Metamath Proof Explorer

Theorem esplyval

Description: The elementary polynomials for a given index I of variables and base ring R . (Contributed by Thierry Arnoux, 18-Jan-2026)

Ref Expression
Hypotheses esplyval.d 
|- D = { h e. ( NN0 ^m I ) | h finSupp 0 }
esplyval.i 
|- ( ph -> I e. V )
esplyval.r 
|- ( ph -> R e. W )
Assertion esplyval 
|- ( ph -> ( I eSymPoly R ) = ( k e. NN0 |-> ( ( ZRHom ` R ) o. ( ( _Ind ` D ) ` ( ( _Ind ` I ) " { c e. ~P I | ( # ` c ) = k } ) ) ) ) )

Proof

Step Hyp Ref Expression
1 esplyval.d 
 |-  D = { h e. ( NN0 ^m I ) | h finSupp 0 }
2 esplyval.i 
 |-  ( ph -> I e. V )
3 esplyval.r 
 |-  ( ph -> R e. W )
4 df-esply 
 |-  eSymPoly = ( i e. _V , r e. _V |-> ( k e. NN0 |-> ( ( ZRHom ` r ) o. ( ( _Ind ` { h e. ( NN0 ^m i ) | h finSupp 0 } ) ` ( ( _Ind ` i ) " { c e. ~P i | ( # ` c ) = k } ) ) ) ) )
5 4 a1i 
 |-  ( ph -> eSymPoly = ( i e. _V , r e. _V |-> ( k e. NN0 |-> ( ( ZRHom ` r ) o. ( ( _Ind ` { h e. ( NN0 ^m i ) | h finSupp 0 } ) ` ( ( _Ind ` i ) " { c e. ~P i | ( # ` c ) = k } ) ) ) ) ) )
6 fveq2 
 |-  ( r = R -> ( ZRHom ` r ) = ( ZRHom ` R ) )
7 6 adantl 
 |-  ( ( i = I /\ r = R ) -> ( ZRHom ` r ) = ( ZRHom ` R ) )
8 oveq2 
 |-  ( i = I -> ( NN0 ^m i ) = ( NN0 ^m I ) )
9 8 rabeqdv 
 |-  ( i = I -> { h e. ( NN0 ^m i ) | h finSupp 0 } = { h e. ( NN0 ^m I ) | h finSupp 0 } )
10 9 1 eqtr4di 
 |-  ( i = I -> { h e. ( NN0 ^m i ) | h finSupp 0 } = D )
11 10 fveq2d 
 |-  ( i = I -> ( _Ind ` { h e. ( NN0 ^m i ) | h finSupp 0 } ) = ( _Ind ` D ) )
12 11 adantr 
 |-  ( ( i = I /\ r = R ) -> ( _Ind ` { h e. ( NN0 ^m i ) | h finSupp 0 } ) = ( _Ind ` D ) )
13 fveq2 
 |-  ( i = I -> ( _Ind ` i ) = ( _Ind ` I ) )
14 13 adantr 
 |-  ( ( i = I /\ r = R ) -> ( _Ind ` i ) = ( _Ind ` I ) )
15 pweq 
 |-  ( i = I -> ~P i = ~P I )
16 15 rabeqdv 
 |-  ( i = I -> { c e. ~P i | ( # ` c ) = k } = { c e. ~P I | ( # ` c ) = k } )
17 16 adantr 
 |-  ( ( i = I /\ r = R ) -> { c e. ~P i | ( # ` c ) = k } = { c e. ~P I | ( # ` c ) = k } )
18 14 17 imaeq12d 
 |-  ( ( i = I /\ r = R ) -> ( ( _Ind ` i ) " { c e. ~P i | ( # ` c ) = k } ) = ( ( _Ind ` I ) " { c e. ~P I | ( # ` c ) = k } ) )
19 12 18 fveq12d 
 |-  ( ( i = I /\ r = R ) -> ( ( _Ind ` { h e. ( NN0 ^m i ) | h finSupp 0 } ) ` ( ( _Ind ` i ) " { c e. ~P i | ( # ` c ) = k } ) ) = ( ( _Ind ` D ) ` ( ( _Ind ` I ) " { c e. ~P I | ( # ` c ) = k } ) ) )
20 7 19 coeq12d 
 |-  ( ( i = I /\ r = R ) -> ( ( ZRHom ` r ) o. ( ( _Ind ` { h e. ( NN0 ^m i ) | h finSupp 0 } ) ` ( ( _Ind ` i ) " { c e. ~P i | ( # ` c ) = k } ) ) ) = ( ( ZRHom ` R ) o. ( ( _Ind ` D ) ` ( ( _Ind ` I ) " { c e. ~P I | ( # ` c ) = k } ) ) ) )
21 20 mpteq2dv 
 |-  ( ( i = I /\ r = R ) -> ( k e. NN0 |-> ( ( ZRHom ` r ) o. ( ( _Ind ` { h e. ( NN0 ^m i ) | h finSupp 0 } ) ` ( ( _Ind ` i ) " { c e. ~P i | ( # ` c ) = k } ) ) ) ) = ( k e. NN0 |-> ( ( ZRHom ` R ) o. ( ( _Ind ` D ) ` ( ( _Ind ` I ) " { c e. ~P I | ( # ` c ) = k } ) ) ) ) )
22 21 adantl 
 |-  ( ( ph /\ ( i = I /\ r = R ) ) -> ( k e. NN0 |-> ( ( ZRHom ` r ) o. ( ( _Ind ` { h e. ( NN0 ^m i ) | h finSupp 0 } ) ` ( ( _Ind ` i ) " { c e. ~P i | ( # ` c ) = k } ) ) ) ) = ( k e. NN0 |-> ( ( ZRHom ` R ) o. ( ( _Ind ` D ) ` ( ( _Ind ` I ) " { c e. ~P I | ( # ` c ) = k } ) ) ) ) )
23 2 elexd 
 |-  ( ph -> I e. _V )
24 3 elexd 
 |-  ( ph -> R e. _V )
25 nn0ex 
 |-  NN0 e. _V
26 25 mptex 
 |-  ( k e. NN0 |-> ( ( ZRHom ` R ) o. ( ( _Ind ` D ) ` ( ( _Ind ` I ) " { c e. ~P I | ( # ` c ) = k } ) ) ) ) e. _V
27 26 a1i 
 |-  ( ph -> ( k e. NN0 |-> ( ( ZRHom ` R ) o. ( ( _Ind ` D ) ` ( ( _Ind ` I ) " { c e. ~P I | ( # ` c ) = k } ) ) ) ) e. _V )
28 5 22 23 24 27 ovmpod 
 |-  ( ph -> ( I eSymPoly R ) = ( k e. NN0 |-> ( ( ZRHom ` R ) o. ( ( _Ind ` D ) ` ( ( _Ind ` I ) " { c e. ~P I | ( # ` c ) = k } ) ) ) ) )