Description: Existential uniqueness of reciprocals. Theorem I.8 of Apostol p. 18. (Contributed by NM, 29-Jan-1995) (Revised by Mario Carneiro, 17-Feb-2014)
Ref | Expression | ||
---|---|---|---|
Assertion | receu | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recex | |
|
2 | 1 | 3adant1 | |
3 | simprl | |
|
4 | simpll | |
|
5 | 3 4 | mulcld | |
6 | oveq1 | |
|
7 | 6 | ad2antll | |
8 | simplr | |
|
9 | 8 3 4 | mulassd | |
10 | 4 | mulid2d | |
11 | 7 9 10 | 3eqtr3d | |
12 | oveq2 | |
|
13 | 12 | eqeq1d | |
14 | 13 | rspcev | |
15 | 5 11 14 | syl2anc | |
16 | 15 | rexlimdvaa | |
17 | 16 | 3adant3 | |
18 | 2 17 | mpd | |
19 | eqtr3 | |
|
20 | mulcan | |
|
21 | 19 20 | syl5ib | |
22 | 21 | 3expa | |
23 | 22 | expcom | |
24 | 23 | 3adant1 | |
25 | 24 | ralrimivv | |
26 | oveq2 | |
|
27 | 26 | eqeq1d | |
28 | 27 | reu4 | |
29 | 18 25 28 | sylanbrc | |