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Metamath Proof Explorer

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 ) )