Step |
Hyp |
Ref |
Expression |
1 |
|
eqwrd |
|- ( ( W e. Word V /\ S e. Word V ) -> ( W = S <-> ( ( # ` W ) = ( # ` S ) /\ A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) ) ) ) |
2 |
1
|
3adant3 |
|- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) -> ( W = S <-> ( ( # ` W ) = ( # ` S ) /\ A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) ) ) ) |
3 |
|
elfzofz |
|- ( I e. ( 0 ..^ ( # ` W ) ) -> I e. ( 0 ... ( # ` W ) ) ) |
4 |
|
fzosplit |
|- ( I e. ( 0 ... ( # ` W ) ) -> ( 0 ..^ ( # ` W ) ) = ( ( 0 ..^ I ) u. ( I ..^ ( # ` W ) ) ) ) |
5 |
3 4
|
syl |
|- ( I e. ( 0 ..^ ( # ` W ) ) -> ( 0 ..^ ( # ` W ) ) = ( ( 0 ..^ I ) u. ( I ..^ ( # ` W ) ) ) ) |
6 |
5
|
3ad2ant3 |
|- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) -> ( 0 ..^ ( # ` W ) ) = ( ( 0 ..^ I ) u. ( I ..^ ( # ` W ) ) ) ) |
7 |
6
|
adantr |
|- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( 0 ..^ ( # ` W ) ) = ( ( 0 ..^ I ) u. ( I ..^ ( # ` W ) ) ) ) |
8 |
7
|
raleqdv |
|- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) <-> A. i e. ( ( 0 ..^ I ) u. ( I ..^ ( # ` W ) ) ) ( W ` i ) = ( S ` i ) ) ) |
9 |
|
ralunb |
|- ( A. i e. ( ( 0 ..^ I ) u. ( I ..^ ( # ` W ) ) ) ( W ` i ) = ( S ` i ) <-> ( A. i e. ( 0 ..^ I ) ( W ` i ) = ( S ` i ) /\ A. i e. ( I ..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) ) ) |
10 |
8 9
|
bitrdi |
|- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) <-> ( A. i e. ( 0 ..^ I ) ( W ` i ) = ( S ` i ) /\ A. i e. ( I ..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) ) ) ) |
11 |
|
eqidd |
|- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> I = I ) |
12 |
|
3simpa |
|- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) -> ( W e. Word V /\ S e. Word V ) ) |
13 |
12
|
adantr |
|- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( W e. Word V /\ S e. Word V ) ) |
14 |
|
elfzonn0 |
|- ( I e. ( 0 ..^ ( # ` W ) ) -> I e. NN0 ) |
15 |
14 14
|
jca |
|- ( I e. ( 0 ..^ ( # ` W ) ) -> ( I e. NN0 /\ I e. NN0 ) ) |
16 |
15
|
3ad2ant3 |
|- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) -> ( I e. NN0 /\ I e. NN0 ) ) |
17 |
16
|
adantr |
|- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( I e. NN0 /\ I e. NN0 ) ) |
18 |
|
elfzo0le |
|- ( I e. ( 0 ..^ ( # ` W ) ) -> I <_ ( # ` W ) ) |
19 |
18
|
3ad2ant3 |
|- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) -> I <_ ( # ` W ) ) |
20 |
19
|
adantr |
|- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> I <_ ( # ` W ) ) |
21 |
|
breq2 |
|- ( ( # ` W ) = ( # ` S ) -> ( I <_ ( # ` W ) <-> I <_ ( # ` S ) ) ) |
22 |
21
|
adantl |
|- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( I <_ ( # ` W ) <-> I <_ ( # ` S ) ) ) |
23 |
20 22
|
mpbid |
|- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> I <_ ( # ` S ) ) |
24 |
|
pfxeq |
|- ( ( ( W e. Word V /\ S e. Word V ) /\ ( I e. NN0 /\ I e. NN0 ) /\ ( I <_ ( # ` W ) /\ I <_ ( # ` S ) ) ) -> ( ( W prefix I ) = ( S prefix I ) <-> ( I = I /\ A. i e. ( 0 ..^ I ) ( W ` i ) = ( S ` i ) ) ) ) |
25 |
13 17 20 23 24
|
syl112anc |
|- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( ( W prefix I ) = ( S prefix I ) <-> ( I = I /\ A. i e. ( 0 ..^ I ) ( W ` i ) = ( S ` i ) ) ) ) |
26 |
11 25
|
mpbirand |
|- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( ( W prefix I ) = ( S prefix I ) <-> A. i e. ( 0 ..^ I ) ( W ` i ) = ( S ` i ) ) ) |
27 |
|
lencl |
|- ( W e. Word V -> ( # ` W ) e. NN0 ) |
28 |
27 14
|
anim12ci |
|- ( ( W e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) -> ( I e. NN0 /\ ( # ` W ) e. NN0 ) ) |
29 |
28
|
3adant2 |
|- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) -> ( I e. NN0 /\ ( # ` W ) e. NN0 ) ) |
30 |
29
|
adantr |
|- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( I e. NN0 /\ ( # ` W ) e. NN0 ) ) |
31 |
27
|
nn0red |
|- ( W e. Word V -> ( # ` W ) e. RR ) |
32 |
31
|
leidd |
|- ( W e. Word V -> ( # ` W ) <_ ( # ` W ) ) |
33 |
32
|
adantr |
|- ( ( W e. Word V /\ ( # ` W ) = ( # ` S ) ) -> ( # ` W ) <_ ( # ` W ) ) |
34 |
|
eqle |
|- ( ( ( # ` W ) e. RR /\ ( # ` W ) = ( # ` S ) ) -> ( # ` W ) <_ ( # ` S ) ) |
35 |
31 34
|
sylan |
|- ( ( W e. Word V /\ ( # ` W ) = ( # ` S ) ) -> ( # ` W ) <_ ( # ` S ) ) |
36 |
33 35
|
jca |
|- ( ( W e. Word V /\ ( # ` W ) = ( # ` S ) ) -> ( ( # ` W ) <_ ( # ` W ) /\ ( # ` W ) <_ ( # ` S ) ) ) |
37 |
36
|
3ad2antl1 |
|- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( ( # ` W ) <_ ( # ` W ) /\ ( # ` W ) <_ ( # ` S ) ) ) |
38 |
|
swrdspsleq |
|- ( ( ( W e. Word V /\ S e. Word V ) /\ ( I e. NN0 /\ ( # ` W ) e. NN0 ) /\ ( ( # ` W ) <_ ( # ` W ) /\ ( # ` W ) <_ ( # ` S ) ) ) -> ( ( W substr <. I , ( # ` W ) >. ) = ( S substr <. I , ( # ` W ) >. ) <-> A. i e. ( I ..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) ) ) |
39 |
13 30 37 38
|
syl3anc |
|- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( ( W substr <. I , ( # ` W ) >. ) = ( S substr <. I , ( # ` W ) >. ) <-> A. i e. ( I ..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) ) ) |
40 |
26 39
|
anbi12d |
|- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( ( ( W prefix I ) = ( S prefix I ) /\ ( W substr <. I , ( # ` W ) >. ) = ( S substr <. I , ( # ` W ) >. ) ) <-> ( A. i e. ( 0 ..^ I ) ( W ` i ) = ( S ` i ) /\ A. i e. ( I ..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) ) ) ) |
41 |
10 40
|
bitr4d |
|- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) <-> ( ( W prefix I ) = ( S prefix I ) /\ ( W substr <. I , ( # ` W ) >. ) = ( S substr <. I , ( # ` W ) >. ) ) ) ) |
42 |
41
|
pm5.32da |
|- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) -> ( ( ( # ` W ) = ( # ` S ) /\ A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) ) <-> ( ( # ` W ) = ( # ` S ) /\ ( ( W prefix I ) = ( S prefix I ) /\ ( W substr <. I , ( # ` W ) >. ) = ( S substr <. I , ( # ` W ) >. ) ) ) ) ) |
43 |
2 42
|
bitrd |
|- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0 ..^ ( # ` W ) ) ) -> ( W = S <-> ( ( # ` W ) = ( # ` S ) /\ ( ( W prefix I ) = ( S prefix I ) /\ ( W substr <. I , ( # ` W ) >. ) = ( S substr <. I , ( # ` W ) >. ) ) ) ) ) |