Theorem dvbssntr

Description: The set of differentiable points is a subset of the interior of the domain of the function. (Contributed by Mario Carneiro, 7-Aug-2014) (Revised by Mario Carneiro, 9-Feb-2015)

	Ref	Expression
Hypotheses	dvcl.s	⊢ ( 𝜑 → 𝑆 ⊆ ℂ )
	dvcl.f	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℂ )
	dvcl.a	⊢ ( 𝜑 → 𝐴 ⊆ 𝑆 )
	dvbssntr.j	⊢ 𝐽 = ( 𝐾 ↾t 𝑆 )
	dvbssntr.k	⊢ 𝐾 = ( TopOpen ‘ ℂfld )
Assertion	dvbssntr	⊢ ( 𝜑 → dom ( 𝑆 D 𝐹 ) ⊆ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		dvcl.s	⊢ ( 𝜑 → 𝑆 ⊆ ℂ )
2		dvcl.f	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℂ )
3		dvcl.a	⊢ ( 𝜑 → 𝐴 ⊆ 𝑆 )
4		dvbssntr.j	⊢ 𝐽 = ( 𝐾 ↾t 𝑆 )
5		dvbssntr.k	⊢ 𝐾 = ( TopOpen ‘ ℂfld )
6	4 5	dvfval	⊢ ( ( 𝑆 ⊆ ℂ ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐴 ⊆ 𝑆 ) → ( ( 𝑆 D 𝐹 ) = ∪ 𝑥 ∈ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ( { 𝑥 } × ( ( 𝑧 ∈ ( 𝐴 ∖ { 𝑥 } ) ↦ ( ( ( 𝐹 ‘ 𝑧 ) − ( 𝐹 ‘ 𝑥 ) ) / ( 𝑧 − 𝑥 ) ) ) limℂ 𝑥 ) ) ∧ ( 𝑆 D 𝐹 ) ⊆ ( ( ( int ‘ 𝐽 ) ‘ 𝐴 ) × ℂ ) ) )
7	1 2 3 6	syl3anc	⊢ ( 𝜑 → ( ( 𝑆 D 𝐹 ) = ∪ 𝑥 ∈ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ( { 𝑥 } × ( ( 𝑧 ∈ ( 𝐴 ∖ { 𝑥 } ) ↦ ( ( ( 𝐹 ‘ 𝑧 ) − ( 𝐹 ‘ 𝑥 ) ) / ( 𝑧 − 𝑥 ) ) ) limℂ 𝑥 ) ) ∧ ( 𝑆 D 𝐹 ) ⊆ ( ( ( int ‘ 𝐽 ) ‘ 𝐴 ) × ℂ ) ) )
8		dmss	⊢ ( ( 𝑆 D 𝐹 ) ⊆ ( ( ( int ‘ 𝐽 ) ‘ 𝐴 ) × ℂ ) → dom ( 𝑆 D 𝐹 ) ⊆ dom ( ( ( int ‘ 𝐽 ) ‘ 𝐴 ) × ℂ ) )
9	7 8	simpl2im	⊢ ( 𝜑 → dom ( 𝑆 D 𝐹 ) ⊆ dom ( ( ( int ‘ 𝐽 ) ‘ 𝐴 ) × ℂ ) )
10		dmxpss	⊢ dom ( ( ( int ‘ 𝐽 ) ‘ 𝐴 ) × ℂ ) ⊆ ( ( int ‘ 𝐽 ) ‘ 𝐴 )
11	9 10	sstrdi	⊢ ( 𝜑 → dom ( 𝑆 D 𝐹 ) ⊆ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) )