Metamath Proof Explorer

Theorem etransclem12

Description: C applied to N . (Contributed by Glauco Siliprandi, 5-Apr-2020)

Ref Expression
Hypotheses etransclem12.1 
|- C = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } )
etransclem12.2 
|- ( ph -> N e. NN0 )
Assertion etransclem12 
|- ( ph -> ( C ` N ) = { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } )

Proof

Step Hyp Ref Expression
1 etransclem12.1 
 |-  C = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } )
2 etransclem12.2 
 |-  ( ph -> N e. NN0 )
3 oveq2 
 |-  ( n = N -> ( 0 ... n ) = ( 0 ... N ) )
4 3 oveq1d 
 |-  ( n = N -> ( ( 0 ... n ) ^m ( 0 ... M ) ) = ( ( 0 ... N ) ^m ( 0 ... M ) ) )
5 eqeq2 
 |-  ( n = N -> ( sum_ j e. ( 0 ... M ) ( c ` j ) = n <-> sum_ j e. ( 0 ... M ) ( c ` j ) = N ) )
6 4 5 rabeqbidv 
 |-  ( n = N -> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } = { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } )
7 ovex 
 |-  ( ( 0 ... N ) ^m ( 0 ... M ) ) e. _V
8 7 rabex 
 |-  { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } e. _V
9 8 a1i 
 |-  ( ph -> { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } e. _V )
10 1 6 2 9 fvmptd3 
 |-  ( ph -> ( C ` N ) = { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } )