Metamath Proof Explorer

Theorem dvnfval

Description: Value of the iterated derivative. (Contributed by Mario Carneiro, 11-Feb-2015)

Ref Expression
Hypothesis dvnfval.1 
|- G = ( x e. _V |-> ( S _D x ) )
Assertion dvnfval 
|- ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> ( S Dn F ) = seq 0 ( ( G o. 1st ) , ( NN0 X. { F } ) ) )

Proof

Step Hyp Ref Expression
1 dvnfval.1 
 |-  G = ( x e. _V |-> ( S _D x ) )
2 df-dvn 
 |-  Dn = ( s e. ~P CC , f e. ( CC ^pm s ) |-> seq 0 ( ( ( x e. _V |-> ( s _D x ) ) o. 1st ) , ( NN0 X. { f } ) ) )
3 2 a1i 
 |-  ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> Dn = ( s e. ~P CC , f e. ( CC ^pm s ) |-> seq 0 ( ( ( x e. _V |-> ( s _D x ) ) o. 1st ) , ( NN0 X. { f } ) ) ) )
4 simprl 
 |-  ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> s = S )
5 4 oveq1d 
 |-  ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> ( s _D x ) = ( S _D x ) )
6 5 mpteq2dv 
 |-  ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> ( x e. _V |-> ( s _D x ) ) = ( x e. _V |-> ( S _D x ) ) )
7 6 1 eqtr4di 
 |-  ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> ( x e. _V |-> ( s _D x ) ) = G )
8 7 coeq1d 
 |-  ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> ( ( x e. _V |-> ( s _D x ) ) o. 1st ) = ( G o. 1st ) )
9 8 seqeq2d 
 |-  ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> seq 0 ( ( ( x e. _V |-> ( s _D x ) ) o. 1st ) , ( NN0 X. { f } ) ) = seq 0 ( ( G o. 1st ) , ( NN0 X. { f } ) ) )
10 simprr 
 |-  ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> f = F )
11 10 sneqd 
 |-  ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> { f } = { F } )
12 11 xpeq2d 
 |-  ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> ( NN0 X. { f } ) = ( NN0 X. { F } ) )
13 12 seqeq3d 
 |-  ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> seq 0 ( ( G o. 1st ) , ( NN0 X. { f } ) ) = seq 0 ( ( G o. 1st ) , ( NN0 X. { F } ) ) )
14 9 13 eqtrd 
 |-  ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> seq 0 ( ( ( x e. _V |-> ( s _D x ) ) o. 1st ) , ( NN0 X. { f } ) ) = seq 0 ( ( G o. 1st ) , ( NN0 X. { F } ) ) )
15 simpr 
 |-  ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ s = S ) -> s = S )
16 15 oveq2d 
 |-  ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ s = S ) -> ( CC ^pm s ) = ( CC ^pm S ) )
17 simpl 
 |-  ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> S C_ CC )
18 cnex 
 |-  CC e. _V
19 18 elpw2 
 |-  ( S e. ~P CC <-> S C_ CC )
20 17 19 sylibr 
 |-  ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> S e. ~P CC )
21 simpr 
 |-  ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> F e. ( CC ^pm S ) )
22 seqex 
 |-  seq 0 ( ( G o. 1st ) , ( NN0 X. { F } ) ) e. _V
23 22 a1i 
 |-  ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> seq 0 ( ( G o. 1st ) , ( NN0 X. { F } ) ) e. _V )
24 3 14 16 20 21 23 ovmpodx 
 |-  ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> ( S Dn F ) = seq 0 ( ( G o. 1st ) , ( NN0 X. { F } ) ) )