Description: The inverse of the identity is the identity. (Contributed by AV, 8-Apr-2020)
Ref | Expression | ||
---|---|---|---|
Hypotheses | invid.b | |- B = ( Base ` C ) |
|
invid.i | |- I = ( Id ` C ) |
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invid.c | |- ( ph -> C e. Cat ) |
||
invid.x | |- ( ph -> X e. B ) |
||
Assertion | invid | |- ( ph -> ( I ` X ) ( X ( Inv ` C ) X ) ( I ` X ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invid.b | |- B = ( Base ` C ) |
|
2 | invid.i | |- I = ( Id ` C ) |
|
3 | invid.c | |- ( ph -> C e. Cat ) |
|
4 | invid.x | |- ( ph -> X e. B ) |
|
5 | 1 2 3 4 | sectid | |- ( ph -> ( I ` X ) ( X ( Sect ` C ) X ) ( I ` X ) ) |
6 | eqid | |- ( Inv ` C ) = ( Inv ` C ) |
|
7 | eqid | |- ( Sect ` C ) = ( Sect ` C ) |
|
8 | 1 6 3 4 4 7 | isinv | |- ( ph -> ( ( I ` X ) ( X ( Inv ` C ) X ) ( I ` X ) <-> ( ( I ` X ) ( X ( Sect ` C ) X ) ( I ` X ) /\ ( I ` X ) ( X ( Sect ` C ) X ) ( I ` X ) ) ) ) |
9 | 5 5 8 | mpbir2and | |- ( ph -> ( I ` X ) ( X ( Inv ` C ) X ) ( I ` X ) ) |