Step |
Hyp |
Ref |
Expression |
1 |
|
efgval.w |
|- W = ( _I ` Word ( I X. 2o ) ) |
2 |
|
efgval.r |
|- .~ = ( ~FG ` I ) |
3 |
|
efgval2.m |
|- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) |
4 |
|
efgval2.t |
|- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) ) |
5 |
|
efgred.d |
|- D = ( W \ U_ x e. W ran ( T ` x ) ) |
6 |
|
efgred.s |
|- S = ( m e. { t e. ( Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1 ..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } |-> ( m ` ( ( # ` m ) - 1 ) ) ) |
7 |
1 2 3 4 5 6
|
efgsdm |
|- ( F e. dom S <-> ( F e. ( Word W \ { (/) } ) /\ ( F ` 0 ) e. D /\ A. a e. ( 1 ..^ ( # ` F ) ) ( F ` a ) e. ran ( T ` ( F ` ( a - 1 ) ) ) ) ) |
8 |
7
|
simp1bi |
|- ( F e. dom S -> F e. ( Word W \ { (/) } ) ) |
9 |
|
eldifsn |
|- ( F e. ( Word W \ { (/) } ) <-> ( F e. Word W /\ F =/= (/) ) ) |
10 |
|
lennncl |
|- ( ( F e. Word W /\ F =/= (/) ) -> ( # ` F ) e. NN ) |
11 |
9 10
|
sylbi |
|- ( F e. ( Word W \ { (/) } ) -> ( # ` F ) e. NN ) |
12 |
|
fzo0end |
|- ( ( # ` F ) e. NN -> ( ( # ` F ) - 1 ) e. ( 0 ..^ ( # ` F ) ) ) |
13 |
8 11 12
|
3syl |
|- ( F e. dom S -> ( ( # ` F ) - 1 ) e. ( 0 ..^ ( # ` F ) ) ) |
14 |
|
nnm1nn0 |
|- ( ( # ` F ) e. NN -> ( ( # ` F ) - 1 ) e. NN0 ) |
15 |
8 11 14
|
3syl |
|- ( F e. dom S -> ( ( # ` F ) - 1 ) e. NN0 ) |
16 |
|
eleq1 |
|- ( a = 0 -> ( a e. ( 0 ..^ ( # ` F ) ) <-> 0 e. ( 0 ..^ ( # ` F ) ) ) ) |
17 |
|
fveq2 |
|- ( a = 0 -> ( F ` a ) = ( F ` 0 ) ) |
18 |
17
|
breq2d |
|- ( a = 0 -> ( ( F ` 0 ) .~ ( F ` a ) <-> ( F ` 0 ) .~ ( F ` 0 ) ) ) |
19 |
16 18
|
imbi12d |
|- ( a = 0 -> ( ( a e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` a ) ) <-> ( 0 e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` 0 ) ) ) ) |
20 |
19
|
imbi2d |
|- ( a = 0 -> ( ( F e. dom S -> ( a e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` a ) ) ) <-> ( F e. dom S -> ( 0 e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` 0 ) ) ) ) ) |
21 |
|
eleq1 |
|- ( a = i -> ( a e. ( 0 ..^ ( # ` F ) ) <-> i e. ( 0 ..^ ( # ` F ) ) ) ) |
22 |
|
fveq2 |
|- ( a = i -> ( F ` a ) = ( F ` i ) ) |
23 |
22
|
breq2d |
|- ( a = i -> ( ( F ` 0 ) .~ ( F ` a ) <-> ( F ` 0 ) .~ ( F ` i ) ) ) |
24 |
21 23
|
imbi12d |
|- ( a = i -> ( ( a e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` a ) ) <-> ( i e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` i ) ) ) ) |
25 |
24
|
imbi2d |
|- ( a = i -> ( ( F e. dom S -> ( a e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` a ) ) ) <-> ( F e. dom S -> ( i e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` i ) ) ) ) ) |
26 |
|
eleq1 |
|- ( a = ( i + 1 ) -> ( a e. ( 0 ..^ ( # ` F ) ) <-> ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) ) |
27 |
|
fveq2 |
|- ( a = ( i + 1 ) -> ( F ` a ) = ( F ` ( i + 1 ) ) ) |
28 |
27
|
breq2d |
|- ( a = ( i + 1 ) -> ( ( F ` 0 ) .~ ( F ` a ) <-> ( F ` 0 ) .~ ( F ` ( i + 1 ) ) ) ) |
29 |
26 28
|
imbi12d |
|- ( a = ( i + 1 ) -> ( ( a e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` a ) ) <-> ( ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` ( i + 1 ) ) ) ) ) |
30 |
29
|
imbi2d |
|- ( a = ( i + 1 ) -> ( ( F e. dom S -> ( a e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` a ) ) ) <-> ( F e. dom S -> ( ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` ( i + 1 ) ) ) ) ) ) |
31 |
|
eleq1 |
|- ( a = ( ( # ` F ) - 1 ) -> ( a e. ( 0 ..^ ( # ` F ) ) <-> ( ( # ` F ) - 1 ) e. ( 0 ..^ ( # ` F ) ) ) ) |
32 |
|
fveq2 |
|- ( a = ( ( # ` F ) - 1 ) -> ( F ` a ) = ( F ` ( ( # ` F ) - 1 ) ) ) |
33 |
32
|
breq2d |
|- ( a = ( ( # ` F ) - 1 ) -> ( ( F ` 0 ) .~ ( F ` a ) <-> ( F ` 0 ) .~ ( F ` ( ( # ` F ) - 1 ) ) ) ) |
34 |
31 33
|
imbi12d |
|- ( a = ( ( # ` F ) - 1 ) -> ( ( a e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` a ) ) <-> ( ( ( # ` F ) - 1 ) e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` ( ( # ` F ) - 1 ) ) ) ) ) |
35 |
34
|
imbi2d |
|- ( a = ( ( # ` F ) - 1 ) -> ( ( F e. dom S -> ( a e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` a ) ) ) <-> ( F e. dom S -> ( ( ( # ` F ) - 1 ) e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` ( ( # ` F ) - 1 ) ) ) ) ) ) |
36 |
1 2
|
efger |
|- .~ Er W |
37 |
36
|
a1i |
|- ( ( F e. dom S /\ 0 e. ( 0 ..^ ( # ` F ) ) ) -> .~ Er W ) |
38 |
|
eldifi |
|- ( F e. ( Word W \ { (/) } ) -> F e. Word W ) |
39 |
|
wrdf |
|- ( F e. Word W -> F : ( 0 ..^ ( # ` F ) ) --> W ) |
40 |
8 38 39
|
3syl |
|- ( F e. dom S -> F : ( 0 ..^ ( # ` F ) ) --> W ) |
41 |
40
|
ffvelrnda |
|- ( ( F e. dom S /\ 0 e. ( 0 ..^ ( # ` F ) ) ) -> ( F ` 0 ) e. W ) |
42 |
37 41
|
erref |
|- ( ( F e. dom S /\ 0 e. ( 0 ..^ ( # ` F ) ) ) -> ( F ` 0 ) .~ ( F ` 0 ) ) |
43 |
42
|
ex |
|- ( F e. dom S -> ( 0 e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` 0 ) ) ) |
44 |
|
elnn0uz |
|- ( i e. NN0 <-> i e. ( ZZ>= ` 0 ) ) |
45 |
|
peano2fzor |
|- ( ( i e. ( ZZ>= ` 0 ) /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> i e. ( 0 ..^ ( # ` F ) ) ) |
46 |
44 45
|
sylanb |
|- ( ( i e. NN0 /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> i e. ( 0 ..^ ( # ` F ) ) ) |
47 |
46
|
3adant1 |
|- ( ( F e. dom S /\ i e. NN0 /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> i e. ( 0 ..^ ( # ` F ) ) ) |
48 |
47
|
3expia |
|- ( ( F e. dom S /\ i e. NN0 ) -> ( ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) -> i e. ( 0 ..^ ( # ` F ) ) ) ) |
49 |
48
|
imim1d |
|- ( ( F e. dom S /\ i e. NN0 ) -> ( ( i e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` i ) ) -> ( ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` i ) ) ) ) |
50 |
40
|
3ad2ant1 |
|- ( ( F e. dom S /\ i e. NN0 /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> F : ( 0 ..^ ( # ` F ) ) --> W ) |
51 |
50 47
|
ffvelrnd |
|- ( ( F e. dom S /\ i e. NN0 /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> ( F ` i ) e. W ) |
52 |
|
fvoveq1 |
|- ( a = ( i + 1 ) -> ( F ` ( a - 1 ) ) = ( F ` ( ( i + 1 ) - 1 ) ) ) |
53 |
52
|
fveq2d |
|- ( a = ( i + 1 ) -> ( T ` ( F ` ( a - 1 ) ) ) = ( T ` ( F ` ( ( i + 1 ) - 1 ) ) ) ) |
54 |
53
|
rneqd |
|- ( a = ( i + 1 ) -> ran ( T ` ( F ` ( a - 1 ) ) ) = ran ( T ` ( F ` ( ( i + 1 ) - 1 ) ) ) ) |
55 |
27 54
|
eleq12d |
|- ( a = ( i + 1 ) -> ( ( F ` a ) e. ran ( T ` ( F ` ( a - 1 ) ) ) <-> ( F ` ( i + 1 ) ) e. ran ( T ` ( F ` ( ( i + 1 ) - 1 ) ) ) ) ) |
56 |
7
|
simp3bi |
|- ( F e. dom S -> A. a e. ( 1 ..^ ( # ` F ) ) ( F ` a ) e. ran ( T ` ( F ` ( a - 1 ) ) ) ) |
57 |
56
|
3ad2ant1 |
|- ( ( F e. dom S /\ i e. NN0 /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> A. a e. ( 1 ..^ ( # ` F ) ) ( F ` a ) e. ran ( T ` ( F ` ( a - 1 ) ) ) ) |
58 |
|
nn0p1nn |
|- ( i e. NN0 -> ( i + 1 ) e. NN ) |
59 |
58
|
3ad2ant2 |
|- ( ( F e. dom S /\ i e. NN0 /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> ( i + 1 ) e. NN ) |
60 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
61 |
59 60
|
eleqtrdi |
|- ( ( F e. dom S /\ i e. NN0 /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> ( i + 1 ) e. ( ZZ>= ` 1 ) ) |
62 |
|
elfzolt2b |
|- ( ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) -> ( i + 1 ) e. ( ( i + 1 ) ..^ ( # ` F ) ) ) |
63 |
62
|
3ad2ant3 |
|- ( ( F e. dom S /\ i e. NN0 /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> ( i + 1 ) e. ( ( i + 1 ) ..^ ( # ` F ) ) ) |
64 |
|
elfzo3 |
|- ( ( i + 1 ) e. ( 1 ..^ ( # ` F ) ) <-> ( ( i + 1 ) e. ( ZZ>= ` 1 ) /\ ( i + 1 ) e. ( ( i + 1 ) ..^ ( # ` F ) ) ) ) |
65 |
61 63 64
|
sylanbrc |
|- ( ( F e. dom S /\ i e. NN0 /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> ( i + 1 ) e. ( 1 ..^ ( # ` F ) ) ) |
66 |
55 57 65
|
rspcdva |
|- ( ( F e. dom S /\ i e. NN0 /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> ( F ` ( i + 1 ) ) e. ran ( T ` ( F ` ( ( i + 1 ) - 1 ) ) ) ) |
67 |
|
nn0cn |
|- ( i e. NN0 -> i e. CC ) |
68 |
67
|
3ad2ant2 |
|- ( ( F e. dom S /\ i e. NN0 /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> i e. CC ) |
69 |
|
ax-1cn |
|- 1 e. CC |
70 |
|
pncan |
|- ( ( i e. CC /\ 1 e. CC ) -> ( ( i + 1 ) - 1 ) = i ) |
71 |
68 69 70
|
sylancl |
|- ( ( F e. dom S /\ i e. NN0 /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> ( ( i + 1 ) - 1 ) = i ) |
72 |
71
|
fveq2d |
|- ( ( F e. dom S /\ i e. NN0 /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> ( F ` ( ( i + 1 ) - 1 ) ) = ( F ` i ) ) |
73 |
72
|
fveq2d |
|- ( ( F e. dom S /\ i e. NN0 /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> ( T ` ( F ` ( ( i + 1 ) - 1 ) ) ) = ( T ` ( F ` i ) ) ) |
74 |
73
|
rneqd |
|- ( ( F e. dom S /\ i e. NN0 /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> ran ( T ` ( F ` ( ( i + 1 ) - 1 ) ) ) = ran ( T ` ( F ` i ) ) ) |
75 |
66 74
|
eleqtrd |
|- ( ( F e. dom S /\ i e. NN0 /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> ( F ` ( i + 1 ) ) e. ran ( T ` ( F ` i ) ) ) |
76 |
1 2 3 4
|
efgi2 |
|- ( ( ( F ` i ) e. W /\ ( F ` ( i + 1 ) ) e. ran ( T ` ( F ` i ) ) ) -> ( F ` i ) .~ ( F ` ( i + 1 ) ) ) |
77 |
51 75 76
|
syl2anc |
|- ( ( F e. dom S /\ i e. NN0 /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> ( F ` i ) .~ ( F ` ( i + 1 ) ) ) |
78 |
36
|
a1i |
|- ( ( F e. dom S /\ i e. NN0 /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> .~ Er W ) |
79 |
78
|
ertr |
|- ( ( F e. dom S /\ i e. NN0 /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> ( ( ( F ` 0 ) .~ ( F ` i ) /\ ( F ` i ) .~ ( F ` ( i + 1 ) ) ) -> ( F ` 0 ) .~ ( F ` ( i + 1 ) ) ) ) |
80 |
77 79
|
mpan2d |
|- ( ( F e. dom S /\ i e. NN0 /\ ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) ) -> ( ( F ` 0 ) .~ ( F ` i ) -> ( F ` 0 ) .~ ( F ` ( i + 1 ) ) ) ) |
81 |
80
|
3expia |
|- ( ( F e. dom S /\ i e. NN0 ) -> ( ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) -> ( ( F ` 0 ) .~ ( F ` i ) -> ( F ` 0 ) .~ ( F ` ( i + 1 ) ) ) ) ) |
82 |
81
|
a2d |
|- ( ( F e. dom S /\ i e. NN0 ) -> ( ( ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` i ) ) -> ( ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` ( i + 1 ) ) ) ) ) |
83 |
49 82
|
syld |
|- ( ( F e. dom S /\ i e. NN0 ) -> ( ( i e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` i ) ) -> ( ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` ( i + 1 ) ) ) ) ) |
84 |
83
|
expcom |
|- ( i e. NN0 -> ( F e. dom S -> ( ( i e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` i ) ) -> ( ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` ( i + 1 ) ) ) ) ) ) |
85 |
84
|
a2d |
|- ( i e. NN0 -> ( ( F e. dom S -> ( i e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` i ) ) ) -> ( F e. dom S -> ( ( i + 1 ) e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` ( i + 1 ) ) ) ) ) ) |
86 |
20 25 30 35 43 85
|
nn0ind |
|- ( ( ( # ` F ) - 1 ) e. NN0 -> ( F e. dom S -> ( ( ( # ` F ) - 1 ) e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` ( ( # ` F ) - 1 ) ) ) ) ) |
87 |
15 86
|
mpcom |
|- ( F e. dom S -> ( ( ( # ` F ) - 1 ) e. ( 0 ..^ ( # ` F ) ) -> ( F ` 0 ) .~ ( F ` ( ( # ` F ) - 1 ) ) ) ) |
88 |
13 87
|
mpd |
|- ( F e. dom S -> ( F ` 0 ) .~ ( F ` ( ( # ` F ) - 1 ) ) ) |
89 |
1 2 3 4 5 6
|
efgsval |
|- ( F e. dom S -> ( S ` F ) = ( F ` ( ( # ` F ) - 1 ) ) ) |
90 |
88 89
|
breqtrrd |
|- ( F e. dom S -> ( F ` 0 ) .~ ( S ` F ) ) |