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Metamath Proof Explorer

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 ) )