Step |
Hyp |
Ref |
Expression |
1 |
|
df-s3 |
|- <" A B C "> = ( <" A B "> ++ <" C "> ) |
2 |
1
|
a1i |
|- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( D e. V /\ E e. V /\ F e. V ) ) -> <" A B C "> = ( <" A B "> ++ <" C "> ) ) |
3 |
|
df-s3 |
|- <" D E F "> = ( <" D E "> ++ <" F "> ) |
4 |
3
|
a1i |
|- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( D e. V /\ E e. V /\ F e. V ) ) -> <" D E F "> = ( <" D E "> ++ <" F "> ) ) |
5 |
2 4
|
eqeq12d |
|- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( D e. V /\ E e. V /\ F e. V ) ) -> ( <" A B C "> = <" D E F "> <-> ( <" A B "> ++ <" C "> ) = ( <" D E "> ++ <" F "> ) ) ) |
6 |
|
s2cl |
|- ( ( A e. V /\ B e. V ) -> <" A B "> e. Word V ) |
7 |
|
s1cl |
|- ( C e. V -> <" C "> e. Word V ) |
8 |
6 7
|
anim12i |
|- ( ( ( A e. V /\ B e. V ) /\ C e. V ) -> ( <" A B "> e. Word V /\ <" C "> e. Word V ) ) |
9 |
8
|
3impa |
|- ( ( A e. V /\ B e. V /\ C e. V ) -> ( <" A B "> e. Word V /\ <" C "> e. Word V ) ) |
10 |
9
|
adantr |
|- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( D e. V /\ E e. V /\ F e. V ) ) -> ( <" A B "> e. Word V /\ <" C "> e. Word V ) ) |
11 |
|
s2cl |
|- ( ( D e. V /\ E e. V ) -> <" D E "> e. Word V ) |
12 |
|
s1cl |
|- ( F e. V -> <" F "> e. Word V ) |
13 |
11 12
|
anim12i |
|- ( ( ( D e. V /\ E e. V ) /\ F e. V ) -> ( <" D E "> e. Word V /\ <" F "> e. Word V ) ) |
14 |
13
|
3impa |
|- ( ( D e. V /\ E e. V /\ F e. V ) -> ( <" D E "> e. Word V /\ <" F "> e. Word V ) ) |
15 |
14
|
adantl |
|- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( D e. V /\ E e. V /\ F e. V ) ) -> ( <" D E "> e. Word V /\ <" F "> e. Word V ) ) |
16 |
|
s2len |
|- ( # ` <" A B "> ) = 2 |
17 |
|
s2len |
|- ( # ` <" D E "> ) = 2 |
18 |
16 17
|
eqtr4i |
|- ( # ` <" A B "> ) = ( # ` <" D E "> ) |
19 |
18
|
a1i |
|- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( D e. V /\ E e. V /\ F e. V ) ) -> ( # ` <" A B "> ) = ( # ` <" D E "> ) ) |
20 |
|
ccatopth |
|- ( ( ( <" A B "> e. Word V /\ <" C "> e. Word V ) /\ ( <" D E "> e. Word V /\ <" F "> e. Word V ) /\ ( # ` <" A B "> ) = ( # ` <" D E "> ) ) -> ( ( <" A B "> ++ <" C "> ) = ( <" D E "> ++ <" F "> ) <-> ( <" A B "> = <" D E "> /\ <" C "> = <" F "> ) ) ) |
21 |
10 15 19 20
|
syl3anc |
|- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( D e. V /\ E e. V /\ F e. V ) ) -> ( ( <" A B "> ++ <" C "> ) = ( <" D E "> ++ <" F "> ) <-> ( <" A B "> = <" D E "> /\ <" C "> = <" F "> ) ) ) |
22 |
5 21
|
bitrd |
|- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( D e. V /\ E e. V /\ F e. V ) ) -> ( <" A B C "> = <" D E F "> <-> ( <" A B "> = <" D E "> /\ <" C "> = <" F "> ) ) ) |