Description: The multiplicative identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014)
Ref | Expression | ||
---|---|---|---|
Hypotheses | irredn0.i | |- I = ( Irred ` R ) |
|
irredn1.o | |- .1. = ( 1r ` R ) |
||
Assertion | irredn1 | |- ( ( R e. Ring /\ X e. I ) -> X =/= .1. ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | irredn0.i | |- I = ( Irred ` R ) |
|
2 | irredn1.o | |- .1. = ( 1r ` R ) |
|
3 | eqid | |- ( Unit ` R ) = ( Unit ` R ) |
|
4 | 3 2 | 1unit | |- ( R e. Ring -> .1. e. ( Unit ` R ) ) |
5 | eleq1 | |- ( X = .1. -> ( X e. ( Unit ` R ) <-> .1. e. ( Unit ` R ) ) ) |
|
6 | 4 5 | syl5ibrcom | |- ( R e. Ring -> ( X = .1. -> X e. ( Unit ` R ) ) ) |
7 | 6 | necon3bd | |- ( R e. Ring -> ( -. X e. ( Unit ` R ) -> X =/= .1. ) ) |
8 | 1 3 | irrednu | |- ( X e. I -> -. X e. ( Unit ` R ) ) |
9 | 7 8 | impel | |- ( ( R e. Ring /\ X e. I ) -> X =/= .1. ) |