Description: A multiplicative inverse in a division ring is nonzero. ( recne0d analog). (Contributed by SN, 14-Aug-2024)
Ref | Expression | ||
---|---|---|---|
Hypotheses | drnginvrn0d.b | |- B = ( Base ` R ) |
|
drnginvrn0d.0 | |- .0. = ( 0g ` R ) |
||
drnginvrn0d.i | |- I = ( invr ` R ) |
||
drnginvrn0d.r | |- ( ph -> R e. DivRing ) |
||
drnginvrn0d.x | |- ( ph -> X e. B ) |
||
drnginvrn0d.1 | |- ( ph -> X =/= .0. ) |
||
Assertion | drnginvrn0d | |- ( ph -> ( I ` X ) =/= .0. ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drnginvrn0d.b | |- B = ( Base ` R ) |
|
2 | drnginvrn0d.0 | |- .0. = ( 0g ` R ) |
|
3 | drnginvrn0d.i | |- I = ( invr ` R ) |
|
4 | drnginvrn0d.r | |- ( ph -> R e. DivRing ) |
|
5 | drnginvrn0d.x | |- ( ph -> X e. B ) |
|
6 | drnginvrn0d.1 | |- ( ph -> X =/= .0. ) |
|
7 | 1 2 3 | drnginvrn0 | |- ( ( R e. DivRing /\ X e. B /\ X =/= .0. ) -> ( I ` X ) =/= .0. ) |
8 | 4 5 6 7 | syl3anc | |- ( ph -> ( I ` X ) =/= .0. ) |