Step |
Hyp |
Ref |
Expression |
1 |
|
1le2 |
|- 1 <_ 2 |
2 |
|
breq2 |
|- ( ( # ` W ) = 2 -> ( 1 <_ ( # ` W ) <-> 1 <_ 2 ) ) |
3 |
1 2
|
mpbiri |
|- ( ( # ` W ) = 2 -> 1 <_ ( # ` W ) ) |
4 |
|
wrdsymb1 |
|- ( ( W e. Word S /\ 1 <_ ( # ` W ) ) -> ( W ` 0 ) e. S ) |
5 |
3 4
|
sylan2 |
|- ( ( W e. Word S /\ ( # ` W ) = 2 ) -> ( W ` 0 ) e. S ) |
6 |
|
lsw |
|- ( W e. Word S -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) ) |
7 |
|
oveq1 |
|- ( ( # ` W ) = 2 -> ( ( # ` W ) - 1 ) = ( 2 - 1 ) ) |
8 |
|
2m1e1 |
|- ( 2 - 1 ) = 1 |
9 |
7 8
|
eqtrdi |
|- ( ( # ` W ) = 2 -> ( ( # ` W ) - 1 ) = 1 ) |
10 |
9
|
fveq2d |
|- ( ( # ` W ) = 2 -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` 1 ) ) |
11 |
6 10
|
sylan9eq |
|- ( ( W e. Word S /\ ( # ` W ) = 2 ) -> ( lastS ` W ) = ( W ` 1 ) ) |
12 |
|
2nn |
|- 2 e. NN |
13 |
|
lswlgt0cl |
|- ( ( 2 e. NN /\ ( W e. Word S /\ ( # ` W ) = 2 ) ) -> ( lastS ` W ) e. S ) |
14 |
12 13
|
mpan |
|- ( ( W e. Word S /\ ( # ` W ) = 2 ) -> ( lastS ` W ) e. S ) |
15 |
11 14
|
eqeltrrd |
|- ( ( W e. Word S /\ ( # ` W ) = 2 ) -> ( W ` 1 ) e. S ) |
16 |
|
wrdlen2s2 |
|- ( ( W e. Word S /\ ( # ` W ) = 2 ) -> W = <" ( W ` 0 ) ( W ` 1 ) "> ) |
17 |
|
id |
|- ( s = ( W ` 0 ) -> s = ( W ` 0 ) ) |
18 |
|
eqidd |
|- ( s = ( W ` 0 ) -> t = t ) |
19 |
17 18
|
s2eqd |
|- ( s = ( W ` 0 ) -> <" s t "> = <" ( W ` 0 ) t "> ) |
20 |
19
|
eqeq2d |
|- ( s = ( W ` 0 ) -> ( W = <" s t "> <-> W = <" ( W ` 0 ) t "> ) ) |
21 |
|
eqidd |
|- ( t = ( W ` 1 ) -> ( W ` 0 ) = ( W ` 0 ) ) |
22 |
|
id |
|- ( t = ( W ` 1 ) -> t = ( W ` 1 ) ) |
23 |
21 22
|
s2eqd |
|- ( t = ( W ` 1 ) -> <" ( W ` 0 ) t "> = <" ( W ` 0 ) ( W ` 1 ) "> ) |
24 |
23
|
eqeq2d |
|- ( t = ( W ` 1 ) -> ( W = <" ( W ` 0 ) t "> <-> W = <" ( W ` 0 ) ( W ` 1 ) "> ) ) |
25 |
20 24
|
rspc2ev |
|- ( ( ( W ` 0 ) e. S /\ ( W ` 1 ) e. S /\ W = <" ( W ` 0 ) ( W ` 1 ) "> ) -> E. s e. S E. t e. S W = <" s t "> ) |
26 |
5 15 16 25
|
syl3anc |
|- ( ( W e. Word S /\ ( # ` W ) = 2 ) -> E. s e. S E. t e. S W = <" s t "> ) |