Step |
Hyp |
Ref |
Expression |
1 |
|
cycpmconjs.c |
|- C = ( M " ( `' # " { P } ) ) |
2 |
|
cycpmconjs.s |
|- S = ( SymGrp ` D ) |
3 |
|
cycpmconjs.n |
|- N = ( # ` D ) |
4 |
|
cycpmconjs.m |
|- M = ( toCyc ` D ) |
5 |
|
cycpmconjs.b |
|- B = ( Base ` S ) |
6 |
|
cycpmconjs.a |
|- .+ = ( +g ` S ) |
7 |
|
cycpmconjs.l |
|- .- = ( -g ` S ) |
8 |
|
cycpmconjs.p |
|- ( ph -> P e. ( 0 ... N ) ) |
9 |
|
cycpmconjs.d |
|- ( ph -> D e. Fin ) |
10 |
|
cycpmconjs.q |
|- ( ph -> Q e. C ) |
11 |
|
fzofi |
|- ( 0 ..^ N ) e. Fin |
12 |
|
diffi |
|- ( ( 0 ..^ N ) e. Fin -> ( ( 0 ..^ N ) \ dom u ) e. Fin ) |
13 |
11 12
|
mp1i |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( ( 0 ..^ N ) \ dom u ) e. Fin ) |
14 |
|
diffi |
|- ( D e. Fin -> ( D \ ran u ) e. Fin ) |
15 |
9 14
|
syl |
|- ( ph -> ( D \ ran u ) e. Fin ) |
16 |
15
|
ad2antrr |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( D \ ran u ) e. Fin ) |
17 |
|
hashcl |
|- ( D e. Fin -> ( # ` D ) e. NN0 ) |
18 |
9 17
|
syl |
|- ( ph -> ( # ` D ) e. NN0 ) |
19 |
3 18
|
eqeltrid |
|- ( ph -> N e. NN0 ) |
20 |
|
hashfzo0 |
|- ( N e. NN0 -> ( # ` ( 0 ..^ N ) ) = N ) |
21 |
19 20
|
syl |
|- ( ph -> ( # ` ( 0 ..^ N ) ) = N ) |
22 |
21 3
|
eqtrdi |
|- ( ph -> ( # ` ( 0 ..^ N ) ) = ( # ` D ) ) |
23 |
22
|
ad2antrr |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( # ` ( 0 ..^ N ) ) = ( # ` D ) ) |
24 |
|
simplr |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) |
25 |
24
|
elin1d |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> u e. { w e. Word D | w : dom w -1-1-> D } ) |
26 |
|
elrabi |
|- ( u e. { w e. Word D | w : dom w -1-1-> D } -> u e. Word D ) |
27 |
|
wrdfin |
|- ( u e. Word D -> u e. Fin ) |
28 |
25 26 27
|
3syl |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> u e. Fin ) |
29 |
|
id |
|- ( w = u -> w = u ) |
30 |
|
dmeq |
|- ( w = u -> dom w = dom u ) |
31 |
|
eqidd |
|- ( w = u -> D = D ) |
32 |
29 30 31
|
f1eq123d |
|- ( w = u -> ( w : dom w -1-1-> D <-> u : dom u -1-1-> D ) ) |
33 |
32
|
elrab |
|- ( u e. { w e. Word D | w : dom w -1-1-> D } <-> ( u e. Word D /\ u : dom u -1-1-> D ) ) |
34 |
33
|
simprbi |
|- ( u e. { w e. Word D | w : dom w -1-1-> D } -> u : dom u -1-1-> D ) |
35 |
25 34
|
syl |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> u : dom u -1-1-> D ) |
36 |
|
f1fun |
|- ( u : dom u -1-1-> D -> Fun u ) |
37 |
35 36
|
syl |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> Fun u ) |
38 |
|
hashfun |
|- ( u e. Fin -> ( Fun u <-> ( # ` u ) = ( # ` dom u ) ) ) |
39 |
38
|
biimpa |
|- ( ( u e. Fin /\ Fun u ) -> ( # ` u ) = ( # ` dom u ) ) |
40 |
28 37 39
|
syl2anc |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( # ` u ) = ( # ` dom u ) ) |
41 |
24
|
dmexd |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> dom u e. _V ) |
42 |
|
hashf1rn |
|- ( ( dom u e. _V /\ u : dom u -1-1-> D ) -> ( # ` u ) = ( # ` ran u ) ) |
43 |
41 35 42
|
syl2anc |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( # ` u ) = ( # ` ran u ) ) |
44 |
40 43
|
eqtr3d |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( # ` dom u ) = ( # ` ran u ) ) |
45 |
23 44
|
oveq12d |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( ( # ` ( 0 ..^ N ) ) - ( # ` dom u ) ) = ( ( # ` D ) - ( # ` ran u ) ) ) |
46 |
11
|
a1i |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( 0 ..^ N ) e. Fin ) |
47 |
|
wrddm |
|- ( u e. Word D -> dom u = ( 0 ..^ ( # ` u ) ) ) |
48 |
25 26 47
|
3syl |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> dom u = ( 0 ..^ ( # ` u ) ) ) |
49 |
|
hashcl |
|- ( u e. Fin -> ( # ` u ) e. NN0 ) |
50 |
25 26 27 49
|
4syl |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( # ` u ) e. NN0 ) |
51 |
50
|
nn0zd |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( # ` u ) e. ZZ ) |
52 |
18
|
nn0zd |
|- ( ph -> ( # ` D ) e. ZZ ) |
53 |
3 52
|
eqeltrid |
|- ( ph -> N e. ZZ ) |
54 |
53
|
ad2antrr |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> N e. ZZ ) |
55 |
9
|
ad2antrr |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> D e. Fin ) |
56 |
|
wrdf |
|- ( u e. Word D -> u : ( 0 ..^ ( # ` u ) ) --> D ) |
57 |
56
|
frnd |
|- ( u e. Word D -> ran u C_ D ) |
58 |
25 26 57
|
3syl |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ran u C_ D ) |
59 |
|
hashss |
|- ( ( D e. Fin /\ ran u C_ D ) -> ( # ` ran u ) <_ ( # ` D ) ) |
60 |
55 58 59
|
syl2anc |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( # ` ran u ) <_ ( # ` D ) ) |
61 |
3
|
a1i |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> N = ( # ` D ) ) |
62 |
60 43 61
|
3brtr4d |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( # ` u ) <_ N ) |
63 |
|
eluz1 |
|- ( ( # ` u ) e. ZZ -> ( N e. ( ZZ>= ` ( # ` u ) ) <-> ( N e. ZZ /\ ( # ` u ) <_ N ) ) ) |
64 |
63
|
biimpar |
|- ( ( ( # ` u ) e. ZZ /\ ( N e. ZZ /\ ( # ` u ) <_ N ) ) -> N e. ( ZZ>= ` ( # ` u ) ) ) |
65 |
51 54 62 64
|
syl12anc |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> N e. ( ZZ>= ` ( # ` u ) ) ) |
66 |
|
fzoss2 |
|- ( N e. ( ZZ>= ` ( # ` u ) ) -> ( 0 ..^ ( # ` u ) ) C_ ( 0 ..^ N ) ) |
67 |
65 66
|
syl |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( 0 ..^ ( # ` u ) ) C_ ( 0 ..^ N ) ) |
68 |
48 67
|
eqsstrd |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> dom u C_ ( 0 ..^ N ) ) |
69 |
|
hashssdif |
|- ( ( ( 0 ..^ N ) e. Fin /\ dom u C_ ( 0 ..^ N ) ) -> ( # ` ( ( 0 ..^ N ) \ dom u ) ) = ( ( # ` ( 0 ..^ N ) ) - ( # ` dom u ) ) ) |
70 |
46 68 69
|
syl2anc |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( # ` ( ( 0 ..^ N ) \ dom u ) ) = ( ( # ` ( 0 ..^ N ) ) - ( # ` dom u ) ) ) |
71 |
|
hashssdif |
|- ( ( D e. Fin /\ ran u C_ D ) -> ( # ` ( D \ ran u ) ) = ( ( # ` D ) - ( # ` ran u ) ) ) |
72 |
55 58 71
|
syl2anc |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( # ` ( D \ ran u ) ) = ( ( # ` D ) - ( # ` ran u ) ) ) |
73 |
45 70 72
|
3eqtr4d |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( # ` ( ( 0 ..^ N ) \ dom u ) ) = ( # ` ( D \ ran u ) ) ) |
74 |
|
hasheqf1o |
|- ( ( ( ( 0 ..^ N ) \ dom u ) e. Fin /\ ( D \ ran u ) e. Fin ) -> ( ( # ` ( ( 0 ..^ N ) \ dom u ) ) = ( # ` ( D \ ran u ) ) <-> E. f f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) ) |
75 |
74
|
biimpa |
|- ( ( ( ( ( 0 ..^ N ) \ dom u ) e. Fin /\ ( D \ ran u ) e. Fin ) /\ ( # ` ( ( 0 ..^ N ) \ dom u ) ) = ( # ` ( D \ ran u ) ) ) -> E. f f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) |
76 |
13 16 73 75
|
syl21anc |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> E. f f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) |
77 |
35
|
adantr |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> u : dom u -1-1-> D ) |
78 |
|
f1f1orn |
|- ( u : dom u -1-1-> D -> u : dom u -1-1-onto-> ran u ) |
79 |
77 78
|
syl |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> u : dom u -1-1-onto-> ran u ) |
80 |
|
simpr |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) |
81 |
|
disjdif |
|- ( dom u i^i ( ( 0 ..^ N ) \ dom u ) ) = (/) |
82 |
81
|
a1i |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( dom u i^i ( ( 0 ..^ N ) \ dom u ) ) = (/) ) |
83 |
|
disjdif |
|- ( ran u i^i ( D \ ran u ) ) = (/) |
84 |
83
|
a1i |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ran u i^i ( D \ ran u ) ) = (/) ) |
85 |
|
f1oun |
|- ( ( ( u : dom u -1-1-onto-> ran u /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) /\ ( ( dom u i^i ( ( 0 ..^ N ) \ dom u ) ) = (/) /\ ( ran u i^i ( D \ ran u ) ) = (/) ) ) -> ( u u. f ) : ( dom u u. ( ( 0 ..^ N ) \ dom u ) ) -1-1-onto-> ( ran u u. ( D \ ran u ) ) ) |
86 |
79 80 82 84 85
|
syl22anc |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( u u. f ) : ( dom u u. ( ( 0 ..^ N ) \ dom u ) ) -1-1-onto-> ( ran u u. ( D \ ran u ) ) ) |
87 |
|
eqidd |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( u u. f ) = ( u u. f ) ) |
88 |
68
|
adantr |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> dom u C_ ( 0 ..^ N ) ) |
89 |
|
undif |
|- ( dom u C_ ( 0 ..^ N ) <-> ( dom u u. ( ( 0 ..^ N ) \ dom u ) ) = ( 0 ..^ N ) ) |
90 |
88 89
|
sylib |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( dom u u. ( ( 0 ..^ N ) \ dom u ) ) = ( 0 ..^ N ) ) |
91 |
|
undif |
|- ( ran u C_ D <-> ( ran u u. ( D \ ran u ) ) = D ) |
92 |
58 91
|
sylib |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( ran u u. ( D \ ran u ) ) = D ) |
93 |
92
|
adantr |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ran u u. ( D \ ran u ) ) = D ) |
94 |
87 90 93
|
f1oeq123d |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( u u. f ) : ( dom u u. ( ( 0 ..^ N ) \ dom u ) ) -1-1-onto-> ( ran u u. ( D \ ran u ) ) <-> ( u u. f ) : ( 0 ..^ N ) -1-1-onto-> D ) ) |
95 |
86 94
|
mpbid |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( u u. f ) : ( 0 ..^ N ) -1-1-onto-> D ) |
96 |
|
f1ocnv |
|- ( ( u u. f ) : ( 0 ..^ N ) -1-1-onto-> D -> `' ( u u. f ) : D -1-1-onto-> ( 0 ..^ N ) ) |
97 |
95 96
|
syl |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> `' ( u u. f ) : D -1-1-onto-> ( 0 ..^ N ) ) |
98 |
1 2 3 4 5
|
cycpmgcl |
|- ( ( D e. Fin /\ P e. ( 0 ... N ) ) -> C C_ B ) |
99 |
9 8 98
|
syl2anc |
|- ( ph -> C C_ B ) |
100 |
99 10
|
sseldd |
|- ( ph -> Q e. B ) |
101 |
2 5
|
symgbasf1o |
|- ( Q e. B -> Q : D -1-1-onto-> D ) |
102 |
100 101
|
syl |
|- ( ph -> Q : D -1-1-onto-> D ) |
103 |
102
|
ad3antrrr |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> Q : D -1-1-onto-> D ) |
104 |
|
f1oco |
|- ( ( `' ( u u. f ) : D -1-1-onto-> ( 0 ..^ N ) /\ Q : D -1-1-onto-> D ) -> ( `' ( u u. f ) o. Q ) : D -1-1-onto-> ( 0 ..^ N ) ) |
105 |
97 103 104
|
syl2anc |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( `' ( u u. f ) o. Q ) : D -1-1-onto-> ( 0 ..^ N ) ) |
106 |
|
f1oco |
|- ( ( ( `' ( u u. f ) o. Q ) : D -1-1-onto-> ( 0 ..^ N ) /\ ( u u. f ) : ( 0 ..^ N ) -1-1-onto-> D ) -> ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) : ( 0 ..^ N ) -1-1-onto-> ( 0 ..^ N ) ) |
107 |
105 95 106
|
syl2anc |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) : ( 0 ..^ N ) -1-1-onto-> ( 0 ..^ N ) ) |
108 |
|
f1ofun |
|- ( ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) : ( 0 ..^ N ) -1-1-onto-> ( 0 ..^ N ) -> Fun ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) ) |
109 |
|
funrel |
|- ( Fun ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) -> Rel ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) ) |
110 |
107 108 109
|
3syl |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> Rel ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) ) |
111 |
|
f1odm |
|- ( ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) : ( 0 ..^ N ) -1-1-onto-> ( 0 ..^ N ) -> dom ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) = ( 0 ..^ N ) ) |
112 |
107 111
|
syl |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> dom ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) = ( 0 ..^ N ) ) |
113 |
|
fzosplit |
|- ( P e. ( 0 ... N ) -> ( 0 ..^ N ) = ( ( 0 ..^ P ) u. ( P ..^ N ) ) ) |
114 |
8 113
|
syl |
|- ( ph -> ( 0 ..^ N ) = ( ( 0 ..^ P ) u. ( P ..^ N ) ) ) |
115 |
114
|
ad3antrrr |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( 0 ..^ N ) = ( ( 0 ..^ P ) u. ( P ..^ N ) ) ) |
116 |
112 115
|
eqtrd |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> dom ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) = ( ( 0 ..^ P ) u. ( P ..^ N ) ) ) |
117 |
|
fzodisj |
|- ( ( 0 ..^ P ) i^i ( P ..^ N ) ) = (/) |
118 |
|
reldisjun |
|- ( ( Rel ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) /\ dom ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) = ( ( 0 ..^ P ) u. ( P ..^ N ) ) /\ ( ( 0 ..^ P ) i^i ( P ..^ N ) ) = (/) ) -> ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) = ( ( ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) |` ( 0 ..^ P ) ) u. ( ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) |` ( P ..^ N ) ) ) ) |
119 |
117 118
|
mp3an3 |
|- ( ( Rel ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) /\ dom ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) = ( ( 0 ..^ P ) u. ( P ..^ N ) ) ) -> ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) = ( ( ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) |` ( 0 ..^ P ) ) u. ( ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) |` ( P ..^ N ) ) ) ) |
120 |
110 116 119
|
syl2anc |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) = ( ( ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) |` ( 0 ..^ P ) ) u. ( ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) |` ( P ..^ N ) ) ) ) |
121 |
|
resco |
|- ( ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) |` ( 0 ..^ P ) ) = ( ( `' ( u u. f ) o. Q ) o. ( ( u u. f ) |` ( 0 ..^ P ) ) ) |
122 |
121
|
a1i |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) |` ( 0 ..^ P ) ) = ( ( `' ( u u. f ) o. Q ) o. ( ( u u. f ) |` ( 0 ..^ P ) ) ) ) |
123 |
25 26
|
syl |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> u e. Word D ) |
124 |
|
wrdfn |
|- ( u e. Word D -> u Fn ( 0 ..^ ( # ` u ) ) ) |
125 |
123 124
|
syl |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> u Fn ( 0 ..^ ( # ` u ) ) ) |
126 |
24
|
elin2d |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> u e. ( `' # " { P } ) ) |
127 |
|
hashf |
|- # : _V --> ( NN0 u. { +oo } ) |
128 |
|
ffn |
|- ( # : _V --> ( NN0 u. { +oo } ) -> # Fn _V ) |
129 |
|
fniniseg |
|- ( # Fn _V -> ( u e. ( `' # " { P } ) <-> ( u e. _V /\ ( # ` u ) = P ) ) ) |
130 |
127 128 129
|
mp2b |
|- ( u e. ( `' # " { P } ) <-> ( u e. _V /\ ( # ` u ) = P ) ) |
131 |
130
|
simprbi |
|- ( u e. ( `' # " { P } ) -> ( # ` u ) = P ) |
132 |
126 131
|
syl |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( # ` u ) = P ) |
133 |
132
|
oveq2d |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( 0 ..^ ( # ` u ) ) = ( 0 ..^ P ) ) |
134 |
133
|
fneq2d |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( u Fn ( 0 ..^ ( # ` u ) ) <-> u Fn ( 0 ..^ P ) ) ) |
135 |
125 134
|
mpbid |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> u Fn ( 0 ..^ P ) ) |
136 |
135
|
adantr |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> u Fn ( 0 ..^ P ) ) |
137 |
|
f1ofn |
|- ( f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) -> f Fn ( ( 0 ..^ N ) \ dom u ) ) |
138 |
80 137
|
syl |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> f Fn ( ( 0 ..^ N ) \ dom u ) ) |
139 |
48 133
|
eqtrd |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> dom u = ( 0 ..^ P ) ) |
140 |
139
|
ineq1d |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( dom u i^i ( ( 0 ..^ N ) \ dom u ) ) = ( ( 0 ..^ P ) i^i ( ( 0 ..^ N ) \ dom u ) ) ) |
141 |
81
|
a1i |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( dom u i^i ( ( 0 ..^ N ) \ dom u ) ) = (/) ) |
142 |
140 141
|
eqtr3d |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> ( ( 0 ..^ P ) i^i ( ( 0 ..^ N ) \ dom u ) ) = (/) ) |
143 |
142
|
adantr |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( 0 ..^ P ) i^i ( ( 0 ..^ N ) \ dom u ) ) = (/) ) |
144 |
|
fnunres1 |
|- ( ( u Fn ( 0 ..^ P ) /\ f Fn ( ( 0 ..^ N ) \ dom u ) /\ ( ( 0 ..^ P ) i^i ( ( 0 ..^ N ) \ dom u ) ) = (/) ) -> ( ( u u. f ) |` ( 0 ..^ P ) ) = u ) |
145 |
136 138 143 144
|
syl3anc |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( u u. f ) |` ( 0 ..^ P ) ) = u ) |
146 |
145
|
coeq2d |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( `' ( u u. f ) o. Q ) o. ( ( u u. f ) |` ( 0 ..^ P ) ) ) = ( ( `' ( u u. f ) o. Q ) o. u ) ) |
147 |
|
resco |
|- ( ( `' ( u u. f ) o. Q ) |` ran u ) = ( `' ( u u. f ) o. ( Q |` ran u ) ) |
148 |
|
resco |
|- ( ( `' u o. ( M ` u ) ) |` ran u ) = ( `' u o. ( ( M ` u ) |` ran u ) ) |
149 |
148
|
a1i |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( `' u o. ( M ` u ) ) |` ran u ) = ( `' u o. ( ( M ` u ) |` ran u ) ) ) |
150 |
|
cnvun |
|- `' ( u u. f ) = ( `' u u. `' f ) |
151 |
150
|
reseq1i |
|- ( `' ( u u. f ) |` ran u ) = ( ( `' u u. `' f ) |` ran u ) |
152 |
|
f1ocnv |
|- ( u : dom u -1-1-onto-> ran u -> `' u : ran u -1-1-onto-> dom u ) |
153 |
|
f1ofn |
|- ( `' u : ran u -1-1-onto-> dom u -> `' u Fn ran u ) |
154 |
79 152 153
|
3syl |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> `' u Fn ran u ) |
155 |
|
f1ocnv |
|- ( f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) -> `' f : ( D \ ran u ) -1-1-onto-> ( ( 0 ..^ N ) \ dom u ) ) |
156 |
|
f1ofn |
|- ( `' f : ( D \ ran u ) -1-1-onto-> ( ( 0 ..^ N ) \ dom u ) -> `' f Fn ( D \ ran u ) ) |
157 |
80 155 156
|
3syl |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> `' f Fn ( D \ ran u ) ) |
158 |
|
fnunres1 |
|- ( ( `' u Fn ran u /\ `' f Fn ( D \ ran u ) /\ ( ran u i^i ( D \ ran u ) ) = (/) ) -> ( ( `' u u. `' f ) |` ran u ) = `' u ) |
159 |
154 157 84 158
|
syl3anc |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( `' u u. `' f ) |` ran u ) = `' u ) |
160 |
151 159
|
eqtr2id |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> `' u = ( `' ( u u. f ) |` ran u ) ) |
161 |
|
simplr |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( M ` u ) = Q ) |
162 |
161
|
reseq1d |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( M ` u ) |` ran u ) = ( Q |` ran u ) ) |
163 |
160 162
|
coeq12d |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( `' u o. ( ( M ` u ) |` ran u ) ) = ( ( `' ( u u. f ) |` ran u ) o. ( Q |` ran u ) ) ) |
164 |
55
|
adantr |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> D e. Fin ) |
165 |
123
|
adantr |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> u e. Word D ) |
166 |
4 164 165 77
|
tocycfvres1 |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( M ` u ) |` ran u ) = ( ( u cyclShift 1 ) o. `' u ) ) |
167 |
162 166
|
eqtr3d |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( Q |` ran u ) = ( ( u cyclShift 1 ) o. `' u ) ) |
168 |
167
|
rneqd |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ran ( Q |` ran u ) = ran ( ( u cyclShift 1 ) o. `' u ) ) |
169 |
|
1zzd |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> 1 e. ZZ ) |
170 |
|
cshf1o |
|- ( ( u e. Word D /\ u : dom u -1-1-> D /\ 1 e. ZZ ) -> ( u cyclShift 1 ) : dom u -1-1-onto-> ran u ) |
171 |
165 77 169 170
|
syl3anc |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( u cyclShift 1 ) : dom u -1-1-onto-> ran u ) |
172 |
79 152
|
syl |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> `' u : ran u -1-1-onto-> dom u ) |
173 |
|
f1oco |
|- ( ( ( u cyclShift 1 ) : dom u -1-1-onto-> ran u /\ `' u : ran u -1-1-onto-> dom u ) -> ( ( u cyclShift 1 ) o. `' u ) : ran u -1-1-onto-> ran u ) |
174 |
171 172 173
|
syl2anc |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( u cyclShift 1 ) o. `' u ) : ran u -1-1-onto-> ran u ) |
175 |
|
f1ofo |
|- ( ( ( u cyclShift 1 ) o. `' u ) : ran u -1-1-onto-> ran u -> ( ( u cyclShift 1 ) o. `' u ) : ran u -onto-> ran u ) |
176 |
|
forn |
|- ( ( ( u cyclShift 1 ) o. `' u ) : ran u -onto-> ran u -> ran ( ( u cyclShift 1 ) o. `' u ) = ran u ) |
177 |
174 175 176
|
3syl |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ran ( ( u cyclShift 1 ) o. `' u ) = ran u ) |
178 |
168 177
|
eqtrd |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ran ( Q |` ran u ) = ran u ) |
179 |
|
ssid |
|- ran u C_ ran u |
180 |
178 179
|
eqsstrdi |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ran ( Q |` ran u ) C_ ran u ) |
181 |
|
cores |
|- ( ran ( Q |` ran u ) C_ ran u -> ( ( `' ( u u. f ) |` ran u ) o. ( Q |` ran u ) ) = ( `' ( u u. f ) o. ( Q |` ran u ) ) ) |
182 |
180 181
|
syl |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( `' ( u u. f ) |` ran u ) o. ( Q |` ran u ) ) = ( `' ( u u. f ) o. ( Q |` ran u ) ) ) |
183 |
149 163 182
|
3eqtrrd |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( `' ( u u. f ) o. ( Q |` ran u ) ) = ( ( `' u o. ( M ` u ) ) |` ran u ) ) |
184 |
147 183
|
syl5eq |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( `' ( u u. f ) o. Q ) |` ran u ) = ( ( `' u o. ( M ` u ) ) |` ran u ) ) |
185 |
184
|
coeq1d |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( ( `' ( u u. f ) o. Q ) |` ran u ) o. u ) = ( ( ( `' u o. ( M ` u ) ) |` ran u ) o. u ) ) |
186 |
|
cores |
|- ( ran u C_ ran u -> ( ( ( `' u o. ( M ` u ) ) |` ran u ) o. u ) = ( ( `' u o. ( M ` u ) ) o. u ) ) |
187 |
179 186
|
mp1i |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( ( `' u o. ( M ` u ) ) |` ran u ) o. u ) = ( ( `' u o. ( M ` u ) ) o. u ) ) |
188 |
185 187
|
eqtrd |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( ( `' ( u u. f ) o. Q ) |` ran u ) o. u ) = ( ( `' u o. ( M ` u ) ) o. u ) ) |
189 |
|
cores |
|- ( ran u C_ ran u -> ( ( ( `' ( u u. f ) o. Q ) |` ran u ) o. u ) = ( ( `' ( u u. f ) o. Q ) o. u ) ) |
190 |
179 189
|
mp1i |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( ( `' ( u u. f ) o. Q ) |` ran u ) o. u ) = ( ( `' ( u u. f ) o. Q ) o. u ) ) |
191 |
132
|
adantr |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( # ` u ) = P ) |
192 |
1 2 3 4 164 165 77 191
|
cycpmconjslem1 |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( `' u o. ( M ` u ) ) o. u ) = ( ( _I |` ( 0 ..^ P ) ) cyclShift 1 ) ) |
193 |
188 190 192
|
3eqtr3d |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( `' ( u u. f ) o. Q ) o. u ) = ( ( _I |` ( 0 ..^ P ) ) cyclShift 1 ) ) |
194 |
122 146 193
|
3eqtrd |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) |` ( 0 ..^ P ) ) = ( ( _I |` ( 0 ..^ P ) ) cyclShift 1 ) ) |
195 |
|
resco |
|- ( ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) |` ( P ..^ N ) ) = ( ( `' ( u u. f ) o. Q ) o. ( ( u u. f ) |` ( P ..^ N ) ) ) |
196 |
139
|
adantr |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> dom u = ( 0 ..^ P ) ) |
197 |
196
|
difeq2d |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( 0 ..^ N ) \ dom u ) = ( ( 0 ..^ N ) \ ( 0 ..^ P ) ) ) |
198 |
|
fzodif1 |
|- ( P e. ( 0 ... N ) -> ( ( 0 ..^ N ) \ ( 0 ..^ P ) ) = ( P ..^ N ) ) |
199 |
8 198
|
syl |
|- ( ph -> ( ( 0 ..^ N ) \ ( 0 ..^ P ) ) = ( P ..^ N ) ) |
200 |
199
|
ad3antrrr |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( 0 ..^ N ) \ ( 0 ..^ P ) ) = ( P ..^ N ) ) |
201 |
197 200
|
eqtrd |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( 0 ..^ N ) \ dom u ) = ( P ..^ N ) ) |
202 |
201
|
reseq2d |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( u u. f ) |` ( ( 0 ..^ N ) \ dom u ) ) = ( ( u u. f ) |` ( P ..^ N ) ) ) |
203 |
|
fnunres2 |
|- ( ( u Fn ( 0 ..^ P ) /\ f Fn ( ( 0 ..^ N ) \ dom u ) /\ ( ( 0 ..^ P ) i^i ( ( 0 ..^ N ) \ dom u ) ) = (/) ) -> ( ( u u. f ) |` ( ( 0 ..^ N ) \ dom u ) ) = f ) |
204 |
136 138 143 203
|
syl3anc |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( u u. f ) |` ( ( 0 ..^ N ) \ dom u ) ) = f ) |
205 |
202 204
|
eqtr3d |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( u u. f ) |` ( P ..^ N ) ) = f ) |
206 |
205
|
coeq2d |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( `' ( u u. f ) o. Q ) o. ( ( u u. f ) |` ( P ..^ N ) ) ) = ( ( `' ( u u. f ) o. Q ) o. f ) ) |
207 |
195 206
|
syl5eq |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) |` ( P ..^ N ) ) = ( ( `' ( u u. f ) o. Q ) o. f ) ) |
208 |
150
|
reseq1i |
|- ( `' ( u u. f ) |` ( D \ ran u ) ) = ( ( `' u u. `' f ) |` ( D \ ran u ) ) |
209 |
|
fnunres2 |
|- ( ( `' u Fn ran u /\ `' f Fn ( D \ ran u ) /\ ( ran u i^i ( D \ ran u ) ) = (/) ) -> ( ( `' u u. `' f ) |` ( D \ ran u ) ) = `' f ) |
210 |
154 157 84 209
|
syl3anc |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( `' u u. `' f ) |` ( D \ ran u ) ) = `' f ) |
211 |
208 210
|
syl5eq |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( `' ( u u. f ) |` ( D \ ran u ) ) = `' f ) |
212 |
161
|
reseq1d |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( M ` u ) |` ( D \ ran u ) ) = ( Q |` ( D \ ran u ) ) ) |
213 |
4 164 165 77
|
tocycfvres2 |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( M ` u ) |` ( D \ ran u ) ) = ( _I |` ( D \ ran u ) ) ) |
214 |
212 213
|
eqtr3d |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( Q |` ( D \ ran u ) ) = ( _I |` ( D \ ran u ) ) ) |
215 |
211 214
|
coeq12d |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( `' ( u u. f ) |` ( D \ ran u ) ) o. ( Q |` ( D \ ran u ) ) ) = ( `' f o. ( _I |` ( D \ ran u ) ) ) ) |
216 |
214
|
rneqd |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ran ( Q |` ( D \ ran u ) ) = ran ( _I |` ( D \ ran u ) ) ) |
217 |
|
rnresi |
|- ran ( _I |` ( D \ ran u ) ) = ( D \ ran u ) |
218 |
217
|
eqimssi |
|- ran ( _I |` ( D \ ran u ) ) C_ ( D \ ran u ) |
219 |
216 218
|
eqsstrdi |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ran ( Q |` ( D \ ran u ) ) C_ ( D \ ran u ) ) |
220 |
|
cores |
|- ( ran ( Q |` ( D \ ran u ) ) C_ ( D \ ran u ) -> ( ( `' ( u u. f ) |` ( D \ ran u ) ) o. ( Q |` ( D \ ran u ) ) ) = ( `' ( u u. f ) o. ( Q |` ( D \ ran u ) ) ) ) |
221 |
219 220
|
syl |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( `' ( u u. f ) |` ( D \ ran u ) ) o. ( Q |` ( D \ ran u ) ) ) = ( `' ( u u. f ) o. ( Q |` ( D \ ran u ) ) ) ) |
222 |
|
resco |
|- ( ( `' ( u u. f ) o. Q ) |` ( D \ ran u ) ) = ( `' ( u u. f ) o. ( Q |` ( D \ ran u ) ) ) |
223 |
221 222
|
eqtr4di |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( `' ( u u. f ) |` ( D \ ran u ) ) o. ( Q |` ( D \ ran u ) ) ) = ( ( `' ( u u. f ) o. Q ) |` ( D \ ran u ) ) ) |
224 |
215 223
|
eqtr3d |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( `' f o. ( _I |` ( D \ ran u ) ) ) = ( ( `' ( u u. f ) o. Q ) |` ( D \ ran u ) ) ) |
225 |
224
|
coeq1d |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( `' f o. ( _I |` ( D \ ran u ) ) ) o. f ) = ( ( ( `' ( u u. f ) o. Q ) |` ( D \ ran u ) ) o. f ) ) |
226 |
|
f1of |
|- ( `' f : ( D \ ran u ) -1-1-onto-> ( ( 0 ..^ N ) \ dom u ) -> `' f : ( D \ ran u ) --> ( ( 0 ..^ N ) \ dom u ) ) |
227 |
80 155 226
|
3syl |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> `' f : ( D \ ran u ) --> ( ( 0 ..^ N ) \ dom u ) ) |
228 |
|
fcoi1 |
|- ( `' f : ( D \ ran u ) --> ( ( 0 ..^ N ) \ dom u ) -> ( `' f o. ( _I |` ( D \ ran u ) ) ) = `' f ) |
229 |
227 228
|
syl |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( `' f o. ( _I |` ( D \ ran u ) ) ) = `' f ) |
230 |
229
|
coeq1d |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( `' f o. ( _I |` ( D \ ran u ) ) ) o. f ) = ( `' f o. f ) ) |
231 |
|
f1ococnv1 |
|- ( f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) -> ( `' f o. f ) = ( _I |` ( ( 0 ..^ N ) \ dom u ) ) ) |
232 |
80 231
|
syl |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( `' f o. f ) = ( _I |` ( ( 0 ..^ N ) \ dom u ) ) ) |
233 |
201
|
reseq2d |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( _I |` ( ( 0 ..^ N ) \ dom u ) ) = ( _I |` ( P ..^ N ) ) ) |
234 |
230 232 233
|
3eqtrd |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( `' f o. ( _I |` ( D \ ran u ) ) ) o. f ) = ( _I |` ( P ..^ N ) ) ) |
235 |
|
f1of |
|- ( f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) -> f : ( ( 0 ..^ N ) \ dom u ) --> ( D \ ran u ) ) |
236 |
|
frn |
|- ( f : ( ( 0 ..^ N ) \ dom u ) --> ( D \ ran u ) -> ran f C_ ( D \ ran u ) ) |
237 |
80 235 236
|
3syl |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ran f C_ ( D \ ran u ) ) |
238 |
|
cores |
|- ( ran f C_ ( D \ ran u ) -> ( ( ( `' ( u u. f ) o. Q ) |` ( D \ ran u ) ) o. f ) = ( ( `' ( u u. f ) o. Q ) o. f ) ) |
239 |
237 238
|
syl |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( ( `' ( u u. f ) o. Q ) |` ( D \ ran u ) ) o. f ) = ( ( `' ( u u. f ) o. Q ) o. f ) ) |
240 |
225 234 239
|
3eqtr3rd |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( `' ( u u. f ) o. Q ) o. f ) = ( _I |` ( P ..^ N ) ) ) |
241 |
207 240
|
eqtrd |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) |` ( P ..^ N ) ) = ( _I |` ( P ..^ N ) ) ) |
242 |
194 241
|
uneq12d |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) |` ( 0 ..^ P ) ) u. ( ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) |` ( P ..^ N ) ) ) = ( ( ( _I |` ( 0 ..^ P ) ) cyclShift 1 ) u. ( _I |` ( P ..^ N ) ) ) ) |
243 |
120 242
|
eqtrd |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) = ( ( ( _I |` ( 0 ..^ P ) ) cyclShift 1 ) u. ( _I |` ( P ..^ N ) ) ) ) |
244 |
|
vex |
|- u e. _V |
245 |
|
vex |
|- f e. _V |
246 |
244 245
|
unex |
|- ( u u. f ) e. _V |
247 |
|
f1oeq1 |
|- ( q = ( u u. f ) -> ( q : ( 0 ..^ N ) -1-1-onto-> D <-> ( u u. f ) : ( 0 ..^ N ) -1-1-onto-> D ) ) |
248 |
|
cnveq |
|- ( q = ( u u. f ) -> `' q = `' ( u u. f ) ) |
249 |
248
|
coeq1d |
|- ( q = ( u u. f ) -> ( `' q o. Q ) = ( `' ( u u. f ) o. Q ) ) |
250 |
|
id |
|- ( q = ( u u. f ) -> q = ( u u. f ) ) |
251 |
249 250
|
coeq12d |
|- ( q = ( u u. f ) -> ( ( `' q o. Q ) o. q ) = ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) ) |
252 |
251
|
eqeq1d |
|- ( q = ( u u. f ) -> ( ( ( `' q o. Q ) o. q ) = ( ( ( _I |` ( 0 ..^ P ) ) cyclShift 1 ) u. ( _I |` ( P ..^ N ) ) ) <-> ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) = ( ( ( _I |` ( 0 ..^ P ) ) cyclShift 1 ) u. ( _I |` ( P ..^ N ) ) ) ) ) |
253 |
247 252
|
anbi12d |
|- ( q = ( u u. f ) -> ( ( q : ( 0 ..^ N ) -1-1-onto-> D /\ ( ( `' q o. Q ) o. q ) = ( ( ( _I |` ( 0 ..^ P ) ) cyclShift 1 ) u. ( _I |` ( P ..^ N ) ) ) ) <-> ( ( u u. f ) : ( 0 ..^ N ) -1-1-onto-> D /\ ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) = ( ( ( _I |` ( 0 ..^ P ) ) cyclShift 1 ) u. ( _I |` ( P ..^ N ) ) ) ) ) ) |
254 |
246 253
|
spcev |
|- ( ( ( u u. f ) : ( 0 ..^ N ) -1-1-onto-> D /\ ( ( `' ( u u. f ) o. Q ) o. ( u u. f ) ) = ( ( ( _I |` ( 0 ..^ P ) ) cyclShift 1 ) u. ( _I |` ( P ..^ N ) ) ) ) -> E. q ( q : ( 0 ..^ N ) -1-1-onto-> D /\ ( ( `' q o. Q ) o. q ) = ( ( ( _I |` ( 0 ..^ P ) ) cyclShift 1 ) u. ( _I |` ( P ..^ N ) ) ) ) ) |
255 |
95 243 254
|
syl2anc |
|- ( ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) /\ f : ( ( 0 ..^ N ) \ dom u ) -1-1-onto-> ( D \ ran u ) ) -> E. q ( q : ( 0 ..^ N ) -1-1-onto-> D /\ ( ( `' q o. Q ) o. q ) = ( ( ( _I |` ( 0 ..^ P ) ) cyclShift 1 ) u. ( _I |` ( P ..^ N ) ) ) ) ) |
256 |
76 255
|
exlimddv |
|- ( ( ( ph /\ u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ) /\ ( M ` u ) = Q ) -> E. q ( q : ( 0 ..^ N ) -1-1-onto-> D /\ ( ( `' q o. Q ) o. q ) = ( ( ( _I |` ( 0 ..^ P ) ) cyclShift 1 ) u. ( _I |` ( P ..^ N ) ) ) ) ) |
257 |
|
nfcv |
|- F/_ u M |
258 |
4 2 5
|
tocycf |
|- ( D e. Fin -> M : { w e. Word D | w : dom w -1-1-> D } --> B ) |
259 |
|
ffn |
|- ( M : { w e. Word D | w : dom w -1-1-> D } --> B -> M Fn { w e. Word D | w : dom w -1-1-> D } ) |
260 |
9 258 259
|
3syl |
|- ( ph -> M Fn { w e. Word D | w : dom w -1-1-> D } ) |
261 |
10 1
|
eleqtrdi |
|- ( ph -> Q e. ( M " ( `' # " { P } ) ) ) |
262 |
257 260 261
|
fvelimad |
|- ( ph -> E. u e. ( { w e. Word D | w : dom w -1-1-> D } i^i ( `' # " { P } ) ) ( M ` u ) = Q ) |
263 |
256 262
|
r19.29a |
|- ( ph -> E. q ( q : ( 0 ..^ N ) -1-1-onto-> D /\ ( ( `' q o. Q ) o. q ) = ( ( ( _I |` ( 0 ..^ P ) ) cyclShift 1 ) u. ( _I |` ( P ..^ N ) ) ) ) ) |