Description: The two solutions to the difference of squares set equal to zero. (Contributed by NM, 25-Apr-2006)
Ref | Expression | ||
---|---|---|---|
Hypotheses | binom2.1 | ⊢ 𝐴 ∈ ℂ | |
binom2.2 | ⊢ 𝐵 ∈ ℂ | ||
Assertion | subsq0i | ⊢ ( ( ( 𝐴 ↑ 2 ) − ( 𝐵 ↑ 2 ) ) = 0 ↔ ( 𝐴 = 𝐵 ∨ 𝐴 = - 𝐵 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | binom2.1 | ⊢ 𝐴 ∈ ℂ | |
2 | binom2.2 | ⊢ 𝐵 ∈ ℂ | |
3 | 1 | sqcli | ⊢ ( 𝐴 ↑ 2 ) ∈ ℂ |
4 | 2 | sqcli | ⊢ ( 𝐵 ↑ 2 ) ∈ ℂ |
5 | 3 4 | subeq0i | ⊢ ( ( ( 𝐴 ↑ 2 ) − ( 𝐵 ↑ 2 ) ) = 0 ↔ ( 𝐴 ↑ 2 ) = ( 𝐵 ↑ 2 ) ) |
6 | 1 2 | sqeqori | ⊢ ( ( 𝐴 ↑ 2 ) = ( 𝐵 ↑ 2 ) ↔ ( 𝐴 = 𝐵 ∨ 𝐴 = - 𝐵 ) ) |
7 | 5 6 | bitri | ⊢ ( ( ( 𝐴 ↑ 2 ) − ( 𝐵 ↑ 2 ) ) = 0 ↔ ( 𝐴 = 𝐵 ∨ 𝐴 = - 𝐵 ) ) |