Step |
Hyp |
Ref |
Expression |
1 |
|
c0ex |
|- 0 e. _V |
2 |
1
|
tpid1 |
|- 0 e. { 0 , 1 , 2 } |
3 |
|
fzo0to3tp |
|- ( 0 ..^ 3 ) = { 0 , 1 , 2 } |
4 |
2 3
|
eleqtrri |
|- 0 e. ( 0 ..^ 3 ) |
5 |
|
oveq2 |
|- ( ( # ` W ) = 3 -> ( 0 ..^ ( # ` W ) ) = ( 0 ..^ 3 ) ) |
6 |
4 5
|
eleqtrrid |
|- ( ( # ` W ) = 3 -> 0 e. ( 0 ..^ ( # ` W ) ) ) |
7 |
|
wrdsymbcl |
|- ( ( W e. Word V /\ 0 e. ( 0 ..^ ( # ` W ) ) ) -> ( W ` 0 ) e. V ) |
8 |
6 7
|
sylan2 |
|- ( ( W e. Word V /\ ( # ` W ) = 3 ) -> ( W ` 0 ) e. V ) |
9 |
|
1ex |
|- 1 e. _V |
10 |
9
|
tpid2 |
|- 1 e. { 0 , 1 , 2 } |
11 |
10 3
|
eleqtrri |
|- 1 e. ( 0 ..^ 3 ) |
12 |
11 5
|
eleqtrrid |
|- ( ( # ` W ) = 3 -> 1 e. ( 0 ..^ ( # ` W ) ) ) |
13 |
|
wrdsymbcl |
|- ( ( W e. Word V /\ 1 e. ( 0 ..^ ( # ` W ) ) ) -> ( W ` 1 ) e. V ) |
14 |
12 13
|
sylan2 |
|- ( ( W e. Word V /\ ( # ` W ) = 3 ) -> ( W ` 1 ) e. V ) |
15 |
|
2ex |
|- 2 e. _V |
16 |
15
|
tpid3 |
|- 2 e. { 0 , 1 , 2 } |
17 |
16 3
|
eleqtrri |
|- 2 e. ( 0 ..^ 3 ) |
18 |
17 5
|
eleqtrrid |
|- ( ( # ` W ) = 3 -> 2 e. ( 0 ..^ ( # ` W ) ) ) |
19 |
|
wrdsymbcl |
|- ( ( W e. Word V /\ 2 e. ( 0 ..^ ( # ` W ) ) ) -> ( W ` 2 ) e. V ) |
20 |
18 19
|
sylan2 |
|- ( ( W e. Word V /\ ( # ` W ) = 3 ) -> ( W ` 2 ) e. V ) |
21 |
|
simpr |
|- ( ( W e. Word V /\ ( # ` W ) = 3 ) -> ( # ` W ) = 3 ) |
22 |
|
eqid |
|- ( W ` 0 ) = ( W ` 0 ) |
23 |
|
eqid |
|- ( W ` 1 ) = ( W ` 1 ) |
24 |
|
eqid |
|- ( W ` 2 ) = ( W ` 2 ) |
25 |
22 23 24
|
3pm3.2i |
|- ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = ( W ` 2 ) ) |
26 |
21 25
|
jctir |
|- ( ( W e. Word V /\ ( # ` W ) = 3 ) -> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = ( W ` 2 ) ) ) ) |
27 |
|
eqeq2 |
|- ( a = ( W ` 0 ) -> ( ( W ` 0 ) = a <-> ( W ` 0 ) = ( W ` 0 ) ) ) |
28 |
27
|
3anbi1d |
|- ( a = ( W ` 0 ) -> ( ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) <-> ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) |
29 |
28
|
anbi2d |
|- ( a = ( W ` 0 ) -> ( ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) ) |
30 |
|
eqeq2 |
|- ( b = ( W ` 1 ) -> ( ( W ` 1 ) = b <-> ( W ` 1 ) = ( W ` 1 ) ) ) |
31 |
30
|
3anbi2d |
|- ( b = ( W ` 1 ) -> ( ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) <-> ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = c ) ) ) |
32 |
31
|
anbi2d |
|- ( b = ( W ` 1 ) -> ( ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = c ) ) ) ) |
33 |
|
eqeq2 |
|- ( c = ( W ` 2 ) -> ( ( W ` 2 ) = c <-> ( W ` 2 ) = ( W ` 2 ) ) ) |
34 |
33
|
3anbi3d |
|- ( c = ( W ` 2 ) -> ( ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = c ) <-> ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = ( W ` 2 ) ) ) ) |
35 |
34
|
anbi2d |
|- ( c = ( W ` 2 ) -> ( ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = c ) ) <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = ( W ` 2 ) ) ) ) ) |
36 |
29 32 35
|
rspc3ev |
|- ( ( ( ( W ` 0 ) e. V /\ ( W ` 1 ) e. V /\ ( W ` 2 ) e. V ) /\ ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = ( W ` 2 ) ) ) ) -> E. a e. V E. b e. V E. c e. V ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) |
37 |
8 14 20 26 36
|
syl31anc |
|- ( ( W e. Word V /\ ( # ` W ) = 3 ) -> E. a e. V E. b e. V E. c e. V ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) |
38 |
|
df-3an |
|- ( ( a e. V /\ b e. V /\ c e. V ) <-> ( ( a e. V /\ b e. V ) /\ c e. V ) ) |
39 |
|
eqwrds3 |
|- ( ( W e. Word V /\ ( a e. V /\ b e. V /\ c e. V ) ) -> ( W = <" a b c "> <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) ) |
40 |
39
|
ex |
|- ( W e. Word V -> ( ( a e. V /\ b e. V /\ c e. V ) -> ( W = <" a b c "> <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) ) ) |
41 |
38 40
|
syl5bir |
|- ( W e. Word V -> ( ( ( a e. V /\ b e. V ) /\ c e. V ) -> ( W = <" a b c "> <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) ) ) |
42 |
41
|
expd |
|- ( W e. Word V -> ( ( a e. V /\ b e. V ) -> ( c e. V -> ( W = <" a b c "> <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) ) ) ) |
43 |
42
|
adantr |
|- ( ( W e. Word V /\ ( # ` W ) = 3 ) -> ( ( a e. V /\ b e. V ) -> ( c e. V -> ( W = <" a b c "> <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) ) ) ) |
44 |
43
|
imp31 |
|- ( ( ( ( W e. Word V /\ ( # ` W ) = 3 ) /\ ( a e. V /\ b e. V ) ) /\ c e. V ) -> ( W = <" a b c "> <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) ) |
45 |
44
|
rexbidva |
|- ( ( ( W e. Word V /\ ( # ` W ) = 3 ) /\ ( a e. V /\ b e. V ) ) -> ( E. c e. V W = <" a b c "> <-> E. c e. V ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) ) |
46 |
45
|
2rexbidva |
|- ( ( W e. Word V /\ ( # ` W ) = 3 ) -> ( E. a e. V E. b e. V E. c e. V W = <" a b c "> <-> E. a e. V E. b e. V E. c e. V ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) ) |
47 |
37 46
|
mpbird |
|- ( ( W e. Word V /\ ( # ` W ) = 3 ) -> E. a e. V E. b e. V E. c e. V W = <" a b c "> ) |
48 |
|
s3cl |
|- ( ( a e. V /\ b e. V /\ c e. V ) -> <" a b c "> e. Word V ) |
49 |
48
|
ad4ant123 |
|- ( ( ( ( a e. V /\ b e. V ) /\ c e. V ) /\ W = <" a b c "> ) -> <" a b c "> e. Word V ) |
50 |
|
s3len |
|- ( # ` <" a b c "> ) = 3 |
51 |
49 50
|
jctir |
|- ( ( ( ( a e. V /\ b e. V ) /\ c e. V ) /\ W = <" a b c "> ) -> ( <" a b c "> e. Word V /\ ( # ` <" a b c "> ) = 3 ) ) |
52 |
|
eleq1 |
|- ( W = <" a b c "> -> ( W e. Word V <-> <" a b c "> e. Word V ) ) |
53 |
|
fveqeq2 |
|- ( W = <" a b c "> -> ( ( # ` W ) = 3 <-> ( # ` <" a b c "> ) = 3 ) ) |
54 |
52 53
|
anbi12d |
|- ( W = <" a b c "> -> ( ( W e. Word V /\ ( # ` W ) = 3 ) <-> ( <" a b c "> e. Word V /\ ( # ` <" a b c "> ) = 3 ) ) ) |
55 |
54
|
adantl |
|- ( ( ( ( a e. V /\ b e. V ) /\ c e. V ) /\ W = <" a b c "> ) -> ( ( W e. Word V /\ ( # ` W ) = 3 ) <-> ( <" a b c "> e. Word V /\ ( # ` <" a b c "> ) = 3 ) ) ) |
56 |
51 55
|
mpbird |
|- ( ( ( ( a e. V /\ b e. V ) /\ c e. V ) /\ W = <" a b c "> ) -> ( W e. Word V /\ ( # ` W ) = 3 ) ) |
57 |
56
|
rexlimdva2 |
|- ( ( a e. V /\ b e. V ) -> ( E. c e. V W = <" a b c "> -> ( W e. Word V /\ ( # ` W ) = 3 ) ) ) |
58 |
57
|
rexlimivv |
|- ( E. a e. V E. b e. V E. c e. V W = <" a b c "> -> ( W e. Word V /\ ( # ` W ) = 3 ) ) |
59 |
47 58
|
impbii |
|- ( ( W e. Word V /\ ( # ` W ) = 3 ) <-> E. a e. V E. b e. V E. c e. V W = <" a b c "> ) |