Metamath Proof Explorer


Definition df-bj-inftyexpitau

Description: Definition of the auxiliary function inftyexpitau parameterizing the circle at infinity CCinfty in CCbar . We use coupling with { R. } to simplify the proof of bj-inftyexpitaudisj . (Contributed by BJ, 22-Jan-2023) The precise definition is irrelevant and should generally not be used. TODO: prove only the necessary lemmas to prove |- ( A e. RR /\ B e. RR ) -> ( ( inftyexpitauA ) = ( inftyexpitauB ) <-> ( A - B ) e. ZZ ) ) . (New usage is discouraged.)

Ref Expression
Assertion df-bj-inftyexpitau
|- inftyexpitau = ( x e. RR |-> <. ( {R ` ( 1st ` x ) ) , { R. } >. )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cinftyexpitau
 |-  inftyexpitau
1 vx
 |-  x
2 cr
 |-  RR
3 cfractemp
 |-  {R
4 c1st
 |-  1st
5 1 cv
 |-  x
6 5 4 cfv
 |-  ( 1st ` x )
7 6 3 cfv
 |-  ( {R ` ( 1st ` x ) )
8 cnr
 |-  R.
9 8 csn
 |-  { R. }
10 7 9 cop
 |-  <. ( {R ` ( 1st ` x ) ) , { R. } >.
11 1 2 10 cmpt
 |-  ( x e. RR |-> <. ( {R ` ( 1st ` x ) ) , { R. } >. )
12 0 11 wceq
 |-  inftyexpitau = ( x e. RR |-> <. ( {R ` ( 1st ` x ) ) , { R. } >. )