Step |
Hyp |
Ref |
Expression |
1 |
|
frgpup.b |
|- B = ( Base ` H ) |
2 |
|
frgpup.n |
|- N = ( invg ` H ) |
3 |
|
frgpup.t |
|- T = ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) ) |
4 |
|
frgpup.h |
|- ( ph -> H e. Grp ) |
5 |
|
frgpup.i |
|- ( ph -> I e. V ) |
6 |
|
frgpup.a |
|- ( ph -> F : I --> B ) |
7 |
|
frgpup.w |
|- W = ( _I ` Word ( I X. 2o ) ) |
8 |
|
frgpup.r |
|- .~ = ( ~FG ` I ) |
9 |
|
frgpup.g |
|- G = ( freeGrp ` I ) |
10 |
|
frgpup.x |
|- X = ( Base ` G ) |
11 |
|
frgpup.e |
|- E = ran ( g e. W |-> <. [ g ] .~ , ( H gsum ( T o. g ) ) >. ) |
12 |
|
frgpup.u |
|- U = ( varFGrp ` I ) |
13 |
|
frgpup.y |
|- ( ph -> A e. I ) |
14 |
8 12
|
vrgpval |
|- ( ( I e. V /\ A e. I ) -> ( U ` A ) = [ <" <. A , (/) >. "> ] .~ ) |
15 |
5 13 14
|
syl2anc |
|- ( ph -> ( U ` A ) = [ <" <. A , (/) >. "> ] .~ ) |
16 |
15
|
fveq2d |
|- ( ph -> ( E ` ( U ` A ) ) = ( E ` [ <" <. A , (/) >. "> ] .~ ) ) |
17 |
|
0ex |
|- (/) e. _V |
18 |
17
|
prid1 |
|- (/) e. { (/) , 1o } |
19 |
|
df2o3 |
|- 2o = { (/) , 1o } |
20 |
18 19
|
eleqtrri |
|- (/) e. 2o |
21 |
|
opelxpi |
|- ( ( A e. I /\ (/) e. 2o ) -> <. A , (/) >. e. ( I X. 2o ) ) |
22 |
13 20 21
|
sylancl |
|- ( ph -> <. A , (/) >. e. ( I X. 2o ) ) |
23 |
22
|
s1cld |
|- ( ph -> <" <. A , (/) >. "> e. Word ( I X. 2o ) ) |
24 |
|
2on |
|- 2o e. On |
25 |
|
xpexg |
|- ( ( I e. V /\ 2o e. On ) -> ( I X. 2o ) e. _V ) |
26 |
5 24 25
|
sylancl |
|- ( ph -> ( I X. 2o ) e. _V ) |
27 |
|
wrdexg |
|- ( ( I X. 2o ) e. _V -> Word ( I X. 2o ) e. _V ) |
28 |
|
fvi |
|- ( Word ( I X. 2o ) e. _V -> ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) ) |
29 |
26 27 28
|
3syl |
|- ( ph -> ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) ) |
30 |
7 29
|
syl5eq |
|- ( ph -> W = Word ( I X. 2o ) ) |
31 |
23 30
|
eleqtrrd |
|- ( ph -> <" <. A , (/) >. "> e. W ) |
32 |
1 2 3 4 5 6 7 8 9 10 11
|
frgpupval |
|- ( ( ph /\ <" <. A , (/) >. "> e. W ) -> ( E ` [ <" <. A , (/) >. "> ] .~ ) = ( H gsum ( T o. <" <. A , (/) >. "> ) ) ) |
33 |
31 32
|
mpdan |
|- ( ph -> ( E ` [ <" <. A , (/) >. "> ] .~ ) = ( H gsum ( T o. <" <. A , (/) >. "> ) ) ) |
34 |
1 2 3 4 5 6
|
frgpuptf |
|- ( ph -> T : ( I X. 2o ) --> B ) |
35 |
|
s1co |
|- ( ( <. A , (/) >. e. ( I X. 2o ) /\ T : ( I X. 2o ) --> B ) -> ( T o. <" <. A , (/) >. "> ) = <" ( T ` <. A , (/) >. ) "> ) |
36 |
22 34 35
|
syl2anc |
|- ( ph -> ( T o. <" <. A , (/) >. "> ) = <" ( T ` <. A , (/) >. ) "> ) |
37 |
|
df-ov |
|- ( A T (/) ) = ( T ` <. A , (/) >. ) |
38 |
|
iftrue |
|- ( z = (/) -> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) = ( F ` y ) ) |
39 |
|
fveq2 |
|- ( y = A -> ( F ` y ) = ( F ` A ) ) |
40 |
38 39
|
sylan9eqr |
|- ( ( y = A /\ z = (/) ) -> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) = ( F ` A ) ) |
41 |
|
fvex |
|- ( F ` A ) e. _V |
42 |
40 3 41
|
ovmpoa |
|- ( ( A e. I /\ (/) e. 2o ) -> ( A T (/) ) = ( F ` A ) ) |
43 |
13 20 42
|
sylancl |
|- ( ph -> ( A T (/) ) = ( F ` A ) ) |
44 |
37 43
|
eqtr3id |
|- ( ph -> ( T ` <. A , (/) >. ) = ( F ` A ) ) |
45 |
44
|
s1eqd |
|- ( ph -> <" ( T ` <. A , (/) >. ) "> = <" ( F ` A ) "> ) |
46 |
36 45
|
eqtrd |
|- ( ph -> ( T o. <" <. A , (/) >. "> ) = <" ( F ` A ) "> ) |
47 |
46
|
oveq2d |
|- ( ph -> ( H gsum ( T o. <" <. A , (/) >. "> ) ) = ( H gsum <" ( F ` A ) "> ) ) |
48 |
6 13
|
ffvelrnd |
|- ( ph -> ( F ` A ) e. B ) |
49 |
1
|
gsumws1 |
|- ( ( F ` A ) e. B -> ( H gsum <" ( F ` A ) "> ) = ( F ` A ) ) |
50 |
48 49
|
syl |
|- ( ph -> ( H gsum <" ( F ` A ) "> ) = ( F ` A ) ) |
51 |
47 50
|
eqtrd |
|- ( ph -> ( H gsum ( T o. <" <. A , (/) >. "> ) ) = ( F ` A ) ) |
52 |
16 33 51
|
3eqtrd |
|- ( ph -> ( E ` ( U ` A ) ) = ( F ` A ) ) |