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Metamath Proof Explorer

Theorem selvcllem1

Description: T is an associative algebra. For simplicity, I stands for ( I \ J ) and we have J e. W instead of J C_ I . TODO-SN: In practice, this "simplification" makes the lemmas harder to use. (Contributed by SN, 15-Dec-2023)

Ref Expression
Hypotheses selvcllem1.u 
|- U = ( I mPoly R )
selvcllem1.t 
|- T = ( J mPoly U )
selvcllem1.i 
|- ( ph -> I e. V )
selvcllem1.j 
|- ( ph -> J e. W )
selvcllem1.r 
|- ( ph -> R e. CRing )
Assertion selvcllem1 
|- ( ph -> T e. AssAlg )

Proof

Step Hyp Ref Expression
1 selvcllem1.u 
 |-  U = ( I mPoly R )
2 selvcllem1.t 
 |-  T = ( J mPoly U )
3 selvcllem1.i 
 |-  ( ph -> I e. V )
4 selvcllem1.j 
 |-  ( ph -> J e. W )
5 selvcllem1.r 
 |-  ( ph -> R e. CRing )
6 1 mplcrng 
 |-  ( ( I e. V /\ R e. CRing ) -> U e. CRing )
7 3 5 6 syl2anc 
 |-  ( ph -> U e. CRing )
8 2 mplassa 
 |-  ( ( J e. W /\ U e. CRing ) -> T e. AssAlg )
9 4 7 8 syl2anc 
 |-  ( ph -> T e. AssAlg )