| Step |
Hyp |
Ref |
Expression |
| 1 |
|
cats1cld.1 |
|- T = ( S ++ <" X "> ) |
| 2 |
|
cats1cli.2 |
|- S e. Word _V |
| 3 |
|
cats1fvn.3 |
|- ( # ` S ) = M |
| 4 |
3
|
oveq2i |
|- ( 0 + ( # ` S ) ) = ( 0 + M ) |
| 5 |
|
lencl |
|- ( S e. Word _V -> ( # ` S ) e. NN0 ) |
| 6 |
2 5
|
ax-mp |
|- ( # ` S ) e. NN0 |
| 7 |
3 6
|
eqeltrri |
|- M e. NN0 |
| 8 |
7
|
nn0cni |
|- M e. CC |
| 9 |
8
|
addlidi |
|- ( 0 + M ) = M |
| 10 |
4 9
|
eqtr2i |
|- M = ( 0 + ( # ` S ) ) |
| 11 |
1 10
|
fveq12i |
|- ( T ` M ) = ( ( S ++ <" X "> ) ` ( 0 + ( # ` S ) ) ) |
| 12 |
|
s1cli |
|- <" X "> e. Word _V |
| 13 |
|
s1len |
|- ( # ` <" X "> ) = 1 |
| 14 |
|
1nn |
|- 1 e. NN |
| 15 |
13 14
|
eqeltri |
|- ( # ` <" X "> ) e. NN |
| 16 |
|
lbfzo0 |
|- ( 0 e. ( 0 ..^ ( # ` <" X "> ) ) <-> ( # ` <" X "> ) e. NN ) |
| 17 |
15 16
|
mpbir |
|- 0 e. ( 0 ..^ ( # ` <" X "> ) ) |
| 18 |
|
ccatval3 |
|- ( ( S e. Word _V /\ <" X "> e. Word _V /\ 0 e. ( 0 ..^ ( # ` <" X "> ) ) ) -> ( ( S ++ <" X "> ) ` ( 0 + ( # ` S ) ) ) = ( <" X "> ` 0 ) ) |
| 19 |
2 12 17 18
|
mp3an |
|- ( ( S ++ <" X "> ) ` ( 0 + ( # ` S ) ) ) = ( <" X "> ` 0 ) |
| 20 |
11 19
|
eqtri |
|- ( T ` M ) = ( <" X "> ` 0 ) |
| 21 |
|
s1fv |
|- ( X e. V -> ( <" X "> ` 0 ) = X ) |
| 22 |
20 21
|
eqtrid |
|- ( X e. V -> ( T ` M ) = X ) |