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
|
simp3d |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑀 ‘ ( ◡ 𝐹 “ { 𝑥 } ) ) ∈ ( 0 [,) +∞ ) ) |
13 |
|
sneq |
⊢ ( 𝑥 = 𝐴 → { 𝑥 } = { 𝐴 } ) |
14 |
13
|
imaeq2d |
⊢ ( 𝑥 = 𝐴 → ( ◡ 𝐹 “ { 𝑥 } ) = ( ◡ 𝐹 “ { 𝐴 } ) ) |
15 |
14
|
fveq2d |
⊢ ( 𝑥 = 𝐴 → ( 𝑀 ‘ ( ◡ 𝐹 “ { 𝑥 } ) ) = ( 𝑀 ‘ ( ◡ 𝐹 “ { 𝐴 } ) ) ) |
16 |
15
|
eleq1d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑀 ‘ ( ◡ 𝐹 “ { 𝑥 } ) ) ∈ ( 0 [,) +∞ ) ↔ ( 𝑀 ‘ ( ◡ 𝐹 “ { 𝐴 } ) ) ∈ ( 0 [,) +∞ ) ) ) |
17 |
16
|
rspcv |
⊢ ( 𝐴 ∈ ( ran 𝐹 ∖ { 0 } ) → ( ∀ 𝑥 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑀 ‘ ( ◡ 𝐹 “ { 𝑥 } ) ) ∈ ( 0 [,) +∞ ) → ( 𝑀 ‘ ( ◡ 𝐹 “ { 𝐴 } ) ) ∈ ( 0 [,) +∞ ) ) ) |
18 |
12 17
|
mpan9 |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑀 ‘ ( ◡ 𝐹 “ { 𝐴 } ) ) ∈ ( 0 [,) +∞ ) ) |