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Metamath Proof Explorer

Theorem cnfldadd

Description: The addition 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	cnfldadd	$⊢ + = +_{ℂ_{fld}}$

Proof

Step	Hyp	Ref	Expression
1		ax-addf	$⊢ + : ℂ \times ℂ ⟶ ℂ$
2		ffn	$⊢ + : ℂ \times ℂ ⟶ ℂ \to + Fn (ℂ \times ℂ)$
3	1 2	ax-mp	$⊢ + Fn (ℂ \times ℂ)$
4		fnov	$⊢ + Fn (ℂ \times ℂ) \leftrightarrow + = (x \in ℂ, y \in ℂ ⟼ x + y)$
5	3 4	mpbi	$⊢ + = (x \in ℂ, y \in ℂ ⟼ x + y)$
6		mpocnfldadd	$⊢ (x \in ℂ, y \in ℂ ⟼ x + y) = +_{ℂ_{fld}}$
7	5 6	eqtri	$⊢ + = +_{ℂ_{fld}}$