Step |
Hyp |
Ref |
Expression |
1 |
|
swrdcl |
|- ( W e. Word S -> ( W substr <. N , ( # ` W ) >. ) e. Word S ) |
2 |
|
simpr |
|- ( ( W e. Word S /\ N e. ( 0 ... ( # ` W ) ) ) -> N e. ( 0 ... ( # ` W ) ) ) |
3 |
|
elfzuz2 |
|- ( N e. ( 0 ... ( # ` W ) ) -> ( # ` W ) e. ( ZZ>= ` 0 ) ) |
4 |
|
eluzfz2 |
|- ( ( # ` W ) e. ( ZZ>= ` 0 ) -> ( # ` W ) e. ( 0 ... ( # ` W ) ) ) |
5 |
2 3 4
|
3syl |
|- ( ( W e. Word S /\ N e. ( 0 ... ( # ` W ) ) ) -> ( # ` W ) e. ( 0 ... ( # ` W ) ) ) |
6 |
|
ccatpfx |
|- ( ( W e. Word S /\ N e. ( 0 ... ( # ` W ) ) /\ ( # ` W ) e. ( 0 ... ( # ` W ) ) ) -> ( ( W prefix N ) ++ ( W substr <. N , ( # ` W ) >. ) ) = ( W prefix ( # ` W ) ) ) |
7 |
5 6
|
mpd3an3 |
|- ( ( W e. Word S /\ N e. ( 0 ... ( # ` W ) ) ) -> ( ( W prefix N ) ++ ( W substr <. N , ( # ` W ) >. ) ) = ( W prefix ( # ` W ) ) ) |
8 |
|
pfxres |
|- ( ( W e. Word S /\ N e. ( 0 ... ( # ` W ) ) ) -> ( W prefix N ) = ( W |` ( 0 ..^ N ) ) ) |
9 |
8
|
oveq1d |
|- ( ( W e. Word S /\ N e. ( 0 ... ( # ` W ) ) ) -> ( ( W prefix N ) ++ ( W substr <. N , ( # ` W ) >. ) ) = ( ( W |` ( 0 ..^ N ) ) ++ ( W substr <. N , ( # ` W ) >. ) ) ) |
10 |
|
pfxid |
|- ( W e. Word S -> ( W prefix ( # ` W ) ) = W ) |
11 |
10
|
adantr |
|- ( ( W e. Word S /\ N e. ( 0 ... ( # ` W ) ) ) -> ( W prefix ( # ` W ) ) = W ) |
12 |
7 9 11
|
3eqtr3rd |
|- ( ( W e. Word S /\ N e. ( 0 ... ( # ` W ) ) ) -> W = ( ( W |` ( 0 ..^ N ) ) ++ ( W substr <. N , ( # ` W ) >. ) ) ) |
13 |
|
oveq2 |
|- ( v = ( W substr <. N , ( # ` W ) >. ) -> ( ( W |` ( 0 ..^ N ) ) ++ v ) = ( ( W |` ( 0 ..^ N ) ) ++ ( W substr <. N , ( # ` W ) >. ) ) ) |
14 |
13
|
rspceeqv |
|- ( ( ( W substr <. N , ( # ` W ) >. ) e. Word S /\ W = ( ( W |` ( 0 ..^ N ) ) ++ ( W substr <. N , ( # ` W ) >. ) ) ) -> E. v e. Word S W = ( ( W |` ( 0 ..^ N ) ) ++ v ) ) |
15 |
1 12 14
|
syl2an2r |
|- ( ( W e. Word S /\ N e. ( 0 ... ( # ` W ) ) ) -> E. v e. Word S W = ( ( W |` ( 0 ..^ N ) ) ++ v ) ) |