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Metamath Proof Explorer

Theorem cnfldmul

Description: The multiplication operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 6-Oct-2015) (Revised by Thierry Arnoux, 17-Dec-2017) Revise df-cnfld . (Revised by GG, 27-Apr-2025)

Ref Expression
Assertion cnfldmul · = ( .r ‘ ℂfld )

Proof

Step Hyp Ref Expression
1 ax-mulf ⊢ · : ( ℂ × ℂ ) ⟶ ℂ
2 ffn ⊢ ( · : ( ℂ × ℂ ) ⟶ ℂ → · Fn ( ℂ × ℂ ) )
3 1 2 ax-mp ⊢ · Fn ( ℂ × ℂ )
4 fnov ⊢ ( · Fn ( ℂ × ℂ ) ↔ · = ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 · 𝑦 ) ) )
5 3 4 mpbi ⊢ · = ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 · 𝑦 ) )
6 mpocnfldmul ⊢ ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 · 𝑦 ) ) = ( .r ‘ ℂfld )
7 5 6 eqtri ⊢ · = ( .r ‘ ℂfld )