Step |
Hyp |
Ref |
Expression |
1 |
|
simp1 |
|- ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> W e. Word V ) |
2 |
1
|
anim1i |
|- ( ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( W e. Word V /\ ( X e. V /\ Y e. V ) ) ) |
3 |
|
3anass |
|- ( ( W e. Word V /\ X e. V /\ Y e. V ) <-> ( W e. Word V /\ ( X e. V /\ Y e. V ) ) ) |
4 |
2 3
|
sylibr |
|- ( ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( W e. Word V /\ X e. V /\ Y e. V ) ) |
5 |
|
ccatw2s1ccatws2OLD |
|- ( ( W e. Word V /\ X e. V /\ Y e. V ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) = ( W ++ <" X Y "> ) ) |
6 |
5
|
fveq1d |
|- ( ( W e. Word V /\ X e. V /\ Y e. V ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( ( W ++ <" X Y "> ) ` I ) ) |
7 |
4 6
|
syl |
|- ( ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( ( W ++ <" X Y "> ) ` I ) ) |
8 |
1
|
adantr |
|- ( ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> W e. Word V ) |
9 |
|
s2cl |
|- ( ( X e. V /\ Y e. V ) -> <" X Y "> e. Word V ) |
10 |
9
|
adantl |
|- ( ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> <" X Y "> e. Word V ) |
11 |
|
simp2 |
|- ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> I e. NN0 ) |
12 |
|
lencl |
|- ( W e. Word V -> ( # ` W ) e. NN0 ) |
13 |
12
|
nn0zd |
|- ( W e. Word V -> ( # ` W ) e. ZZ ) |
14 |
13
|
3ad2ant1 |
|- ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> ( # ` W ) e. ZZ ) |
15 |
|
simp3 |
|- ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> I < ( # ` W ) ) |
16 |
|
elfzo0z |
|- ( I e. ( 0 ..^ ( # ` W ) ) <-> ( I e. NN0 /\ ( # ` W ) e. ZZ /\ I < ( # ` W ) ) ) |
17 |
11 14 15 16
|
syl3anbrc |
|- ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> I e. ( 0 ..^ ( # ` W ) ) ) |
18 |
17
|
adantr |
|- ( ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> I e. ( 0 ..^ ( # ` W ) ) ) |
19 |
|
ccatval1 |
|- ( ( W e. Word V /\ <" X Y "> e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) -> ( ( W ++ <" X Y "> ) ` I ) = ( W ` I ) ) |
20 |
8 10 18 19
|
syl3anc |
|- ( ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( W ++ <" X Y "> ) ` I ) = ( W ` I ) ) |
21 |
7 20
|
eqtrd |
|- ( ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( W ` I ) ) |