dvcnre

Description: From complex differentiation to real differentiation. (Contributed by Glauco Siliprandi, 11-Dec-2019)

		Ref	Expression
	Assertion	dvcnre	$⊢ (F : ℂ ⟶ ℂ \land ℝ \subseteq dom F_{ℂ}^{'}) \to ℝ D ({F ↾}_{ℝ}) = {F_{ℂ}^{'} ↾}_{ℝ}$

Step	Hyp	Ref	Expression
1		reelprrecn	$⊢ ℝ \in \{ℝ, ℂ\}$
2	1	a1i	$⊢ (F : ℂ ⟶ ℂ \land ℝ \subseteq dom F_{ℂ}^{'}) \to ℝ \in \{ℝ, ℂ\}$
3		simpl	$⊢ (F : ℂ ⟶ ℂ \land ℝ \subseteq dom F_{ℂ}^{'}) \to F : ℂ ⟶ ℂ$
4		ssidd	$⊢ (F : ℂ ⟶ ℂ \land ℝ \subseteq dom F_{ℂ}^{'}) \to ℂ \subseteq ℂ$
5		simpr	$⊢ (F : ℂ ⟶ ℂ \land ℝ \subseteq dom F_{ℂ}^{'}) \to ℝ \subseteq dom F_{ℂ}^{'}$
6		dvres3	$⊢ ((ℝ \in \{ℝ, ℂ\} \land F : ℂ ⟶ ℂ) \land (ℂ \subseteq ℂ \land ℝ \subseteq dom F_{ℂ}^{'})) \to ℝ D ({F ↾}_{ℝ}) = {F_{ℂ}^{'} ↾}_{ℝ}$
7	2 3 4 5 6	syl22anc	$⊢ (F : ℂ ⟶ ℂ \land ℝ \subseteq dom F_{ℂ}^{'}) \to ℝ D ({F ↾}_{ℝ}) = {F_{ℂ}^{'} ↾}_{ℝ}$

Metamath Proof Explorer