Step |
Hyp |
Ref |
Expression |
1 |
|
ccatcl |
|- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) e. Word B ) |
2 |
|
lencl |
|- ( S e. Word B -> ( # ` S ) e. NN0 ) |
3 |
|
lencl |
|- ( T e. Word B -> ( # ` T ) e. NN0 ) |
4 |
2 3
|
anim12i |
|- ( ( S e. Word B /\ T e. Word B ) -> ( ( # ` S ) e. NN0 /\ ( # ` T ) e. NN0 ) ) |
5 |
|
nn0fz0 |
|- ( ( # ` S ) e. NN0 <-> ( # ` S ) e. ( 0 ... ( # ` S ) ) ) |
6 |
2 5
|
sylib |
|- ( S e. Word B -> ( # ` S ) e. ( 0 ... ( # ` S ) ) ) |
7 |
6
|
adantr |
|- ( ( S e. Word B /\ T e. Word B ) -> ( # ` S ) e. ( 0 ... ( # ` S ) ) ) |
8 |
|
elfz0add |
|- ( ( ( # ` S ) e. NN0 /\ ( # ` T ) e. NN0 ) -> ( ( # ` S ) e. ( 0 ... ( # ` S ) ) -> ( # ` S ) e. ( 0 ... ( ( # ` S ) + ( # ` T ) ) ) ) ) |
9 |
4 7 8
|
sylc |
|- ( ( S e. Word B /\ T e. Word B ) -> ( # ` S ) e. ( 0 ... ( ( # ` S ) + ( # ` T ) ) ) ) |
10 |
|
ccatlen |
|- ( ( S e. Word B /\ T e. Word B ) -> ( # ` ( S ++ T ) ) = ( ( # ` S ) + ( # ` T ) ) ) |
11 |
10
|
oveq2d |
|- ( ( S e. Word B /\ T e. Word B ) -> ( 0 ... ( # ` ( S ++ T ) ) ) = ( 0 ... ( ( # ` S ) + ( # ` T ) ) ) ) |
12 |
9 11
|
eleqtrrd |
|- ( ( S e. Word B /\ T e. Word B ) -> ( # ` S ) e. ( 0 ... ( # ` ( S ++ T ) ) ) ) |
13 |
|
pfxres |
|- ( ( ( S ++ T ) e. Word B /\ ( # ` S ) e. ( 0 ... ( # ` ( S ++ T ) ) ) ) -> ( ( S ++ T ) prefix ( # ` S ) ) = ( ( S ++ T ) |` ( 0 ..^ ( # ` S ) ) ) ) |
14 |
1 12 13
|
syl2anc |
|- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) prefix ( # ` S ) ) = ( ( S ++ T ) |` ( 0 ..^ ( # ` S ) ) ) ) |
15 |
|
ccatvalfn |
|- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) Fn ( 0 ..^ ( ( # ` S ) + ( # ` T ) ) ) ) |
16 |
2
|
nn0zd |
|- ( S e. Word B -> ( # ` S ) e. ZZ ) |
17 |
16
|
uzidd |
|- ( S e. Word B -> ( # ` S ) e. ( ZZ>= ` ( # ` S ) ) ) |
18 |
|
uzaddcl |
|- ( ( ( # ` S ) e. ( ZZ>= ` ( # ` S ) ) /\ ( # ` T ) e. NN0 ) -> ( ( # ` S ) + ( # ` T ) ) e. ( ZZ>= ` ( # ` S ) ) ) |
19 |
17 3 18
|
syl2an |
|- ( ( S e. Word B /\ T e. Word B ) -> ( ( # ` S ) + ( # ` T ) ) e. ( ZZ>= ` ( # ` S ) ) ) |
20 |
|
fzoss2 |
|- ( ( ( # ` S ) + ( # ` T ) ) e. ( ZZ>= ` ( # ` S ) ) -> ( 0 ..^ ( # ` S ) ) C_ ( 0 ..^ ( ( # ` S ) + ( # ` T ) ) ) ) |
21 |
19 20
|
syl |
|- ( ( S e. Word B /\ T e. Word B ) -> ( 0 ..^ ( # ` S ) ) C_ ( 0 ..^ ( ( # ` S ) + ( # ` T ) ) ) ) |
22 |
15 21
|
fnssresd |
|- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) |` ( 0 ..^ ( # ` S ) ) ) Fn ( 0 ..^ ( # ` S ) ) ) |
23 |
|
wrdfn |
|- ( S e. Word B -> S Fn ( 0 ..^ ( # ` S ) ) ) |
24 |
23
|
adantr |
|- ( ( S e. Word B /\ T e. Word B ) -> S Fn ( 0 ..^ ( # ` S ) ) ) |
25 |
|
fvres |
|- ( k e. ( 0 ..^ ( # ` S ) ) -> ( ( ( S ++ T ) |` ( 0 ..^ ( # ` S ) ) ) ` k ) = ( ( S ++ T ) ` k ) ) |
26 |
25
|
adantl |
|- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0 ..^ ( # ` S ) ) ) -> ( ( ( S ++ T ) |` ( 0 ..^ ( # ` S ) ) ) ` k ) = ( ( S ++ T ) ` k ) ) |
27 |
|
ccatval1 |
|- ( ( S e. Word B /\ T e. Word B /\ k e. ( 0 ..^ ( # ` S ) ) ) -> ( ( S ++ T ) ` k ) = ( S ` k ) ) |
28 |
27
|
3expa |
|- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0 ..^ ( # ` S ) ) ) -> ( ( S ++ T ) ` k ) = ( S ` k ) ) |
29 |
26 28
|
eqtrd |
|- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0 ..^ ( # ` S ) ) ) -> ( ( ( S ++ T ) |` ( 0 ..^ ( # ` S ) ) ) ` k ) = ( S ` k ) ) |
30 |
22 24 29
|
eqfnfvd |
|- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) |` ( 0 ..^ ( # ` S ) ) ) = S ) |
31 |
14 30
|
eqtrd |
|- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) prefix ( # ` S ) ) = S ) |