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Metamath Proof Explorer

Theorem evlval

Description: Value of the simple/same ring evaluation map. (Contributed by Stefan O'Rear, 19-Mar-2015) (Revised by Mario Carneiro, 12-Jun-2015)

Ref Expression
Hypotheses evlval.q 
|- Q = ( I eval R )
evlval.b 
|- B = ( Base ` R )
Assertion evlval 
|- Q = ( ( I evalSub R ) ` B )

Proof

Step Hyp Ref Expression
1 evlval.q 
 |-  Q = ( I eval R )
2 evlval.b 
 |-  B = ( Base ` R )
3 oveq12 
 |-  ( ( i = I /\ r = R ) -> ( i evalSub r ) = ( I evalSub R ) )
4 fveq2 
 |-  ( r = R -> ( Base ` r ) = ( Base ` R ) )
5 4 2 eqtr4di 
 |-  ( r = R -> ( Base ` r ) = B )
6 5 adantl 
 |-  ( ( i = I /\ r = R ) -> ( Base ` r ) = B )
7 3 6 fveq12d 
 |-  ( ( i = I /\ r = R ) -> ( ( i evalSub r ) ` ( Base ` r ) ) = ( ( I evalSub R ) ` B ) )
8 df-evl 
 |-  eval = ( i e. _V , r e. _V |-> ( ( i evalSub r ) ` ( Base ` r ) ) )
9 fvex 
 |-  ( ( I evalSub R ) ` B ) e. _V
10 7 8 9 ovmpoa 
 |-  ( ( I e. _V /\ R e. _V ) -> ( I eval R ) = ( ( I evalSub R ) ` B ) )
11 8 mpondm0 
 |-  ( -. ( I e. _V /\ R e. _V ) -> ( I eval R ) = (/) )
12 0fv 
 |-  ( (/) ` B ) = (/)
13 11 12 eqtr4di 
 |-  ( -. ( I e. _V /\ R e. _V ) -> ( I eval R ) = ( (/) ` B ) )
14 reldmevls 
 |-  Rel dom evalSub
15 14 ovprc 
 |-  ( -. ( I e. _V /\ R e. _V ) -> ( I evalSub R ) = (/) )
16 15 fveq1d 
 |-  ( -. ( I e. _V /\ R e. _V ) -> ( ( I evalSub R ) ` B ) = ( (/) ` B ) )
17 13 16 eqtr4d 
 |-  ( -. ( I e. _V /\ R e. _V ) -> ( I eval R ) = ( ( I evalSub R ) ` B ) )
18 10 17 pm2.61i 
 |-  ( I eval R ) = ( ( I evalSub R ) ` B )
19 1 18 eqtri 
 |-  Q = ( ( I evalSub R ) ` B )