈ ℂ ∃ 𝑏 ∈ ℂ ( 𝑎 + ( 𝑏 ↑ 2 ) ) = 𝐶 ) |
47 |
1 5 11 20 38 46
|
syl32anc |
⊢ ( 𝐶 ∈ ℂ → ¬ ∃! 𝑎 ∈ ℂ ∃ 𝑏 ∈ ℂ ( 𝑎 + ( 𝑏 ↑ 2 ) ) = 𝐶 ) |