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Metamath Proof Explorer

Theorem opsrcrng

Description: The ring of ordered power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015)

Ref Expression
Hypotheses opsrcrng.o 
|- O = ( ( I ordPwSer R ) ` T )
opsrcrng.i 
|- ( ph -> I e. V )
opsrcrng.r 
|- ( ph -> R e. CRing )
opsrcrng.t 
|- ( ph -> T C_ ( I X. I ) )
Assertion opsrcrng 
|- ( ph -> O e. CRing )

Proof

Step Hyp Ref Expression
1 opsrcrng.o 
 |-  O = ( ( I ordPwSer R ) ` T )
2 opsrcrng.i 
 |-  ( ph -> I e. V )
3 opsrcrng.r 
 |-  ( ph -> R e. CRing )
4 opsrcrng.t 
 |-  ( ph -> T C_ ( I X. I ) )
5 eqid 
 |-  ( I mPwSer R ) = ( I mPwSer R )
6 5 2 3 psrcrng 
 |-  ( ph -> ( I mPwSer R ) e. CRing )
7 eqidd 
 |-  ( ph -> ( Base ` ( I mPwSer R ) ) = ( Base ` ( I mPwSer R ) ) )
8 5 1 4 opsrbas 
 |-  ( ph -> ( Base ` ( I mPwSer R ) ) = ( Base ` O ) )
9 5 1 4 opsrplusg 
 |-  ( ph -> ( +g ` ( I mPwSer R ) ) = ( +g ` O ) )
10 9 oveqdr 
 |-  ( ( ph /\ ( x e. ( Base ` ( I mPwSer R ) ) /\ y e. ( Base ` ( I mPwSer R ) ) ) ) -> ( x ( +g ` ( I mPwSer R ) ) y ) = ( x ( +g ` O ) y ) )
11 5 1 4 opsrmulr 
 |-  ( ph -> ( .r ` ( I mPwSer R ) ) = ( .r ` O ) )
12 11 oveqdr 
 |-  ( ( ph /\ ( x e. ( Base ` ( I mPwSer R ) ) /\ y e. ( Base ` ( I mPwSer R ) ) ) ) -> ( x ( .r ` ( I mPwSer R ) ) y ) = ( x ( .r ` O ) y ) )
13 7 8 10 12 crngpropd 
 |-  ( ph -> ( ( I mPwSer R ) e. CRing <-> O e. CRing ) )
14 6 13 mpbid 
 |-  ( ph -> O e. CRing )