Description: The Lebesgue outer measure function is shift-invariant. (Contributed by Mario Carneiro, 22-Mar-2014) (Proof shortened by AV, 17-Sep-2020)
Ref | Expression | ||
---|---|---|---|
Hypotheses | ovolshft.1 | |
|
ovolshft.2 | |
||
ovolshft.3 | |
||
Assertion | ovolshft | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovolshft.1 | |
|
2 | ovolshft.2 | |
|
3 | ovolshft.3 | |
|
4 | eqid | |
|
5 | 1 2 3 4 | ovolshftlem2 | |
6 | ssrab2 | |
|
7 | 3 6 | eqsstrdi | |
8 | 2 | renegcld | |
9 | 1 2 3 | shft2rab | |
10 | eqid | |
|
11 | 7 8 9 10 | ovolshftlem2 | |
12 | 5 11 | eqssd | |
13 | 12 | infeq1d | |
14 | 10 | ovolval | |
15 | 1 14 | syl | |
16 | 4 | ovolval | |
17 | 7 16 | syl | |
18 | 13 15 17 | 3eqtr4d | |