Step |
Hyp |
Ref |
Expression |
1 |
|
s1cli |
|- <" X "> e. Word _V |
2 |
|
s1cli |
|- <" Y "> e. Word _V |
3 |
|
1z |
|- 1 e. ZZ |
4 |
|
2z |
|- 2 e. ZZ |
5 |
|
1lt2 |
|- 1 < 2 |
6 |
|
fzolb |
|- ( 1 e. ( 1 ..^ 2 ) <-> ( 1 e. ZZ /\ 2 e. ZZ /\ 1 < 2 ) ) |
7 |
3 4 5 6
|
mpbir3an |
|- 1 e. ( 1 ..^ 2 ) |
8 |
|
s1len |
|- ( # ` <" X "> ) = 1 |
9 |
|
s1len |
|- ( # ` <" Y "> ) = 1 |
10 |
8 9
|
oveq12i |
|- ( ( # ` <" X "> ) + ( # ` <" Y "> ) ) = ( 1 + 1 ) |
11 |
|
1p1e2 |
|- ( 1 + 1 ) = 2 |
12 |
10 11
|
eqtri |
|- ( ( # ` <" X "> ) + ( # ` <" Y "> ) ) = 2 |
13 |
8 12
|
oveq12i |
|- ( ( # ` <" X "> ) ..^ ( ( # ` <" X "> ) + ( # ` <" Y "> ) ) ) = ( 1 ..^ 2 ) |
14 |
7 13
|
eleqtrri |
|- 1 e. ( ( # ` <" X "> ) ..^ ( ( # ` <" X "> ) + ( # ` <" Y "> ) ) ) |
15 |
|
ccatval2 |
|- ( ( <" X "> e. Word _V /\ <" Y "> e. Word _V /\ 1 e. ( ( # ` <" X "> ) ..^ ( ( # ` <" X "> ) + ( # ` <" Y "> ) ) ) ) -> ( ( <" X "> ++ <" Y "> ) ` 1 ) = ( <" Y "> ` ( 1 - ( # ` <" X "> ) ) ) ) |
16 |
1 2 14 15
|
mp3an |
|- ( ( <" X "> ++ <" Y "> ) ` 1 ) = ( <" Y "> ` ( 1 - ( # ` <" X "> ) ) ) |
17 |
8
|
oveq2i |
|- ( 1 - ( # ` <" X "> ) ) = ( 1 - 1 ) |
18 |
|
1m1e0 |
|- ( 1 - 1 ) = 0 |
19 |
17 18
|
eqtri |
|- ( 1 - ( # ` <" X "> ) ) = 0 |
20 |
19
|
fveq2i |
|- ( <" Y "> ` ( 1 - ( # ` <" X "> ) ) ) = ( <" Y "> ` 0 ) |
21 |
16 20
|
eqtri |
|- ( ( <" X "> ++ <" Y "> ) ` 1 ) = ( <" Y "> ` 0 ) |
22 |
|
s1fv |
|- ( Y e. V -> ( <" Y "> ` 0 ) = Y ) |
23 |
21 22
|
eqtrid |
|- ( Y e. V -> ( ( <" X "> ++ <" Y "> ) ` 1 ) = Y ) |