Step |
Hyp |
Ref |
Expression |
1 |
|
itcovalpc.f |
|- F = ( n e. NN0 |-> ( n + C ) ) |
2 |
|
fveq2 |
|- ( x = 0 -> ( ( IterComp ` F ) ` x ) = ( ( IterComp ` F ) ` 0 ) ) |
3 |
|
oveq2 |
|- ( x = 0 -> ( C x. x ) = ( C x. 0 ) ) |
4 |
3
|
oveq2d |
|- ( x = 0 -> ( n + ( C x. x ) ) = ( n + ( C x. 0 ) ) ) |
5 |
4
|
mpteq2dv |
|- ( x = 0 -> ( n e. NN0 |-> ( n + ( C x. x ) ) ) = ( n e. NN0 |-> ( n + ( C x. 0 ) ) ) ) |
6 |
2 5
|
eqeq12d |
|- ( x = 0 -> ( ( ( IterComp ` F ) ` x ) = ( n e. NN0 |-> ( n + ( C x. x ) ) ) <-> ( ( IterComp ` F ) ` 0 ) = ( n e. NN0 |-> ( n + ( C x. 0 ) ) ) ) ) |
7 |
|
fveq2 |
|- ( x = y -> ( ( IterComp ` F ) ` x ) = ( ( IterComp ` F ) ` y ) ) |
8 |
|
oveq2 |
|- ( x = y -> ( C x. x ) = ( C x. y ) ) |
9 |
8
|
oveq2d |
|- ( x = y -> ( n + ( C x. x ) ) = ( n + ( C x. y ) ) ) |
10 |
9
|
mpteq2dv |
|- ( x = y -> ( n e. NN0 |-> ( n + ( C x. x ) ) ) = ( n e. NN0 |-> ( n + ( C x. y ) ) ) ) |
11 |
7 10
|
eqeq12d |
|- ( x = y -> ( ( ( IterComp ` F ) ` x ) = ( n e. NN0 |-> ( n + ( C x. x ) ) ) <-> ( ( IterComp ` F ) ` y ) = ( n e. NN0 |-> ( n + ( C x. y ) ) ) ) ) |
12 |
|
fveq2 |
|- ( x = ( y + 1 ) -> ( ( IterComp ` F ) ` x ) = ( ( IterComp ` F ) ` ( y + 1 ) ) ) |
13 |
|
oveq2 |
|- ( x = ( y + 1 ) -> ( C x. x ) = ( C x. ( y + 1 ) ) ) |
14 |
13
|
oveq2d |
|- ( x = ( y + 1 ) -> ( n + ( C x. x ) ) = ( n + ( C x. ( y + 1 ) ) ) ) |
15 |
14
|
mpteq2dv |
|- ( x = ( y + 1 ) -> ( n e. NN0 |-> ( n + ( C x. x ) ) ) = ( n e. NN0 |-> ( n + ( C x. ( y + 1 ) ) ) ) ) |
16 |
12 15
|
eqeq12d |
|- ( x = ( y + 1 ) -> ( ( ( IterComp ` F ) ` x ) = ( n e. NN0 |-> ( n + ( C x. x ) ) ) <-> ( ( IterComp ` F ) ` ( y + 1 ) ) = ( n e. NN0 |-> ( n + ( C x. ( y + 1 ) ) ) ) ) ) |
17 |
|
fveq2 |
|- ( x = I -> ( ( IterComp ` F ) ` x ) = ( ( IterComp ` F ) ` I ) ) |
18 |
|
oveq2 |
|- ( x = I -> ( C x. x ) = ( C x. I ) ) |
19 |
18
|
oveq2d |
|- ( x = I -> ( n + ( C x. x ) ) = ( n + ( C x. I ) ) ) |
20 |
19
|
mpteq2dv |
|- ( x = I -> ( n e. NN0 |-> ( n + ( C x. x ) ) ) = ( n e. NN0 |-> ( n + ( C x. I ) ) ) ) |
21 |
17 20
|
eqeq12d |
|- ( x = I -> ( ( ( IterComp ` F ) ` x ) = ( n e. NN0 |-> ( n + ( C x. x ) ) ) <-> ( ( IterComp ` F ) ` I ) = ( n e. NN0 |-> ( n + ( C x. I ) ) ) ) ) |
22 |
1
|
itcovalpclem1 |
|- ( C e. NN0 -> ( ( IterComp ` F ) ` 0 ) = ( n e. NN0 |-> ( n + ( C x. 0 ) ) ) ) |
23 |
1
|
itcovalpclem2 |
|- ( ( y e. NN0 /\ C e. NN0 ) -> ( ( ( IterComp ` F ) ` y ) = ( n e. NN0 |-> ( n + ( C x. y ) ) ) -> ( ( IterComp ` F ) ` ( y + 1 ) ) = ( n e. NN0 |-> ( n + ( C x. ( y + 1 ) ) ) ) ) ) |
24 |
23
|
ancoms |
|- ( ( C e. NN0 /\ y e. NN0 ) -> ( ( ( IterComp ` F ) ` y ) = ( n e. NN0 |-> ( n + ( C x. y ) ) ) -> ( ( IterComp ` F ) ` ( y + 1 ) ) = ( n e. NN0 |-> ( n + ( C x. ( y + 1 ) ) ) ) ) ) |
25 |
24
|
imp |
|- ( ( ( C e. NN0 /\ y e. NN0 ) /\ ( ( IterComp ` F ) ` y ) = ( n e. NN0 |-> ( n + ( C x. y ) ) ) ) -> ( ( IterComp ` F ) ` ( y + 1 ) ) = ( n e. NN0 |-> ( n + ( C x. ( y + 1 ) ) ) ) ) |
26 |
6 11 16 21 22 25
|
nn0indd |
|- ( ( C e. NN0 /\ I e. NN0 ) -> ( ( IterComp ` F ) ` I ) = ( n e. NN0 |-> ( n + ( C x. I ) ) ) ) |
27 |
26
|
ancoms |
|- ( ( I e. NN0 /\ C e. NN0 ) -> ( ( IterComp ` F ) ` I ) = ( n e. NN0 |-> ( n + ( C x. I ) ) ) ) |