| Step |
Hyp |
Ref |
Expression |
| 1 |
|
3wlkd.p |
|- P = <" A B C D "> |
| 2 |
|
3wlkd.f |
|- F = <" J K L "> |
| 3 |
|
3wlkd.s |
|- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) ) |
| 4 |
|
3wlkd.n |
|- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) |
| 5 |
|
3wlkd.e |
|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) ) |
| 6 |
1 2 3 4 5
|
3wlkdlem7 |
|- ( ph -> ( J e. _V /\ K e. _V /\ L e. _V ) ) |
| 7 |
|
s3fv0 |
|- ( J e. _V -> ( <" J K L "> ` 0 ) = J ) |
| 8 |
|
s3fv1 |
|- ( K e. _V -> ( <" J K L "> ` 1 ) = K ) |
| 9 |
|
s3fv2 |
|- ( L e. _V -> ( <" J K L "> ` 2 ) = L ) |
| 10 |
7 8 9
|
3anim123i |
|- ( ( J e. _V /\ K e. _V /\ L e. _V ) -> ( ( <" J K L "> ` 0 ) = J /\ ( <" J K L "> ` 1 ) = K /\ ( <" J K L "> ` 2 ) = L ) ) |
| 11 |
6 10
|
syl |
|- ( ph -> ( ( <" J K L "> ` 0 ) = J /\ ( <" J K L "> ` 1 ) = K /\ ( <" J K L "> ` 2 ) = L ) ) |
| 12 |
2
|
fveq1i |
|- ( F ` 0 ) = ( <" J K L "> ` 0 ) |
| 13 |
12
|
eqeq1i |
|- ( ( F ` 0 ) = J <-> ( <" J K L "> ` 0 ) = J ) |
| 14 |
2
|
fveq1i |
|- ( F ` 1 ) = ( <" J K L "> ` 1 ) |
| 15 |
14
|
eqeq1i |
|- ( ( F ` 1 ) = K <-> ( <" J K L "> ` 1 ) = K ) |
| 16 |
2
|
fveq1i |
|- ( F ` 2 ) = ( <" J K L "> ` 2 ) |
| 17 |
16
|
eqeq1i |
|- ( ( F ` 2 ) = L <-> ( <" J K L "> ` 2 ) = L ) |
| 18 |
13 15 17
|
3anbi123i |
|- ( ( ( F ` 0 ) = J /\ ( F ` 1 ) = K /\ ( F ` 2 ) = L ) <-> ( ( <" J K L "> ` 0 ) = J /\ ( <" J K L "> ` 1 ) = K /\ ( <" J K L "> ` 2 ) = L ) ) |
| 19 |
11 18
|
sylibr |
|- ( ph -> ( ( F ` 0 ) = J /\ ( F ` 1 ) = K /\ ( F ` 2 ) = L ) ) |