Step |
Hyp |
Ref |
Expression |
1 |
|
quad1.a |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
quad1.z |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
3 |
|
quad1.b |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
4 |
|
quad1.c |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
5 |
|
quad1.d |
⊢ ( 𝜑 → 𝐷 = ( ( 𝐵 ↑ 2 ) − ( 4 · ( 𝐴 · 𝐶 ) ) ) ) |
6 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℂ ) → 𝐴 ∈ ℂ ) |
7 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℂ ) → 𝐴 ≠ 0 ) |
8 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℂ ) → 𝐵 ∈ ℂ ) |
9 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℂ ) → 𝐶 ∈ ℂ ) |
10 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℂ ) → 𝑥 ∈ ℂ ) |
11 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℂ ) → 𝐷 = ( ( 𝐵 ↑ 2 ) − ( 4 · ( 𝐴 · 𝐶 ) ) ) ) |
12 |
6 7 8 9 10 11
|
quad |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℂ ) → ( ( ( 𝐴 · ( 𝑥 ↑ 2 ) ) + ( ( 𝐵 · 𝑥 ) + 𝐶 ) ) = 0 ↔ ( 𝑥 = ( ( - 𝐵 + ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) ∨ 𝑥 = ( ( - 𝐵 − ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) ) ) ) |
13 |
12
|
reubidva |
⊢ ( 𝜑 → ( ∃! 𝑥 ∈ ℂ ( ( 𝐴 · ( 𝑥 ↑ 2 ) ) + ( ( 𝐵 · 𝑥 ) + 𝐶 ) ) = 0 ↔ ∃! 𝑥 ∈ ℂ ( 𝑥 = ( ( - 𝐵 + ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) ∨ 𝑥 = ( ( - 𝐵 − ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) ) ) ) |
14 |
3
|
negcld |
⊢ ( 𝜑 → - 𝐵 ∈ ℂ ) |
15 |
3
|
sqcld |
⊢ ( 𝜑 → ( 𝐵 ↑ 2 ) ∈ ℂ ) |
16 |
|
4cn |
⊢ 4 ∈ ℂ |
17 |
16
|
a1i |
⊢ ( 𝜑 → 4 ∈ ℂ ) |
18 |
1 4
|
mulcld |
⊢ ( 𝜑 → ( 𝐴 · 𝐶 ) ∈ ℂ ) |
19 |
17 18
|
mulcld |
⊢ ( 𝜑 → ( 4 · ( 𝐴 · 𝐶 ) ) ∈ ℂ ) |
20 |
15 19
|
subcld |
⊢ ( 𝜑 → ( ( 𝐵 ↑ 2 ) − ( 4 · ( 𝐴 · 𝐶 ) ) ) ∈ ℂ ) |
21 |
5 20
|
eqeltrd |
⊢ ( 𝜑 → 𝐷 ∈ ℂ ) |
22 |
21
|
sqrtcld |
⊢ ( 𝜑 → ( √ ‘ 𝐷 ) ∈ ℂ ) |
23 |
14 22
|
addcld |
⊢ ( 𝜑 → ( - 𝐵 + ( √ ‘ 𝐷 ) ) ∈ ℂ ) |
24 |
|
2cnd |
⊢ ( 𝜑 → 2 ∈ ℂ ) |
25 |
24 1
|
mulcld |
⊢ ( 𝜑 → ( 2 · 𝐴 ) ∈ ℂ ) |
26 |
|
2ne0 |
⊢ 2 ≠ 0 |
27 |
26
|
a1i |
⊢ ( 𝜑 → 2 ≠ 0 ) |
28 |
24 1 27 2
|
mulne0d |
⊢ ( 𝜑 → ( 2 · 𝐴 ) ≠ 0 ) |
29 |
23 25 28
|
divcld |
⊢ ( 𝜑 → ( ( - 𝐵 + ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) ∈ ℂ ) |
30 |
14 22
|
subcld |
⊢ ( 𝜑 → ( - 𝐵 − ( √ ‘ 𝐷 ) ) ∈ ℂ ) |
31 |
30 25 28
|
divcld |
⊢ ( 𝜑 → ( ( - 𝐵 − ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) ∈ ℂ ) |
32 |
|
euoreqb |
⊢ ( ( ( ( - 𝐵 + ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) ∈ ℂ ∧ ( ( - 𝐵 − ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) ∈ ℂ ) → ( ∃! 𝑥 ∈ ℂ ( 𝑥 = ( ( - 𝐵 + ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) ∨ 𝑥 = ( ( - 𝐵 − ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) ) ↔ ( ( - 𝐵 + ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) = ( ( - 𝐵 − ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) ) ) |
33 |
29 31 32
|
syl2anc |
⊢ ( 𝜑 → ( ∃! 𝑥 ∈ ℂ ( 𝑥 = ( ( - 𝐵 + ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) ∨ 𝑥 = ( ( - 𝐵 − ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) ) ↔ ( ( - 𝐵 + ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) = ( ( - 𝐵 − ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) ) ) |
34 |
14 22 25 28
|
divdird |
⊢ ( 𝜑 → ( ( - 𝐵 + ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) = ( ( - 𝐵 / ( 2 · 𝐴 ) ) + ( ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) ) ) |
35 |
14 22 25 28
|
divsubdird |
⊢ ( 𝜑 → ( ( - 𝐵 − ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) = ( ( - 𝐵 / ( 2 · 𝐴 ) ) − ( ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) ) ) |
36 |
14 25 28
|
divcld |
⊢ ( 𝜑 → ( - 𝐵 / ( 2 · 𝐴 ) ) ∈ ℂ ) |
37 |
22 25 28
|
divcld |
⊢ ( 𝜑 → ( ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) ∈ ℂ ) |
38 |
36 37
|
negsubd |
⊢ ( 𝜑 → ( ( - 𝐵 / ( 2 · 𝐴 ) ) + - ( ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) ) = ( ( - 𝐵 / ( 2 · 𝐴 ) ) − ( ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) ) ) |
39 |
22 25 28
|
divnegd |
⊢ ( 𝜑 → - ( ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) = ( - ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) ) |
40 |
39
|
oveq2d |
⊢ ( 𝜑 → ( ( - 𝐵 / ( 2 · 𝐴 ) ) + - ( ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) ) = ( ( - 𝐵 / ( 2 · 𝐴 ) ) + ( - ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) ) ) |
41 |
35 38 40
|
3eqtr2d |
⊢ ( 𝜑 → ( ( - 𝐵 − ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) = ( ( - 𝐵 / ( 2 · 𝐴 ) ) + ( - ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) ) ) |
42 |
34 41
|
eqeq12d |
⊢ ( 𝜑 → ( ( ( - 𝐵 + ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) = ( ( - 𝐵 − ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) ↔ ( ( - 𝐵 / ( 2 · 𝐴 ) ) + ( ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) ) = ( ( - 𝐵 / ( 2 · 𝐴 ) ) + ( - ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) ) ) ) |
43 |
22
|
negcld |
⊢ ( 𝜑 → - ( √ ‘ 𝐷 ) ∈ ℂ ) |
44 |
43 25 28
|
divcld |
⊢ ( 𝜑 → ( - ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) ∈ ℂ ) |
45 |
36 37 44
|
addcand |
⊢ ( 𝜑 → ( ( ( - 𝐵 / ( 2 · 𝐴 ) ) + ( ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) ) = ( ( - 𝐵 / ( 2 · 𝐴 ) ) + ( - ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) ) ↔ ( ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) = ( - ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) ) ) |
46 |
|
div11 |
⊢ ( ( ( √ ‘ 𝐷 ) ∈ ℂ ∧ - ( √ ‘ 𝐷 ) ∈ ℂ ∧ ( ( 2 · 𝐴 ) ∈ ℂ ∧ ( 2 · 𝐴 ) ≠ 0 ) ) → ( ( ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) = ( - ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) ↔ ( √ ‘ 𝐷 ) = - ( √ ‘ 𝐷 ) ) ) |
47 |
22 43 25 28 46
|
syl112anc |
⊢ ( 𝜑 → ( ( ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) = ( - ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) ↔ ( √ ‘ 𝐷 ) = - ( √ ‘ 𝐷 ) ) ) |
48 |
22
|
eqnegd |
⊢ ( 𝜑 → ( ( √ ‘ 𝐷 ) = - ( √ ‘ 𝐷 ) ↔ ( √ ‘ 𝐷 ) = 0 ) ) |
49 |
|
cnsqrt00 |
⊢ ( 𝐷 ∈ ℂ → ( ( √ ‘ 𝐷 ) = 0 ↔ 𝐷 = 0 ) ) |
50 |
21 49
|
syl |
⊢ ( 𝜑 → ( ( √ ‘ 𝐷 ) = 0 ↔ 𝐷 = 0 ) ) |
51 |
48 50
|
bitrd |
⊢ ( 𝜑 → ( ( √ ‘ 𝐷 ) = - ( √ ‘ 𝐷 ) ↔ 𝐷 = 0 ) ) |
52 |
45 47 51
|
3bitrd |
⊢ ( 𝜑 → ( ( ( - 𝐵 / ( 2 · 𝐴 ) ) + ( ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) ) = ( ( - 𝐵 / ( 2 · 𝐴 ) ) + ( - ( √ ‘ 𝐷 ) / ( 2 · 𝐴 ) ) ) ↔ 𝐷 = 0 ) ) |
53 |
42 52
|
bitrd |
⊢ ( 𝜑 → ( ( ( - 𝐵 + ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) = ( ( - 𝐵 − ( √ ‘ 𝐷 ) ) / ( 2 · 𝐴 ) ) ↔ 𝐷 = 0 ) ) |
54 |
13 33 53
|
3bitrd |
⊢ ( 𝜑 → ( ∃! 𝑥 ∈ ℂ ( ( 𝐴 · ( 𝑥 ↑ 2 ) ) + ( ( 𝐵 · 𝑥 ) + 𝐶 ) ) = 0 ↔ 𝐷 = 0 ) ) |