Description: Subtract the reciprocal of 1 minus a number from 1 results in the number divided by the number minus 1. (Contributed by AV, 15-Feb-2023)
Ref | Expression | ||
---|---|---|---|
Assertion | 1subrec1sub | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1cnd | |
|
2 | simpl | |
|
3 | 1 2 | subcld | |
4 | simpr | |
|
5 | 4 | necomd | |
6 | 1 2 5 | subne0d | |
7 | 1 3 6 | divcan4d | |
8 | 7 | eqcomd | |
9 | 8 | oveq1d | |
10 | 1 3 | mulcld | |
11 | 10 1 3 6 | divsubdird | |
12 | 3 | mulid2d | |
13 | 12 | oveq1d | |
14 | negcl | |
|
15 | 14 | adantr | |
16 | 1 2 | negsubd | |
17 | 16 | eqcomd | |
18 | 1 15 17 | mvrladdd | |
19 | 13 18 | eqtrd | |
20 | 19 | oveq1d | |
21 | 2 3 6 | divneg2d | |
22 | 2 3 6 | divnegd | |
23 | 1 2 | negsubdi2d | |
24 | 23 | oveq2d | |
25 | 21 22 24 | 3eqtr3d | |
26 | 20 25 | eqtrd | |
27 | 9 11 26 | 3eqtr2d | |