Description: Any ring with an absolute value is a domain, which is to say that it contains no zero divisors. (Contributed by Mario Carneiro, 10-Sep-2014)
Ref | Expression | ||
---|---|---|---|
Hypotheses | abv0.a | |
|
abvneg.b | |
||
abvrec.z | |
||
abvdom.t | |
||
Assertion | abvdom | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abv0.a | |
|
2 | abvneg.b | |
|
3 | abvrec.z | |
|
4 | abvdom.t | |
|
5 | simp1 | |
|
6 | simp2l | |
|
7 | simp3l | |
|
8 | 1 2 4 | abvmul | |
9 | 5 6 7 8 | syl3anc | |
10 | 1 2 | abvcl | |
11 | 5 6 10 | syl2anc | |
12 | 11 | recnd | |
13 | 1 2 | abvcl | |
14 | 5 7 13 | syl2anc | |
15 | 14 | recnd | |
16 | simp2r | |
|
17 | 1 2 3 | abvne0 | |
18 | 5 6 16 17 | syl3anc | |
19 | simp3r | |
|
20 | 1 2 3 | abvne0 | |
21 | 5 7 19 20 | syl3anc | |
22 | 12 15 18 21 | mulne0d | |
23 | 9 22 | eqnetrd | |
24 | 1 3 | abv0 | |
25 | 5 24 | syl | |
26 | fveqeq2 | |
|
27 | 25 26 | syl5ibrcom | |
28 | 27 | necon3d | |
29 | 23 28 | mpd | |