Metamath Proof Explorer

Theorem seq1

Description: Value of the sequence builder function at its initial value. (Contributed by Mario Carneiro, 24-Jun-2013) (Revised by Mario Carneiro, 15-Sep-2013)

Ref Expression
Assertion seq1 
|- ( M e. ZZ -> ( seq M ( .+ , F ) ` M ) = ( F ` M ) )

Proof

Step Hyp Ref Expression
1 seqeq1 
 |-  ( M = if ( M e. ZZ , M , 0 ) -> seq M ( .+ , F ) = seq if ( M e. ZZ , M , 0 ) ( .+ , F ) )
2 id 
 |-  ( M = if ( M e. ZZ , M , 0 ) -> M = if ( M e. ZZ , M , 0 ) )
3 1 2 fveq12d 
 |-  ( M = if ( M e. ZZ , M , 0 ) -> ( seq M ( .+ , F ) ` M ) = ( seq if ( M e. ZZ , M , 0 ) ( .+ , F ) ` if ( M e. ZZ , M , 0 ) ) )
4 fveq2 
 |-  ( M = if ( M e. ZZ , M , 0 ) -> ( F ` M ) = ( F ` if ( M e. ZZ , M , 0 ) ) )
5 3 4 eqeq12d 
 |-  ( M = if ( M e. ZZ , M , 0 ) -> ( ( seq M ( .+ , F ) ` M ) = ( F ` M ) <-> ( seq if ( M e. ZZ , M , 0 ) ( .+ , F ) ` if ( M e. ZZ , M , 0 ) ) = ( F ` if ( M e. ZZ , M , 0 ) ) ) )
6 0z 
 |-  0 e. ZZ
7 6 elimel 
 |-  if ( M e. ZZ , M , 0 ) e. ZZ
8 eqid 
 |-  ( rec ( ( x e. _V |-> ( x + 1 ) ) , if ( M e. ZZ , M , 0 ) ) |` _om ) = ( rec ( ( x e. _V |-> ( x + 1 ) ) , if ( M e. ZZ , M , 0 ) ) |` _om )
9 fvex 
 |-  ( F ` if ( M e. ZZ , M , 0 ) ) e. _V
10 eqid 
 |-  ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x ( z e. _V , w e. _V |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. if ( M e. ZZ , M , 0 ) , ( F ` if ( M e. ZZ , M , 0 ) ) >. ) |` _om ) = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x ( z e. _V , w e. _V |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. if ( M e. ZZ , M , 0 ) , ( F ` if ( M e. ZZ , M , 0 ) ) >. ) |` _om )
11 10 seqval 
 |-  seq if ( M e. ZZ , M , 0 ) ( .+ , F ) = ran ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x ( z e. _V , w e. _V |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. if ( M e. ZZ , M , 0 ) , ( F ` if ( M e. ZZ , M , 0 ) ) >. ) |` _om )
12 7 8 9 10 11 uzrdg0i 
 |-  ( seq if ( M e. ZZ , M , 0 ) ( .+ , F ) ` if ( M e. ZZ , M , 0 ) ) = ( F ` if ( M e. ZZ , M , 0 ) )
13 5 12 dedth 
 |-  ( M e. ZZ -> ( seq M ( .+ , F ) ` M ) = ( F ` M ) )