Theorem sqeqori

Description: The squares of two complex numbers are equal iff one number equals the other or its negative. Lemma 15-4.7 of Gleason p. 311 and its converse. (Contributed by NM, 15-Jan-2006)

	Ref	Expression
Hypotheses	binom2.1	⊢ 𝐴 ∈ ℂ
	binom2.2	⊢ 𝐵 ∈ ℂ
Assertion	sqeqori	⊢ ( ( 𝐴 ↑ 2 ) = ( 𝐵 ↑ 2 ) ↔ ( 𝐴 = 𝐵 ∨ 𝐴 = - 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		binom2.1	⊢ 𝐴 ∈ ℂ
2		binom2.2	⊢ 𝐵 ∈ ℂ
3	1 2	subsqi	⊢ ( ( 𝐴 ↑ 2 ) − ( 𝐵 ↑ 2 ) ) = ( ( 𝐴 + 𝐵 ) · ( 𝐴 − 𝐵 ) )
4	3	eqeq1i	⊢ ( ( ( 𝐴 ↑ 2 ) − ( 𝐵 ↑ 2 ) ) = 0 ↔ ( ( 𝐴 + 𝐵 ) · ( 𝐴 − 𝐵 ) ) = 0 )
5	1	sqcli	⊢ ( 𝐴 ↑ 2 ) ∈ ℂ
6	2	sqcli	⊢ ( 𝐵 ↑ 2 ) ∈ ℂ
7	5 6	subeq0i	⊢ ( ( ( 𝐴 ↑ 2 ) − ( 𝐵 ↑ 2 ) ) = 0 ↔ ( 𝐴 ↑ 2 ) = ( 𝐵 ↑ 2 ) )
8	1 2	addcli	⊢ ( 𝐴 + 𝐵 ) ∈ ℂ
9	1 2	subcli	⊢ ( 𝐴 − 𝐵 ) ∈ ℂ
10	8 9	mul0ori	⊢ ( ( ( 𝐴 + 𝐵 ) · ( 𝐴 − 𝐵 ) ) = 0 ↔ ( ( 𝐴 + 𝐵 ) = 0 ∨ ( 𝐴 − 𝐵 ) = 0 ) )
11	4 7 10	3bitr3i	⊢ ( ( 𝐴 ↑ 2 ) = ( 𝐵 ↑ 2 ) ↔ ( ( 𝐴 + 𝐵 ) = 0 ∨ ( 𝐴 − 𝐵 ) = 0 ) )
12		orcom	⊢ ( ( ( 𝐴 + 𝐵 ) = 0 ∨ ( 𝐴 − 𝐵 ) = 0 ) ↔ ( ( 𝐴 − 𝐵 ) = 0 ∨ ( 𝐴 + 𝐵 ) = 0 ) )
13	1 2	subeq0i	⊢ ( ( 𝐴 − 𝐵 ) = 0 ↔ 𝐴 = 𝐵 )
14	1 2	subnegi	⊢ ( 𝐴 − - 𝐵 ) = ( 𝐴 + 𝐵 )
15	14	eqeq1i	⊢ ( ( 𝐴 − - 𝐵 ) = 0 ↔ ( 𝐴 + 𝐵 ) = 0 )
16	2	negcli	⊢ - 𝐵 ∈ ℂ
17	1 16	subeq0i	⊢ ( ( 𝐴 − - 𝐵 ) = 0 ↔ 𝐴 = - 𝐵 )
18	15 17	bitr3i	⊢ ( ( 𝐴 + 𝐵 ) = 0 ↔ 𝐴 = - 𝐵 )
19	13 18	orbi12i	⊢ ( ( ( 𝐴 − 𝐵 ) = 0 ∨ ( 𝐴 + 𝐵 ) = 0 ) ↔ ( 𝐴 = 𝐵 ∨ 𝐴 = - 𝐵 ) )
20	11 12 19	3bitri	⊢ ( ( 𝐴 ↑ 2 ) = ( 𝐵 ↑ 2 ) ↔ ( 𝐴 = 𝐵 ∨ 𝐴 = - 𝐵 ) )