| Step |
Hyp |
Ref |
Expression |
| 1 |
|
frmdval.m |
|- M = ( freeMnd ` I ) |
| 2 |
|
frmdval.b |
|- ( I e. V -> B = Word I ) |
| 3 |
|
frmdval.p |
|- .+ = ( ++ |` ( B X. B ) ) |
| 4 |
|
df-frmd |
|- freeMnd = ( i e. _V |-> { <. ( Base ` ndx ) , Word i >. , <. ( +g ` ndx ) , ( ++ |` ( Word i X. Word i ) ) >. } ) |
| 5 |
|
wrdeq |
|- ( i = I -> Word i = Word I ) |
| 6 |
2
|
eqcomd |
|- ( I e. V -> Word I = B ) |
| 7 |
5 6
|
sylan9eqr |
|- ( ( I e. V /\ i = I ) -> Word i = B ) |
| 8 |
7
|
opeq2d |
|- ( ( I e. V /\ i = I ) -> <. ( Base ` ndx ) , Word i >. = <. ( Base ` ndx ) , B >. ) |
| 9 |
7
|
sqxpeqd |
|- ( ( I e. V /\ i = I ) -> ( Word i X. Word i ) = ( B X. B ) ) |
| 10 |
9
|
reseq2d |
|- ( ( I e. V /\ i = I ) -> ( ++ |` ( Word i X. Word i ) ) = ( ++ |` ( B X. B ) ) ) |
| 11 |
10 3
|
eqtr4di |
|- ( ( I e. V /\ i = I ) -> ( ++ |` ( Word i X. Word i ) ) = .+ ) |
| 12 |
11
|
opeq2d |
|- ( ( I e. V /\ i = I ) -> <. ( +g ` ndx ) , ( ++ |` ( Word i X. Word i ) ) >. = <. ( +g ` ndx ) , .+ >. ) |
| 13 |
8 12
|
preq12d |
|- ( ( I e. V /\ i = I ) -> { <. ( Base ` ndx ) , Word i >. , <. ( +g ` ndx ) , ( ++ |` ( Word i X. Word i ) ) >. } = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. } ) |
| 14 |
|
elex |
|- ( I e. V -> I e. _V ) |
| 15 |
|
prex |
|- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. } e. _V |
| 16 |
15
|
a1i |
|- ( I e. V -> { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. } e. _V ) |
| 17 |
4 13 14 16
|
fvmptd2 |
|- ( I e. V -> ( freeMnd ` I ) = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. } ) |
| 18 |
1 17
|
eqtrid |
|- ( I e. V -> M = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. } ) |