Step |
Hyp |
Ref |
Expression |
1 |
|
sitgval.b |
⊢ 𝐵 = ( Base ‘ 𝑊 ) |
2 |
|
sitgval.j |
⊢ 𝐽 = ( TopOpen ‘ 𝑊 ) |
3 |
|
sitgval.s |
⊢ 𝑆 = ( sigaGen ‘ 𝐽 ) |
4 |
|
sitgval.0 |
⊢ 0 = ( 0g ‘ 𝑊 ) |
5 |
|
sitgval.x |
⊢ · = ( ·𝑠 ‘ 𝑊 ) |
6 |
|
sitgval.h |
⊢ 𝐻 = ( ℝHom ‘ ( Scalar ‘ 𝑊 ) ) |
7 |
|
sitgval.1 |
⊢ ( 𝜑 → 𝑊 ∈ 𝑉 ) |
8 |
|
sitgval.2 |
⊢ ( 𝜑 → 𝑀 ∈ ∪ ran measures ) |
9 |
|
sibfmbl.1 |
⊢ ( 𝜑 → 𝐹 ∈ dom ( 𝑊 sitg 𝑀 ) ) |
10 |
1 2 3 4 5 6 7 8
|
issibf |
⊢ ( 𝜑 → ( 𝐹 ∈ dom ( 𝑊 sitg 𝑀 ) ↔ ( 𝐹 ∈ ( dom 𝑀 MblFnM 𝑆 ) ∧ ran 𝐹 ∈ Fin ∧ ∀ 𝑥 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑀 ‘ ( ◡ 𝐹 “ { 𝑥 } ) ) ∈ ( 0 [,) +∞ ) ) ) ) |
11 |
9 10
|
mpbid |
⊢ ( 𝜑 → ( 𝐹 ∈ ( dom 𝑀 MblFnM 𝑆 ) ∧ ran 𝐹 ∈ Fin ∧ ∀ 𝑥 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑀 ‘ ( ◡ 𝐹 “ { 𝑥 } ) ) ∈ ( 0 [,) +∞ ) ) ) |
12 |
11
|
simp1d |
⊢ ( 𝜑 → 𝐹 ∈ ( dom 𝑀 MblFnM 𝑆 ) ) |