Description: Closure for the 2-cycles. (Contributed by Thierry Arnoux, 24-Sep-2023)
Ref | Expression | ||
---|---|---|---|
Hypotheses | cycpm2.c | |- C = ( toCyc ` D ) |
|
cycpm2.d | |- ( ph -> D e. V ) |
||
cycpm2.i | |- ( ph -> I e. D ) |
||
cycpm2.j | |- ( ph -> J e. D ) |
||
cycpm2.1 | |- ( ph -> I =/= J ) |
||
cycpm2cl.s | |- S = ( SymGrp ` D ) |
||
Assertion | cycpm2cl | |- ( ph -> ( C ` <" I J "> ) e. ( Base ` S ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cycpm2.c | |- C = ( toCyc ` D ) |
|
2 | cycpm2.d | |- ( ph -> D e. V ) |
|
3 | cycpm2.i | |- ( ph -> I e. D ) |
|
4 | cycpm2.j | |- ( ph -> J e. D ) |
|
5 | cycpm2.1 | |- ( ph -> I =/= J ) |
|
6 | cycpm2cl.s | |- S = ( SymGrp ` D ) |
|
7 | 3 4 | s2cld | |- ( ph -> <" I J "> e. Word D ) |
8 | 3 4 5 | s2f1 | |- ( ph -> <" I J "> : dom <" I J "> -1-1-> D ) |
9 | 1 2 7 8 6 | cycpmcl | |- ( ph -> ( C ` <" I J "> ) e. ( Base ` S ) ) |