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Metamath Proof Explorer

Theorem plyid

Description: The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014)

Ref Expression
Assertion plyid 
|- ( ( S C_ CC /\ 1 e. S ) -> Xp e. ( Poly ` S ) )

Proof

Step Hyp Ref Expression
1 mptresid 
 |-  ( _I |` CC ) = ( z e. CC |-> z )
2 df-idp 
 |-  Xp = ( _I |` CC )
3 exp1 
 |-  ( z e. CC -> ( z ^ 1 ) = z )
4 3 mpteq2ia 
 |-  ( z e. CC |-> ( z ^ 1 ) ) = ( z e. CC |-> z )
5 1 2 4 3eqtr4i 
 |-  Xp = ( z e. CC |-> ( z ^ 1 ) )
6 1nn0 
 |-  1 e. NN0
7 plypow 
 |-  ( ( S C_ CC /\ 1 e. S /\ 1 e. NN0 ) -> ( z e. CC |-> ( z ^ 1 ) ) e. ( Poly ` S ) )
8 6 7 mp3an3 
 |-  ( ( S C_ CC /\ 1 e. S ) -> ( z e. CC |-> ( z ^ 1 ) ) e. ( Poly ` S ) )
9 5 8 eqeltrid 
 |-  ( ( S C_ CC /\ 1 e. S ) -> Xp e. ( Poly ` S ) )