Metamath Proof Explorer


Theorem dvun

Description: Condition for the union of the derivatives of two disjoint functions to be equal to the derivative of the union of the two functions. If A and B are open sets, this condition (dvun.n) is satisfied by isopn3i . (Contributed by SN, 30-Sep-2025)

Ref Expression
Hypotheses dvun.j 𝐽 = ( 𝐾t 𝑆 )
dvun.k 𝐾 = ( TopOpen ‘ ℂfld )
dvun.s ( 𝜑𝑆 ⊆ ℂ )
dvun.f ( 𝜑𝐹 : 𝐴 ⟶ ℂ )
dvun.g ( 𝜑𝐺 : 𝐵 ⟶ ℂ )
dvun.a ( 𝜑𝐴𝑆 )
dvun.b ( 𝜑𝐵𝑆 )
dvun.d ( 𝜑 → ( 𝐴𝐵 ) = ∅ )
dvun.n ( 𝜑 → ( ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ∪ ( ( int ‘ 𝐽 ) ‘ 𝐵 ) ) = ( ( int ‘ 𝐽 ) ‘ ( 𝐴𝐵 ) ) )
Assertion dvun ( 𝜑 → ( ( 𝑆 D 𝐹 ) ∪ ( 𝑆 D 𝐺 ) ) = ( 𝑆 D ( 𝐹𝐺 ) ) )

Proof

Step Hyp Ref Expression
1 dvun.j 𝐽 = ( 𝐾t 𝑆 )
2 dvun.k 𝐾 = ( TopOpen ‘ ℂfld )
3 dvun.s ( 𝜑𝑆 ⊆ ℂ )
4 dvun.f ( 𝜑𝐹 : 𝐴 ⟶ ℂ )
5 dvun.g ( 𝜑𝐺 : 𝐵 ⟶ ℂ )
6 dvun.a ( 𝜑𝐴𝑆 )
7 dvun.b ( 𝜑𝐵𝑆 )
8 dvun.d ( 𝜑 → ( 𝐴𝐵 ) = ∅ )
9 dvun.n ( 𝜑 → ( ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ∪ ( ( int ‘ 𝐽 ) ‘ 𝐵 ) ) = ( ( int ‘ 𝐽 ) ‘ ( 𝐴𝐵 ) ) )
10 resundi ( ( 𝑆 D ( 𝐹𝐺 ) ) ↾ ( ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ∪ ( ( int ‘ 𝐽 ) ‘ 𝐵 ) ) ) = ( ( ( 𝑆 D ( 𝐹𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) ∪ ( ( 𝑆 D ( 𝐹𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐵 ) ) )
11 9 reseq2d ( 𝜑 → ( ( 𝑆 D ( 𝐹𝐺 ) ) ↾ ( ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ∪ ( ( int ‘ 𝐽 ) ‘ 𝐵 ) ) ) = ( ( 𝑆 D ( 𝐹𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ ( 𝐴𝐵 ) ) ) )
12 10 11 eqtr3id ( 𝜑 → ( ( ( 𝑆 D ( 𝐹𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) ∪ ( ( 𝑆 D ( 𝐹𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐵 ) ) ) = ( ( 𝑆 D ( 𝐹𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ ( 𝐴𝐵 ) ) ) )
13 4 5 8 fun2d ( 𝜑 → ( 𝐹𝐺 ) : ( 𝐴𝐵 ) ⟶ ℂ )
14 6 7 unssd ( 𝜑 → ( 𝐴𝐵 ) ⊆ 𝑆 )
15 2 1 dvres ( ( ( 𝑆 ⊆ ℂ ∧ ( 𝐹𝐺 ) : ( 𝐴𝐵 ) ⟶ ℂ ) ∧ ( ( 𝐴𝐵 ) ⊆ 𝑆𝐴𝑆 ) ) → ( 𝑆 D ( ( 𝐹𝐺 ) ↾ 𝐴 ) ) = ( ( 𝑆 D ( 𝐹𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) )
16 3 13 14 6 15 syl22anc ( 𝜑 → ( 𝑆 D ( ( 𝐹𝐺 ) ↾ 𝐴 ) ) = ( ( 𝑆 D ( 𝐹𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) )
17 4 ffnd ( 𝜑𝐹 Fn 𝐴 )
18 5 ffnd ( 𝜑𝐺 Fn 𝐵 )
19 fnunres1 ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ( 𝐴𝐵 ) = ∅ ) → ( ( 𝐹𝐺 ) ↾ 𝐴 ) = 𝐹 )
20 17 18 8 19 syl3anc ( 𝜑 → ( ( 𝐹𝐺 ) ↾ 𝐴 ) = 𝐹 )
21 20 oveq2d ( 𝜑 → ( 𝑆 D ( ( 𝐹𝐺 ) ↾ 𝐴 ) ) = ( 𝑆 D 𝐹 ) )
22 16 21 eqtr3d ( 𝜑 → ( ( 𝑆 D ( 𝐹𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) = ( 𝑆 D 𝐹 ) )
23 2 1 dvres ( ( ( 𝑆 ⊆ ℂ ∧ ( 𝐹𝐺 ) : ( 𝐴𝐵 ) ⟶ ℂ ) ∧ ( ( 𝐴𝐵 ) ⊆ 𝑆𝐵𝑆 ) ) → ( 𝑆 D ( ( 𝐹𝐺 ) ↾ 𝐵 ) ) = ( ( 𝑆 D ( 𝐹𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐵 ) ) )
24 3 13 14 7 23 syl22anc ( 𝜑 → ( 𝑆 D ( ( 𝐹𝐺 ) ↾ 𝐵 ) ) = ( ( 𝑆 D ( 𝐹𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐵 ) ) )
25 fnunres2 ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ( 𝐴𝐵 ) = ∅ ) → ( ( 𝐹𝐺 ) ↾ 𝐵 ) = 𝐺 )
26 17 18 8 25 syl3anc ( 𝜑 → ( ( 𝐹𝐺 ) ↾ 𝐵 ) = 𝐺 )
27 26 oveq2d ( 𝜑 → ( 𝑆 D ( ( 𝐹𝐺 ) ↾ 𝐵 ) ) = ( 𝑆 D 𝐺 ) )
28 24 27 eqtr3d ( 𝜑 → ( ( 𝑆 D ( 𝐹𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐵 ) ) = ( 𝑆 D 𝐺 ) )
29 22 28 uneq12d ( 𝜑 → ( ( ( 𝑆 D ( 𝐹𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) ∪ ( ( 𝑆 D ( 𝐹𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ 𝐵 ) ) ) = ( ( 𝑆 D 𝐹 ) ∪ ( 𝑆 D 𝐺 ) ) )
30 2 1 dvres ( ( ( 𝑆 ⊆ ℂ ∧ ( 𝐹𝐺 ) : ( 𝐴𝐵 ) ⟶ ℂ ) ∧ ( ( 𝐴𝐵 ) ⊆ 𝑆 ∧ ( 𝐴𝐵 ) ⊆ 𝑆 ) ) → ( 𝑆 D ( ( 𝐹𝐺 ) ↾ ( 𝐴𝐵 ) ) ) = ( ( 𝑆 D ( 𝐹𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ ( 𝐴𝐵 ) ) ) )
31 3 13 14 14 30 syl22anc ( 𝜑 → ( 𝑆 D ( ( 𝐹𝐺 ) ↾ ( 𝐴𝐵 ) ) ) = ( ( 𝑆 D ( 𝐹𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ ( 𝐴𝐵 ) ) ) )
32 13 ffnd ( 𝜑 → ( 𝐹𝐺 ) Fn ( 𝐴𝐵 ) )
33 fnresdm ( ( 𝐹𝐺 ) Fn ( 𝐴𝐵 ) → ( ( 𝐹𝐺 ) ↾ ( 𝐴𝐵 ) ) = ( 𝐹𝐺 ) )
34 32 33 syl ( 𝜑 → ( ( 𝐹𝐺 ) ↾ ( 𝐴𝐵 ) ) = ( 𝐹𝐺 ) )
35 34 oveq2d ( 𝜑 → ( 𝑆 D ( ( 𝐹𝐺 ) ↾ ( 𝐴𝐵 ) ) ) = ( 𝑆 D ( 𝐹𝐺 ) ) )
36 31 35 eqtr3d ( 𝜑 → ( ( 𝑆 D ( 𝐹𝐺 ) ) ↾ ( ( int ‘ 𝐽 ) ‘ ( 𝐴𝐵 ) ) ) = ( 𝑆 D ( 𝐹𝐺 ) ) )
37 12 29 36 3eqtr3d ( 𝜑 → ( ( 𝑆 D 𝐹 ) ∪ ( 𝑆 D 𝐺 ) ) = ( 𝑆 D ( 𝐹𝐺 ) ) )