Metamath Proof Explorer

Theorem efgsval

Description: Value of the auxiliary function S defining a sequence of extensions. (Contributed by Mario Carneiro, 27-Sep-2015)

Ref Expression
Hypotheses efgval.w 
|- W = ( _I ` Word ( I X. 2o ) )
efgval.r 
|- .~ = ( ~FG ` I )
efgval2.m 
|- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
efgval2.t 
|- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
efgred.d 
|- D = ( W \ U_ x e. W ran ( T ` x ) )
efgred.s 
|- S = ( m e. { t e. ( Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1 ..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } |-> ( m ` ( ( # ` m ) - 1 ) ) )
Assertion efgsval 
|- ( F e. dom S -> ( S ` F ) = ( F ` ( ( # ` F ) - 1 ) ) )

Proof

Step Hyp Ref Expression
1 efgval.w 
 |-  W = ( _I ` Word ( I X. 2o ) )
2 efgval.r 
 |-  .~ = ( ~FG ` I )
3 efgval2.m 
 |-  M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
4 efgval2.t 
 |-  T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
5 efgred.d 
 |-  D = ( W \ U_ x e. W ran ( T ` x ) )
6 efgred.s 
 |-  S = ( m e. { t e. ( Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1 ..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } |-> ( m ` ( ( # ` m ) - 1 ) ) )
7 id 
 |-  ( f = F -> f = F )
8 fveq2 
 |-  ( f = F -> ( # ` f ) = ( # ` F ) )
9 8 oveq1d 
 |-  ( f = F -> ( ( # ` f ) - 1 ) = ( ( # ` F ) - 1 ) )
10 7 9 fveq12d 
 |-  ( f = F -> ( f ` ( ( # ` f ) - 1 ) ) = ( F ` ( ( # ` F ) - 1 ) ) )
11 id 
 |-  ( m = f -> m = f )
12 fveq2 
 |-  ( m = f -> ( # ` m ) = ( # ` f ) )
13 12 oveq1d 
 |-  ( m = f -> ( ( # ` m ) - 1 ) = ( ( # ` f ) - 1 ) )
14 11 13 fveq12d 
 |-  ( m = f -> ( m ` ( ( # ` m ) - 1 ) ) = ( f ` ( ( # ` f ) - 1 ) ) )
15 14 cbvmptv 
 |-  ( m e. { t e. ( Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1 ..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } |-> ( m ` ( ( # ` m ) - 1 ) ) ) = ( f e. { t e. ( Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1 ..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } |-> ( f ` ( ( # ` f ) - 1 ) ) )
16 6 15 eqtri 
 |-  S = ( f e. { t e. ( Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1 ..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } |-> ( f ` ( ( # ` f ) - 1 ) ) )
17 fvex 
 |-  ( F ` ( ( # ` F ) - 1 ) ) e. _V
18 10 16 17 fvmpt 
 |-  ( F e. { t e. ( Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1 ..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } -> ( S ` F ) = ( F ` ( ( # ` F ) - 1 ) ) )
19 1 2 3 4 5 6 efgsf 
 |-  S : { t e. ( Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1 ..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } --> W
20 19 fdmi 
 |-  dom S = { t e. ( Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1 ..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) }
21 18 20 eleq2s 
 |-  ( F e. dom S -> ( S ` F ) = ( F ` ( ( # ` F ) - 1 ) ) )