Step |
Hyp |
Ref |
Expression |
1 |
|
simp3 |
|- ( ( S C_ CC /\ F e. ( CC ^pm S ) /\ N e. NN0 ) -> N e. NN0 ) |
2 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
3 |
1 2
|
eleqtrdi |
|- ( ( S C_ CC /\ F e. ( CC ^pm S ) /\ N e. NN0 ) -> N e. ( ZZ>= ` 0 ) ) |
4 |
|
seqp1 |
|- ( N e. ( ZZ>= ` 0 ) -> ( seq 0 ( ( ( x e. _V |-> ( S _D x ) ) o. 1st ) , ( NN0 X. { F } ) ) ` ( N + 1 ) ) = ( ( seq 0 ( ( ( x e. _V |-> ( S _D x ) ) o. 1st ) , ( NN0 X. { F } ) ) ` N ) ( ( x e. _V |-> ( S _D x ) ) o. 1st ) ( ( NN0 X. { F } ) ` ( N + 1 ) ) ) ) |
5 |
3 4
|
syl |
|- ( ( S C_ CC /\ F e. ( CC ^pm S ) /\ N e. NN0 ) -> ( seq 0 ( ( ( x e. _V |-> ( S _D x ) ) o. 1st ) , ( NN0 X. { F } ) ) ` ( N + 1 ) ) = ( ( seq 0 ( ( ( x e. _V |-> ( S _D x ) ) o. 1st ) , ( NN0 X. { F } ) ) ` N ) ( ( x e. _V |-> ( S _D x ) ) o. 1st ) ( ( NN0 X. { F } ) ` ( N + 1 ) ) ) ) |
6 |
|
fvex |
|- ( seq 0 ( ( ( x e. _V |-> ( S _D x ) ) o. 1st ) , ( NN0 X. { F } ) ) ` N ) e. _V |
7 |
|
fvex |
|- ( ( NN0 X. { F } ) ` ( N + 1 ) ) e. _V |
8 |
6 7
|
algrflem |
|- ( ( seq 0 ( ( ( x e. _V |-> ( S _D x ) ) o. 1st ) , ( NN0 X. { F } ) ) ` N ) ( ( x e. _V |-> ( S _D x ) ) o. 1st ) ( ( NN0 X. { F } ) ` ( N + 1 ) ) ) = ( ( x e. _V |-> ( S _D x ) ) ` ( seq 0 ( ( ( x e. _V |-> ( S _D x ) ) o. 1st ) , ( NN0 X. { F } ) ) ` N ) ) |
9 |
5 8
|
eqtrdi |
|- ( ( S C_ CC /\ F e. ( CC ^pm S ) /\ N e. NN0 ) -> ( seq 0 ( ( ( x e. _V |-> ( S _D x ) ) o. 1st ) , ( NN0 X. { F } ) ) ` ( N + 1 ) ) = ( ( x e. _V |-> ( S _D x ) ) ` ( seq 0 ( ( ( x e. _V |-> ( S _D x ) ) o. 1st ) , ( NN0 X. { F } ) ) ` N ) ) ) |
10 |
|
eqid |
|- ( x e. _V |-> ( S _D x ) ) = ( x e. _V |-> ( S _D x ) ) |
11 |
10
|
dvnfval |
|- ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> ( S Dn F ) = seq 0 ( ( ( x e. _V |-> ( S _D x ) ) o. 1st ) , ( NN0 X. { F } ) ) ) |
12 |
11
|
3adant3 |
|- ( ( S C_ CC /\ F e. ( CC ^pm S ) /\ N e. NN0 ) -> ( S Dn F ) = seq 0 ( ( ( x e. _V |-> ( S _D x ) ) o. 1st ) , ( NN0 X. { F } ) ) ) |
13 |
12
|
fveq1d |
|- ( ( S C_ CC /\ F e. ( CC ^pm S ) /\ N e. NN0 ) -> ( ( S Dn F ) ` ( N + 1 ) ) = ( seq 0 ( ( ( x e. _V |-> ( S _D x ) ) o. 1st ) , ( NN0 X. { F } ) ) ` ( N + 1 ) ) ) |
14 |
|
fvex |
|- ( ( S Dn F ) ` N ) e. _V |
15 |
|
oveq2 |
|- ( x = ( ( S Dn F ) ` N ) -> ( S _D x ) = ( S _D ( ( S Dn F ) ` N ) ) ) |
16 |
|
ovex |
|- ( S _D ( ( S Dn F ) ` N ) ) e. _V |
17 |
15 10 16
|
fvmpt |
|- ( ( ( S Dn F ) ` N ) e. _V -> ( ( x e. _V |-> ( S _D x ) ) ` ( ( S Dn F ) ` N ) ) = ( S _D ( ( S Dn F ) ` N ) ) ) |
18 |
14 17
|
ax-mp |
|- ( ( x e. _V |-> ( S _D x ) ) ` ( ( S Dn F ) ` N ) ) = ( S _D ( ( S Dn F ) ` N ) ) |
19 |
12
|
fveq1d |
|- ( ( S C_ CC /\ F e. ( CC ^pm S ) /\ N e. NN0 ) -> ( ( S Dn F ) ` N ) = ( seq 0 ( ( ( x e. _V |-> ( S _D x ) ) o. 1st ) , ( NN0 X. { F } ) ) ` N ) ) |
20 |
19
|
fveq2d |
|- ( ( S C_ CC /\ F e. ( CC ^pm S ) /\ N e. NN0 ) -> ( ( x e. _V |-> ( S _D x ) ) ` ( ( S Dn F ) ` N ) ) = ( ( x e. _V |-> ( S _D x ) ) ` ( seq 0 ( ( ( x e. _V |-> ( S _D x ) ) o. 1st ) , ( NN0 X. { F } ) ) ` N ) ) ) |
21 |
18 20
|
syl5eqr |
|- ( ( S C_ CC /\ F e. ( CC ^pm S ) /\ N e. NN0 ) -> ( S _D ( ( S Dn F ) ` N ) ) = ( ( x e. _V |-> ( S _D x ) ) ` ( seq 0 ( ( ( x e. _V |-> ( S _D x ) ) o. 1st ) , ( NN0 X. { F } ) ) ` N ) ) ) |
22 |
9 13 21
|
3eqtr4d |
|- ( ( S C_ CC /\ F e. ( CC ^pm S ) /\ N e. NN0 ) -> ( ( S Dn F ) ` ( N + 1 ) ) = ( S _D ( ( S Dn F ) ` N ) ) ) |