Step |
Hyp |
Ref |
Expression |
0 |
|
csitm |
⊢ sitm |
1 |
|
vw |
⊢ 𝑤 |
2 |
|
cvv |
⊢ V |
3 |
|
vm |
⊢ 𝑚 |
4 |
|
cmeas |
⊢ measures |
5 |
4
|
crn |
⊢ ran measures |
6 |
5
|
cuni |
⊢ ∪ ran measures |
7 |
|
vf |
⊢ 𝑓 |
8 |
1
|
cv |
⊢ 𝑤 |
9 |
|
csitg |
⊢ sitg |
10 |
3
|
cv |
⊢ 𝑚 |
11 |
8 10 9
|
co |
⊢ ( 𝑤 sitg 𝑚 ) |
12 |
11
|
cdm |
⊢ dom ( 𝑤 sitg 𝑚 ) |
13 |
|
vg |
⊢ 𝑔 |
14 |
|
cxrs |
⊢ ℝ*𝑠 |
15 |
|
cress |
⊢ ↾s |
16 |
|
cc0 |
⊢ 0 |
17 |
|
cicc |
⊢ [,] |
18 |
|
cpnf |
⊢ +∞ |
19 |
16 18 17
|
co |
⊢ ( 0 [,] +∞ ) |
20 |
14 19 15
|
co |
⊢ ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) |
21 |
20 10 9
|
co |
⊢ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) sitg 𝑚 ) |
22 |
7
|
cv |
⊢ 𝑓 |
23 |
|
cds |
⊢ dist |
24 |
8 23
|
cfv |
⊢ ( dist ‘ 𝑤 ) |
25 |
24
|
cof |
⊢ ∘f ( dist ‘ 𝑤 ) |
26 |
13
|
cv |
⊢ 𝑔 |
27 |
22 26 25
|
co |
⊢ ( 𝑓 ∘f ( dist ‘ 𝑤 ) 𝑔 ) |
28 |
27 21
|
cfv |
⊢ ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) sitg 𝑚 ) ‘ ( 𝑓 ∘f ( dist ‘ 𝑤 ) 𝑔 ) ) |
29 |
7 13 12 12 28
|
cmpo |
⊢ ( 𝑓 ∈ dom ( 𝑤 sitg 𝑚 ) , 𝑔 ∈ dom ( 𝑤 sitg 𝑚 ) ↦ ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) sitg 𝑚 ) ‘ ( 𝑓 ∘f ( dist ‘ 𝑤 ) 𝑔 ) ) ) |
30 |
1 3 2 6 29
|
cmpo |
⊢ ( 𝑤 ∈ V , 𝑚 ∈ ∪ ran measures ↦ ( 𝑓 ∈ dom ( 𝑤 sitg 𝑚 ) , 𝑔 ∈ dom ( 𝑤 sitg 𝑚 ) ↦ ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) sitg 𝑚 ) ‘ ( 𝑓 ∘f ( dist ‘ 𝑤 ) 𝑔 ) ) ) ) |
31 |
0 30
|
wceq |
⊢ sitm = ( 𝑤 ∈ V , 𝑚 ∈ ∪ ran measures ↦ ( 𝑓 ∈ dom ( 𝑤 sitg 𝑚 ) , 𝑔 ∈ dom ( 𝑤 sitg 𝑚 ) ↦ ( ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) sitg 𝑚 ) ‘ ( 𝑓 ∘f ( dist ‘ 𝑤 ) 𝑔 ) ) ) ) |