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Metamath Proof Explorer

Theorem cjaddd

Description: Complex conjugate distributes over addition. Proposition 10-3.4(a) of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses recld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
readdd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion cjaddd ( 𝜑 → ( ∗ ‘ ( 𝐴 + 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) + ( ∗ ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 recld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 readdd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3 cjadd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ∗ ‘ ( 𝐴 + 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) + ( ∗ ‘ 𝐵 ) ) )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( ∗ ‘ ( 𝐴 + 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) + ( ∗ ‘ 𝐵 ) ) )