Step |
Hyp |
Ref |
Expression |
1 |
|
ccatfval |
|- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) = ( x e. ( 0 ..^ ( ( # ` S ) + ( # ` T ) ) ) |-> if ( x e. ( 0 ..^ ( # ` S ) ) , ( S ` x ) , ( T ` ( x - ( # ` S ) ) ) ) ) ) |
2 |
1
|
3adant3 |
|- ( ( S e. Word B /\ T e. Word B /\ I e. ( 0 ..^ ( # ` S ) ) ) -> ( S ++ T ) = ( x e. ( 0 ..^ ( ( # ` S ) + ( # ` T ) ) ) |-> if ( x e. ( 0 ..^ ( # ` S ) ) , ( S ` x ) , ( T ` ( x - ( # ` S ) ) ) ) ) ) |
3 |
|
eleq1 |
|- ( x = I -> ( x e. ( 0 ..^ ( # ` S ) ) <-> I e. ( 0 ..^ ( # ` S ) ) ) ) |
4 |
|
fveq2 |
|- ( x = I -> ( S ` x ) = ( S ` I ) ) |
5 |
|
fvoveq1 |
|- ( x = I -> ( T ` ( x - ( # ` S ) ) ) = ( T ` ( I - ( # ` S ) ) ) ) |
6 |
3 4 5
|
ifbieq12d |
|- ( x = I -> if ( x e. ( 0 ..^ ( # ` S ) ) , ( S ` x ) , ( T ` ( x - ( # ` S ) ) ) ) = if ( I e. ( 0 ..^ ( # ` S ) ) , ( S ` I ) , ( T ` ( I - ( # ` S ) ) ) ) ) |
7 |
|
iftrue |
|- ( I e. ( 0 ..^ ( # ` S ) ) -> if ( I e. ( 0 ..^ ( # ` S ) ) , ( S ` I ) , ( T ` ( I - ( # ` S ) ) ) ) = ( S ` I ) ) |
8 |
7
|
3ad2ant3 |
|- ( ( S e. Word B /\ T e. Word B /\ I e. ( 0 ..^ ( # ` S ) ) ) -> if ( I e. ( 0 ..^ ( # ` S ) ) , ( S ` I ) , ( T ` ( I - ( # ` S ) ) ) ) = ( S ` I ) ) |
9 |
6 8
|
sylan9eqr |
|- ( ( ( S e. Word B /\ T e. Word B /\ I e. ( 0 ..^ ( # ` S ) ) ) /\ x = I ) -> if ( x e. ( 0 ..^ ( # ` S ) ) , ( S ` x ) , ( T ` ( x - ( # ` S ) ) ) ) = ( S ` I ) ) |
10 |
|
simp3 |
|- ( ( S e. Word B /\ T e. Word B /\ I e. ( 0 ..^ ( # ` S ) ) ) -> I e. ( 0 ..^ ( # ` S ) ) ) |
11 |
|
lencl |
|- ( T e. Word B -> ( # ` T ) e. NN0 ) |
12 |
11
|
3ad2ant2 |
|- ( ( S e. Word B /\ T e. Word B /\ I e. ( 0 ..^ ( # ` S ) ) ) -> ( # ` T ) e. NN0 ) |
13 |
|
elfzoext |
|- ( ( I e. ( 0 ..^ ( # ` S ) ) /\ ( # ` T ) e. NN0 ) -> I e. ( 0 ..^ ( ( # ` S ) + ( # ` T ) ) ) ) |
14 |
10 12 13
|
syl2anc |
|- ( ( S e. Word B /\ T e. Word B /\ I e. ( 0 ..^ ( # ` S ) ) ) -> I e. ( 0 ..^ ( ( # ` S ) + ( # ` T ) ) ) ) |
15 |
|
fvexd |
|- ( ( S e. Word B /\ T e. Word B /\ I e. ( 0 ..^ ( # ` S ) ) ) -> ( S ` I ) e. _V ) |
16 |
2 9 14 15
|
fvmptd |
|- ( ( S e. Word B /\ T e. Word B /\ I e. ( 0 ..^ ( # ` S ) ) ) -> ( ( S ++ T ) ` I ) = ( S ` I ) ) |