Theorem abs2dif2

Description: Difference of absolute values. (Contributed by Mario Carneiro, 14-Apr-2016)

	Ref	Expression
Assertion	abs2dif2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( abs ‘ ( 𝐴 − 𝐵 ) ) ≤ ( ( abs ‘ 𝐴 ) + ( abs ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		negcl	⊢ ( 𝐵 ∈ ℂ → - 𝐵 ∈ ℂ )
2		abstri	⊢ ( ( 𝐴 ∈ ℂ ∧ - 𝐵 ∈ ℂ ) → ( abs ‘ ( 𝐴 + - 𝐵 ) ) ≤ ( ( abs ‘ 𝐴 ) + ( abs ‘ - 𝐵 ) ) )
3	1 2	sylan2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( abs ‘ ( 𝐴 + - 𝐵 ) ) ≤ ( ( abs ‘ 𝐴 ) + ( abs ‘ - 𝐵 ) ) )
4		negsub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 ) )
5	4	fveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( abs ‘ ( 𝐴 + - 𝐵 ) ) = ( abs ‘ ( 𝐴 − 𝐵 ) ) )
6		absneg	⊢ ( 𝐵 ∈ ℂ → ( abs ‘ - 𝐵 ) = ( abs ‘ 𝐵 ) )
7	6	adantl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( abs ‘ - 𝐵 ) = ( abs ‘ 𝐵 ) )
8	7	oveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( abs ‘ 𝐴 ) + ( abs ‘ - 𝐵 ) ) = ( ( abs ‘ 𝐴 ) + ( abs ‘ 𝐵 ) ) )
9	3 5 8	3brtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( abs ‘ ( 𝐴 − 𝐵 ) ) ≤ ( ( abs ‘ 𝐴 ) + ( abs ‘ 𝐵 ) ) )