Theorem abslt2sqd

Description: Comparison of the square of two numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020)

	Ref	Expression
Hypotheses	abslt2sqd.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	abslt2sqd.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	abslt2sqd.l	⊢ ( 𝜑 → ( abs ‘ 𝐴 ) < ( abs ‘ 𝐵 ) )
Assertion	abslt2sqd	⊢ ( 𝜑 → ( 𝐴 ↑ 2 ) < ( 𝐵 ↑ 2 ) )

Proof

Step	Hyp	Ref	Expression
1		abslt2sqd.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		abslt2sqd.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		abslt2sqd.l	⊢ ( 𝜑 → ( abs ‘ 𝐴 ) < ( abs ‘ 𝐵 ) )
4	1	recnd	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
5	4	abscld	⊢ ( 𝜑 → ( abs ‘ 𝐴 ) ∈ ℝ )
6	4	absge0d	⊢ ( 𝜑 → 0 ≤ ( abs ‘ 𝐴 ) )
7	2	recnd	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
8	7	abscld	⊢ ( 𝜑 → ( abs ‘ 𝐵 ) ∈ ℝ )
9	7	absge0d	⊢ ( 𝜑 → 0 ≤ ( abs ‘ 𝐵 ) )
10		lt2sq	⊢ ( ( ( ( abs ‘ 𝐴 ) ∈ ℝ ∧ 0 ≤ ( abs ‘ 𝐴 ) ) ∧ ( ( abs ‘ 𝐵 ) ∈ ℝ ∧ 0 ≤ ( abs ‘ 𝐵 ) ) ) → ( ( abs ‘ 𝐴 ) < ( abs ‘ 𝐵 ) ↔ ( ( abs ‘ 𝐴 ) ↑ 2 ) < ( ( abs ‘ 𝐵 ) ↑ 2 ) ) )
11	5 6 8 9 10	syl22anc	⊢ ( 𝜑 → ( ( abs ‘ 𝐴 ) < ( abs ‘ 𝐵 ) ↔ ( ( abs ‘ 𝐴 ) ↑ 2 ) < ( ( abs ‘ 𝐵 ) ↑ 2 ) ) )
12	3 11	mpbid	⊢ ( 𝜑 → ( ( abs ‘ 𝐴 ) ↑ 2 ) < ( ( abs ‘ 𝐵 ) ↑ 2 ) )
13		absresq	⊢ ( 𝐴 ∈ ℝ → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 ↑ 2 ) )
14	1 13	syl	⊢ ( 𝜑 → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 ↑ 2 ) )
15		absresq	⊢ ( 𝐵 ∈ ℝ → ( ( abs ‘ 𝐵 ) ↑ 2 ) = ( 𝐵 ↑ 2 ) )
16	2 15	syl	⊢ ( 𝜑 → ( ( abs ‘ 𝐵 ) ↑ 2 ) = ( 𝐵 ↑ 2 ) )
17	14 16	breq12d	⊢ ( 𝜑 → ( ( ( abs ‘ 𝐴 ) ↑ 2 ) < ( ( abs ‘ 𝐵 ) ↑ 2 ) ↔ ( 𝐴 ↑ 2 ) < ( 𝐵 ↑ 2 ) ) )
18	12 17	mpbid	⊢ ( 𝜑 → ( 𝐴 ↑ 2 ) < ( 𝐵 ↑ 2 ) )