Step |
Hyp |
Ref |
Expression |
1 |
|
frgpval.m |
|- G = ( freeGrp ` I ) |
2 |
|
frgpval.b |
|- M = ( freeMnd ` ( I X. 2o ) ) |
3 |
|
frgpval.r |
|- .~ = ( ~FG ` I ) |
4 |
|
frgpcpbl.p |
|- .+ = ( +g ` M ) |
5 |
|
eqid |
|- ( _I ` Word ( I X. 2o ) ) = ( _I ` Word ( I X. 2o ) ) |
6 |
|
eqid |
|- ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) |
7 |
|
eqid |
|- ( v e. ( _I ` Word ( I X. 2o ) ) |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) ` w ) "> >. ) ) ) = ( v e. ( _I ` Word ( I X. 2o ) ) |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) ` w ) "> >. ) ) ) |
8 |
|
eqid |
|- ( ( _I ` Word ( I X. 2o ) ) \ U_ x e. ( _I ` Word ( I X. 2o ) ) ran ( ( v e. ( _I ` Word ( I X. 2o ) ) |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) ` w ) "> >. ) ) ) ` x ) ) = ( ( _I ` Word ( I X. 2o ) ) \ U_ x e. ( _I ` Word ( I X. 2o ) ) ran ( ( v e. ( _I ` Word ( I X. 2o ) ) |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) ` w ) "> >. ) ) ) ` x ) ) |
9 |
|
eqid |
|- ( m e. { t e. ( Word ( _I ` Word ( I X. 2o ) ) \ { (/) } ) | ( ( t ` 0 ) e. ( ( _I ` Word ( I X. 2o ) ) \ U_ x e. ( _I ` Word ( I X. 2o ) ) ran ( ( v e. ( _I ` Word ( I X. 2o ) ) |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) ` w ) "> >. ) ) ) ` x ) ) /\ A. k e. ( 1 ..^ ( # ` t ) ) ( t ` k ) e. ran ( ( v e. ( _I ` Word ( I X. 2o ) ) |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) ` w ) "> >. ) ) ) ` ( t ` ( k - 1 ) ) ) ) } |-> ( m ` ( ( # ` m ) - 1 ) ) ) = ( m e. { t e. ( Word ( _I ` Word ( I X. 2o ) ) \ { (/) } ) | ( ( t ` 0 ) e. ( ( _I ` Word ( I X. 2o ) ) \ U_ x e. ( _I ` Word ( I X. 2o ) ) ran ( ( v e. ( _I ` Word ( I X. 2o ) ) |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) ` w ) "> >. ) ) ) ` x ) ) /\ A. k e. ( 1 ..^ ( # ` t ) ) ( t ` k ) e. ran ( ( v e. ( _I ` Word ( I X. 2o ) ) |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. ) ` w ) "> >. ) ) ) ` ( t ` ( k - 1 ) ) ) ) } |-> ( m ` ( ( # ` m ) - 1 ) ) ) |
10 |
5 3 6 7 8 9
|
efgcpbl2 |
|- ( ( A .~ C /\ B .~ D ) -> ( A ++ B ) .~ ( C ++ D ) ) |
11 |
5 3
|
efger |
|- .~ Er ( _I ` Word ( I X. 2o ) ) |
12 |
11
|
a1i |
|- ( ( A .~ C /\ B .~ D ) -> .~ Er ( _I ` Word ( I X. 2o ) ) ) |
13 |
|
simpl |
|- ( ( A .~ C /\ B .~ D ) -> A .~ C ) |
14 |
12 13
|
ercl |
|- ( ( A .~ C /\ B .~ D ) -> A e. ( _I ` Word ( I X. 2o ) ) ) |
15 |
5
|
efgrcl |
|- ( A e. ( _I ` Word ( I X. 2o ) ) -> ( I e. _V /\ ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) ) ) |
16 |
14 15
|
syl |
|- ( ( A .~ C /\ B .~ D ) -> ( I e. _V /\ ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) ) ) |
17 |
16
|
simprd |
|- ( ( A .~ C /\ B .~ D ) -> ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) ) |
18 |
16
|
simpld |
|- ( ( A .~ C /\ B .~ D ) -> I e. _V ) |
19 |
|
2on |
|- 2o e. On |
20 |
|
xpexg |
|- ( ( I e. _V /\ 2o e. On ) -> ( I X. 2o ) e. _V ) |
21 |
18 19 20
|
sylancl |
|- ( ( A .~ C /\ B .~ D ) -> ( I X. 2o ) e. _V ) |
22 |
|
eqid |
|- ( Base ` M ) = ( Base ` M ) |
23 |
2 22
|
frmdbas |
|- ( ( I X. 2o ) e. _V -> ( Base ` M ) = Word ( I X. 2o ) ) |
24 |
21 23
|
syl |
|- ( ( A .~ C /\ B .~ D ) -> ( Base ` M ) = Word ( I X. 2o ) ) |
25 |
17 24
|
eqtr4d |
|- ( ( A .~ C /\ B .~ D ) -> ( _I ` Word ( I X. 2o ) ) = ( Base ` M ) ) |
26 |
14 25
|
eleqtrd |
|- ( ( A .~ C /\ B .~ D ) -> A e. ( Base ` M ) ) |
27 |
|
simpr |
|- ( ( A .~ C /\ B .~ D ) -> B .~ D ) |
28 |
12 27
|
ercl |
|- ( ( A .~ C /\ B .~ D ) -> B e. ( _I ` Word ( I X. 2o ) ) ) |
29 |
28 25
|
eleqtrd |
|- ( ( A .~ C /\ B .~ D ) -> B e. ( Base ` M ) ) |
30 |
2 22 4
|
frmdadd |
|- ( ( A e. ( Base ` M ) /\ B e. ( Base ` M ) ) -> ( A .+ B ) = ( A ++ B ) ) |
31 |
26 29 30
|
syl2anc |
|- ( ( A .~ C /\ B .~ D ) -> ( A .+ B ) = ( A ++ B ) ) |
32 |
12 13
|
ercl2 |
|- ( ( A .~ C /\ B .~ D ) -> C e. ( _I ` Word ( I X. 2o ) ) ) |
33 |
32 25
|
eleqtrd |
|- ( ( A .~ C /\ B .~ D ) -> C e. ( Base ` M ) ) |
34 |
12 27
|
ercl2 |
|- ( ( A .~ C /\ B .~ D ) -> D e. ( _I ` Word ( I X. 2o ) ) ) |
35 |
34 25
|
eleqtrd |
|- ( ( A .~ C /\ B .~ D ) -> D e. ( Base ` M ) ) |
36 |
2 22 4
|
frmdadd |
|- ( ( C e. ( Base ` M ) /\ D e. ( Base ` M ) ) -> ( C .+ D ) = ( C ++ D ) ) |
37 |
33 35 36
|
syl2anc |
|- ( ( A .~ C /\ B .~ D ) -> ( C .+ D ) = ( C ++ D ) ) |
38 |
10 31 37
|
3brtr4d |
|- ( ( A .~ C /\ B .~ D ) -> ( A .+ B ) .~ ( C .+ D ) ) |