Step |
Hyp |
Ref |
Expression |
1 |
|
frmdbas.m |
|- M = ( freeMnd ` I ) |
2 |
|
frmdbas.b |
|- B = ( Base ` M ) |
3 |
|
frmdplusg.p |
|- .+ = ( +g ` M ) |
4 |
1 2
|
frmdbas |
|- ( I e. _V -> B = Word I ) |
5 |
|
eqid |
|- ( ++ |` ( B X. B ) ) = ( ++ |` ( B X. B ) ) |
6 |
1 4 5
|
frmdval |
|- ( I e. _V -> M = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( ++ |` ( B X. B ) ) >. } ) |
7 |
6
|
fveq2d |
|- ( I e. _V -> ( +g ` M ) = ( +g ` { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( ++ |` ( B X. B ) ) >. } ) ) |
8 |
3 7
|
eqtrid |
|- ( I e. _V -> .+ = ( +g ` { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( ++ |` ( B X. B ) ) >. } ) ) |
9 |
|
wrdexg |
|- ( I e. _V -> Word I e. _V ) |
10 |
|
ccatfn |
|- ++ Fn ( _V X. _V ) |
11 |
|
xpss |
|- ( B X. B ) C_ ( _V X. _V ) |
12 |
|
fnssres |
|- ( ( ++ Fn ( _V X. _V ) /\ ( B X. B ) C_ ( _V X. _V ) ) -> ( ++ |` ( B X. B ) ) Fn ( B X. B ) ) |
13 |
10 11 12
|
mp2an |
|- ( ++ |` ( B X. B ) ) Fn ( B X. B ) |
14 |
|
ovres |
|- ( ( x e. B /\ y e. B ) -> ( x ( ++ |` ( B X. B ) ) y ) = ( x ++ y ) ) |
15 |
1 2
|
frmdelbas |
|- ( x e. B -> x e. Word I ) |
16 |
1 2
|
frmdelbas |
|- ( y e. B -> y e. Word I ) |
17 |
|
ccatcl |
|- ( ( x e. Word I /\ y e. Word I ) -> ( x ++ y ) e. Word I ) |
18 |
15 16 17
|
syl2an |
|- ( ( x e. B /\ y e. B ) -> ( x ++ y ) e. Word I ) |
19 |
14 18
|
eqeltrd |
|- ( ( x e. B /\ y e. B ) -> ( x ( ++ |` ( B X. B ) ) y ) e. Word I ) |
20 |
19
|
rgen2 |
|- A. x e. B A. y e. B ( x ( ++ |` ( B X. B ) ) y ) e. Word I |
21 |
|
ffnov |
|- ( ( ++ |` ( B X. B ) ) : ( B X. B ) --> Word I <-> ( ( ++ |` ( B X. B ) ) Fn ( B X. B ) /\ A. x e. B A. y e. B ( x ( ++ |` ( B X. B ) ) y ) e. Word I ) ) |
22 |
13 20 21
|
mpbir2an |
|- ( ++ |` ( B X. B ) ) : ( B X. B ) --> Word I |
23 |
2
|
fvexi |
|- B e. _V |
24 |
23 23
|
xpex |
|- ( B X. B ) e. _V |
25 |
|
fex2 |
|- ( ( ( ++ |` ( B X. B ) ) : ( B X. B ) --> Word I /\ ( B X. B ) e. _V /\ Word I e. _V ) -> ( ++ |` ( B X. B ) ) e. _V ) |
26 |
22 24 25
|
mp3an12 |
|- ( Word I e. _V -> ( ++ |` ( B X. B ) ) e. _V ) |
27 |
|
eqid |
|- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( ++ |` ( B X. B ) ) >. } = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( ++ |` ( B X. B ) ) >. } |
28 |
27
|
grpplusg |
|- ( ( ++ |` ( B X. B ) ) e. _V -> ( ++ |` ( B X. B ) ) = ( +g ` { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( ++ |` ( B X. B ) ) >. } ) ) |
29 |
9 26 28
|
3syl |
|- ( I e. _V -> ( ++ |` ( B X. B ) ) = ( +g ` { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( ++ |` ( B X. B ) ) >. } ) ) |
30 |
8 29
|
eqtr4d |
|- ( I e. _V -> .+ = ( ++ |` ( B X. B ) ) ) |
31 |
|
fvprc |
|- ( -. I e. _V -> ( freeMnd ` I ) = (/) ) |
32 |
1 31
|
eqtrid |
|- ( -. I e. _V -> M = (/) ) |
33 |
32
|
fveq2d |
|- ( -. I e. _V -> ( +g ` M ) = ( +g ` (/) ) ) |
34 |
3 33
|
eqtrid |
|- ( -. I e. _V -> .+ = ( +g ` (/) ) ) |
35 |
|
res0 |
|- ( ++ |` (/) ) = (/) |
36 |
|
df-plusg |
|- +g = Slot 2 |
37 |
36
|
str0 |
|- (/) = ( +g ` (/) ) |
38 |
35 37
|
eqtr2i |
|- ( +g ` (/) ) = ( ++ |` (/) ) |
39 |
34 38
|
eqtrdi |
|- ( -. I e. _V -> .+ = ( ++ |` (/) ) ) |
40 |
32
|
fveq2d |
|- ( -. I e. _V -> ( Base ` M ) = ( Base ` (/) ) ) |
41 |
|
base0 |
|- (/) = ( Base ` (/) ) |
42 |
40 2 41
|
3eqtr4g |
|- ( -. I e. _V -> B = (/) ) |
43 |
42
|
xpeq2d |
|- ( -. I e. _V -> ( B X. B ) = ( B X. (/) ) ) |
44 |
|
xp0 |
|- ( B X. (/) ) = (/) |
45 |
43 44
|
eqtrdi |
|- ( -. I e. _V -> ( B X. B ) = (/) ) |
46 |
45
|
reseq2d |
|- ( -. I e. _V -> ( ++ |` ( B X. B ) ) = ( ++ |` (/) ) ) |
47 |
39 46
|
eqtr4d |
|- ( -. I e. _V -> .+ = ( ++ |` ( B X. B ) ) ) |
48 |
30 47
|
pm2.61i |
|- .+ = ( ++ |` ( B X. B ) ) |