Metamath Proof Explorer


Theorem cycpm2cl

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 ) )

Proof

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 ) )