Step |
Hyp |
Ref |
Expression |
1 |
|
resum2sqcl.q |
⊢ 𝑄 = ( ( 𝐴 ↑ 2 ) + ( 𝐵 ↑ 2 ) ) |
2 |
1
|
resum2sqgt0 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℝ ) → 0 < 𝑄 ) |
3 |
2
|
ex |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → ( 𝐵 ∈ ℝ → 0 < 𝑄 ) ) |
4 |
3
|
expcom |
⊢ ( 𝐴 ≠ 0 → ( 𝐴 ∈ ℝ → ( 𝐵 ∈ ℝ → 0 < 𝑄 ) ) ) |
5 |
4
|
com23 |
⊢ ( 𝐴 ≠ 0 → ( 𝐵 ∈ ℝ → ( 𝐴 ∈ ℝ → 0 < 𝑄 ) ) ) |
6 |
|
eqid |
⊢ ( ( 𝐵 ↑ 2 ) + ( 𝐴 ↑ 2 ) ) = ( ( 𝐵 ↑ 2 ) + ( 𝐴 ↑ 2 ) ) |
7 |
6
|
resum2sqgt0 |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) ∧ 𝐴 ∈ ℝ ) → 0 < ( ( 𝐵 ↑ 2 ) + ( 𝐴 ↑ 2 ) ) ) |
8 |
1
|
breq2i |
⊢ ( 0 < 𝑄 ↔ 0 < ( ( 𝐴 ↑ 2 ) + ( 𝐵 ↑ 2 ) ) ) |
9 |
|
resqcl |
⊢ ( 𝐴 ∈ ℝ → ( 𝐴 ↑ 2 ) ∈ ℝ ) |
10 |
9
|
adantl |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) ∧ 𝐴 ∈ ℝ ) → ( 𝐴 ↑ 2 ) ∈ ℝ ) |
11 |
10
|
recnd |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) ∧ 𝐴 ∈ ℝ ) → ( 𝐴 ↑ 2 ) ∈ ℂ ) |
12 |
|
resqcl |
⊢ ( 𝐵 ∈ ℝ → ( 𝐵 ↑ 2 ) ∈ ℝ ) |
13 |
12
|
ad2antrr |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) ∧ 𝐴 ∈ ℝ ) → ( 𝐵 ↑ 2 ) ∈ ℝ ) |
14 |
13
|
recnd |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) ∧ 𝐴 ∈ ℝ ) → ( 𝐵 ↑ 2 ) ∈ ℂ ) |
15 |
11 14
|
addcomd |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) ∧ 𝐴 ∈ ℝ ) → ( ( 𝐴 ↑ 2 ) + ( 𝐵 ↑ 2 ) ) = ( ( 𝐵 ↑ 2 ) + ( 𝐴 ↑ 2 ) ) ) |
16 |
15
|
breq2d |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) ∧ 𝐴 ∈ ℝ ) → ( 0 < ( ( 𝐴 ↑ 2 ) + ( 𝐵 ↑ 2 ) ) ↔ 0 < ( ( 𝐵 ↑ 2 ) + ( 𝐴 ↑ 2 ) ) ) ) |
17 |
8 16
|
syl5bb |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) ∧ 𝐴 ∈ ℝ ) → ( 0 < 𝑄 ↔ 0 < ( ( 𝐵 ↑ 2 ) + ( 𝐴 ↑ 2 ) ) ) ) |
18 |
7 17
|
mpbird |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) ∧ 𝐴 ∈ ℝ ) → 0 < 𝑄 ) |
19 |
18
|
ex |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( 𝐴 ∈ ℝ → 0 < 𝑄 ) ) |
20 |
19
|
expcom |
⊢ ( 𝐵 ≠ 0 → ( 𝐵 ∈ ℝ → ( 𝐴 ∈ ℝ → 0 < 𝑄 ) ) ) |
21 |
5 20
|
jaoi |
⊢ ( ( 𝐴 ≠ 0 ∨ 𝐵 ≠ 0 ) → ( 𝐵 ∈ ℝ → ( 𝐴 ∈ ℝ → 0 < 𝑄 ) ) ) |
22 |
21
|
3imp31 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐴 ≠ 0 ∨ 𝐵 ≠ 0 ) ) → 0 < 𝑄 ) |