Step |
Hyp |
Ref |
Expression |
1 |
|
itcovalt2.f |
|- F = ( n e. NN0 |-> ( ( 2 x. n ) + C ) ) |
2 |
|
fveq2 |
|- ( x = 0 -> ( ( IterComp ` F ) ` x ) = ( ( IterComp ` F ) ` 0 ) ) |
3 |
|
oveq2 |
|- ( x = 0 -> ( 2 ^ x ) = ( 2 ^ 0 ) ) |
4 |
3
|
oveq2d |
|- ( x = 0 -> ( ( n + C ) x. ( 2 ^ x ) ) = ( ( n + C ) x. ( 2 ^ 0 ) ) ) |
5 |
4
|
oveq1d |
|- ( x = 0 -> ( ( ( n + C ) x. ( 2 ^ x ) ) - C ) = ( ( ( n + C ) x. ( 2 ^ 0 ) ) - C ) ) |
6 |
5
|
mpteq2dv |
|- ( x = 0 -> ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ x ) ) - C ) ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ 0 ) ) - C ) ) ) |
7 |
2 6
|
eqeq12d |
|- ( x = 0 -> ( ( ( IterComp ` F ) ` x ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ x ) ) - C ) ) <-> ( ( IterComp ` F ) ` 0 ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ 0 ) ) - C ) ) ) ) |
8 |
7
|
imbi2d |
|- ( x = 0 -> ( ( C e. NN0 -> ( ( IterComp ` F ) ` x ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ x ) ) - C ) ) ) <-> ( C e. NN0 -> ( ( IterComp ` F ) ` 0 ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ 0 ) ) - C ) ) ) ) ) |
9 |
|
fveq2 |
|- ( x = y -> ( ( IterComp ` F ) ` x ) = ( ( IterComp ` F ) ` y ) ) |
10 |
|
oveq2 |
|- ( x = y -> ( 2 ^ x ) = ( 2 ^ y ) ) |
11 |
10
|
oveq2d |
|- ( x = y -> ( ( n + C ) x. ( 2 ^ x ) ) = ( ( n + C ) x. ( 2 ^ y ) ) ) |
12 |
11
|
oveq1d |
|- ( x = y -> ( ( ( n + C ) x. ( 2 ^ x ) ) - C ) = ( ( ( n + C ) x. ( 2 ^ y ) ) - C ) ) |
13 |
12
|
mpteq2dv |
|- ( x = y -> ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ x ) ) - C ) ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ y ) ) - C ) ) ) |
14 |
9 13
|
eqeq12d |
|- ( x = y -> ( ( ( IterComp ` F ) ` x ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ x ) ) - C ) ) <-> ( ( IterComp ` F ) ` y ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ y ) ) - C ) ) ) ) |
15 |
14
|
imbi2d |
|- ( x = y -> ( ( C e. NN0 -> ( ( IterComp ` F ) ` x ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ x ) ) - C ) ) ) <-> ( C e. NN0 -> ( ( IterComp ` F ) ` y ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ y ) ) - C ) ) ) ) ) |
16 |
|
fveq2 |
|- ( x = ( y + 1 ) -> ( ( IterComp ` F ) ` x ) = ( ( IterComp ` F ) ` ( y + 1 ) ) ) |
17 |
|
oveq2 |
|- ( x = ( y + 1 ) -> ( 2 ^ x ) = ( 2 ^ ( y + 1 ) ) ) |
18 |
17
|
oveq2d |
|- ( x = ( y + 1 ) -> ( ( n + C ) x. ( 2 ^ x ) ) = ( ( n + C ) x. ( 2 ^ ( y + 1 ) ) ) ) |
19 |
18
|
oveq1d |
|- ( x = ( y + 1 ) -> ( ( ( n + C ) x. ( 2 ^ x ) ) - C ) = ( ( ( n + C ) x. ( 2 ^ ( y + 1 ) ) ) - C ) ) |
20 |
19
|
mpteq2dv |
|- ( x = ( y + 1 ) -> ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ x ) ) - C ) ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ ( y + 1 ) ) ) - C ) ) ) |
21 |
16 20
|
eqeq12d |
|- ( x = ( y + 1 ) -> ( ( ( IterComp ` F ) ` x ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ x ) ) - C ) ) <-> ( ( IterComp ` F ) ` ( y + 1 ) ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ ( y + 1 ) ) ) - C ) ) ) ) |
22 |
21
|
imbi2d |
|- ( x = ( y + 1 ) -> ( ( C e. NN0 -> ( ( IterComp ` F ) ` x ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ x ) ) - C ) ) ) <-> ( C e. NN0 -> ( ( IterComp ` F ) ` ( y + 1 ) ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ ( y + 1 ) ) ) - C ) ) ) ) ) |
23 |
|
fveq2 |
|- ( x = I -> ( ( IterComp ` F ) ` x ) = ( ( IterComp ` F ) ` I ) ) |
24 |
|
oveq2 |
|- ( x = I -> ( 2 ^ x ) = ( 2 ^ I ) ) |
25 |
24
|
oveq2d |
|- ( x = I -> ( ( n + C ) x. ( 2 ^ x ) ) = ( ( n + C ) x. ( 2 ^ I ) ) ) |
26 |
25
|
oveq1d |
|- ( x = I -> ( ( ( n + C ) x. ( 2 ^ x ) ) - C ) = ( ( ( n + C ) x. ( 2 ^ I ) ) - C ) ) |
27 |
26
|
mpteq2dv |
|- ( x = I -> ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ x ) ) - C ) ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ I ) ) - C ) ) ) |
28 |
23 27
|
eqeq12d |
|- ( x = I -> ( ( ( IterComp ` F ) ` x ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ x ) ) - C ) ) <-> ( ( IterComp ` F ) ` I ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ I ) ) - C ) ) ) ) |
29 |
28
|
imbi2d |
|- ( x = I -> ( ( C e. NN0 -> ( ( IterComp ` F ) ` x ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ x ) ) - C ) ) ) <-> ( C e. NN0 -> ( ( IterComp ` F ) ` I ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ I ) ) - C ) ) ) ) ) |
30 |
1
|
itcovalt2lem1 |
|- ( C e. NN0 -> ( ( IterComp ` F ) ` 0 ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ 0 ) ) - C ) ) ) |
31 |
|
pm2.27 |
|- ( C e. NN0 -> ( ( C e. NN0 -> ( ( IterComp ` F ) ` y ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ y ) ) - C ) ) ) -> ( ( IterComp ` F ) ` y ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ y ) ) - C ) ) ) ) |
32 |
31
|
adantl |
|- ( ( y e. NN0 /\ C e. NN0 ) -> ( ( C e. NN0 -> ( ( IterComp ` F ) ` y ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ y ) ) - C ) ) ) -> ( ( IterComp ` F ) ` y ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ y ) ) - C ) ) ) ) |
33 |
1
|
itcovalt2lem2 |
|- ( ( y e. NN0 /\ C e. NN0 ) -> ( ( ( IterComp ` F ) ` y ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ y ) ) - C ) ) -> ( ( IterComp ` F ) ` ( y + 1 ) ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ ( y + 1 ) ) ) - C ) ) ) ) |
34 |
32 33
|
syld |
|- ( ( y e. NN0 /\ C e. NN0 ) -> ( ( C e. NN0 -> ( ( IterComp ` F ) ` y ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ y ) ) - C ) ) ) -> ( ( IterComp ` F ) ` ( y + 1 ) ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ ( y + 1 ) ) ) - C ) ) ) ) |
35 |
34
|
ex |
|- ( y e. NN0 -> ( C e. NN0 -> ( ( C e. NN0 -> ( ( IterComp ` F ) ` y ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ y ) ) - C ) ) ) -> ( ( IterComp ` F ) ` ( y + 1 ) ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ ( y + 1 ) ) ) - C ) ) ) ) ) |
36 |
35
|
com23 |
|- ( y e. NN0 -> ( ( C e. NN0 -> ( ( IterComp ` F ) ` y ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ y ) ) - C ) ) ) -> ( C e. NN0 -> ( ( IterComp ` F ) ` ( y + 1 ) ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ ( y + 1 ) ) ) - C ) ) ) ) ) |
37 |
8 15 22 29 30 36
|
nn0ind |
|- ( I e. NN0 -> ( C e. NN0 -> ( ( IterComp ` F ) ` I ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ I ) ) - C ) ) ) ) |
38 |
37
|
imp |
|- ( ( I e. NN0 /\ C e. NN0 ) -> ( ( IterComp ` F ) ` I ) = ( n e. NN0 |-> ( ( ( n + C ) x. ( 2 ^ I ) ) - C ) ) ) |