Theorem abscj

Description: The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of Gleason p. 133. (Contributed by NM, 28-Apr-2005)

	Ref	Expression
Assertion	abscj	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( ∗ ‘ 𝐴 ) ) = ( abs ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		cjcl	⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ 𝐴 ) ∈ ℂ )
2		absval	⊢ ( ( ∗ ‘ 𝐴 ) ∈ ℂ → ( abs ‘ ( ∗ ‘ 𝐴 ) ) = ( √ ‘ ( ( ∗ ‘ 𝐴 ) · ( ∗ ‘ ( ∗ ‘ 𝐴 ) ) ) ) )
3	1 2	syl	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( ∗ ‘ 𝐴 ) ) = ( √ ‘ ( ( ∗ ‘ 𝐴 ) · ( ∗ ‘ ( ∗ ‘ 𝐴 ) ) ) ) )
4		mulcom	⊢ ( ( 𝐴 ∈ ℂ ∧ ( ∗ ‘ 𝐴 ) ∈ ℂ ) → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ∗ ‘ 𝐴 ) · 𝐴 ) )
5	1 4	mpdan	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ∗ ‘ 𝐴 ) · 𝐴 ) )
6		cjcj	⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ ( ∗ ‘ 𝐴 ) ) = 𝐴 )
7	6	oveq2d	⊢ ( 𝐴 ∈ ℂ → ( ( ∗ ‘ 𝐴 ) · ( ∗ ‘ ( ∗ ‘ 𝐴 ) ) ) = ( ( ∗ ‘ 𝐴 ) · 𝐴 ) )
8	5 7	eqtr4d	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ∗ ‘ 𝐴 ) · ( ∗ ‘ ( ∗ ‘ 𝐴 ) ) ) )
9	8	fveq2d	⊢ ( 𝐴 ∈ ℂ → ( √ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) = ( √ ‘ ( ( ∗ ‘ 𝐴 ) · ( ∗ ‘ ( ∗ ‘ 𝐴 ) ) ) ) )
10	3 9	eqtr4d	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( ∗ ‘ 𝐴 ) ) = ( √ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) )
11		absval	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ 𝐴 ) = ( √ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) )
12	10 11	eqtr4d	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( ∗ ‘ 𝐴 ) ) = ( abs ‘ 𝐴 ) )