Step |
Hyp |
Ref |
Expression |
1 |
|
irredn0.i |
|- I = ( Irred ` R ) |
2 |
|
irredrmul.u |
|- U = ( Unit ` R ) |
3 |
|
irredrmul.t |
|- .x. = ( .r ` R ) |
4 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
5 |
|
eqid |
|- ( oppR ` R ) = ( oppR ` R ) |
6 |
|
eqid |
|- ( .r ` ( oppR ` R ) ) = ( .r ` ( oppR ` R ) ) |
7 |
4 3 5 6
|
opprmul |
|- ( Y ( .r ` ( oppR ` R ) ) X ) = ( X .x. Y ) |
8 |
5
|
opprring |
|- ( R e. Ring -> ( oppR ` R ) e. Ring ) |
9 |
5 1
|
opprirred |
|- I = ( Irred ` ( oppR ` R ) ) |
10 |
2 5
|
opprunit |
|- U = ( Unit ` ( oppR ` R ) ) |
11 |
9 10 6
|
irredrmul |
|- ( ( ( oppR ` R ) e. Ring /\ Y e. I /\ X e. U ) -> ( Y ( .r ` ( oppR ` R ) ) X ) e. I ) |
12 |
8 11
|
syl3an1 |
|- ( ( R e. Ring /\ Y e. I /\ X e. U ) -> ( Y ( .r ` ( oppR ` R ) ) X ) e. I ) |
13 |
12
|
3com23 |
|- ( ( R e. Ring /\ X e. U /\ Y e. I ) -> ( Y ( .r ` ( oppR ` R ) ) X ) e. I ) |
14 |
7 13
|
eqeltrrid |
|- ( ( R e. Ring /\ X e. U /\ Y e. I ) -> ( X .x. Y ) e. I ) |