| Step |
Hyp |
Ref |
Expression |
| 1 |
|
frlmvplusgvalc.f |
|- F = ( R freeLMod I ) |
| 2 |
|
frlmvplusgvalc.b |
|- B = ( Base ` F ) |
| 3 |
|
frlmvplusgvalc.r |
|- ( ph -> R e. V ) |
| 4 |
|
frlmvplusgvalc.i |
|- ( ph -> I e. W ) |
| 5 |
|
frlmvplusgvalc.x |
|- ( ph -> X e. B ) |
| 6 |
|
frlmvplusgvalc.y |
|- ( ph -> Y e. B ) |
| 7 |
|
frlmvplusgvalc.j |
|- ( ph -> J e. I ) |
| 8 |
|
frlmvplusgvalc.a |
|- .+ = ( +g ` R ) |
| 9 |
|
frlmvplusgvalc.p |
|- .+b = ( +g ` F ) |
| 10 |
1 2 3 4 5 6 8 9
|
frlmplusgval |
|- ( ph -> ( X .+b Y ) = ( X oF .+ Y ) ) |
| 11 |
10
|
fveq1d |
|- ( ph -> ( ( X .+b Y ) ` J ) = ( ( X oF .+ Y ) ` J ) ) |
| 12 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
| 13 |
1 12 2
|
frlmbasmap |
|- ( ( I e. W /\ X e. B ) -> X e. ( ( Base ` R ) ^m I ) ) |
| 14 |
4 5 13
|
syl2anc |
|- ( ph -> X e. ( ( Base ` R ) ^m I ) ) |
| 15 |
|
fvexd |
|- ( ph -> ( Base ` R ) e. _V ) |
| 16 |
15 4
|
elmapd |
|- ( ph -> ( X e. ( ( Base ` R ) ^m I ) <-> X : I --> ( Base ` R ) ) ) |
| 17 |
14 16
|
mpbid |
|- ( ph -> X : I --> ( Base ` R ) ) |
| 18 |
17
|
ffnd |
|- ( ph -> X Fn I ) |
| 19 |
1 12 2
|
frlmbasmap |
|- ( ( I e. W /\ Y e. B ) -> Y e. ( ( Base ` R ) ^m I ) ) |
| 20 |
4 6 19
|
syl2anc |
|- ( ph -> Y e. ( ( Base ` R ) ^m I ) ) |
| 21 |
15 4
|
elmapd |
|- ( ph -> ( Y e. ( ( Base ` R ) ^m I ) <-> Y : I --> ( Base ` R ) ) ) |
| 22 |
20 21
|
mpbid |
|- ( ph -> Y : I --> ( Base ` R ) ) |
| 23 |
22
|
ffnd |
|- ( ph -> Y Fn I ) |
| 24 |
|
fnfvof |
|- ( ( ( X Fn I /\ Y Fn I ) /\ ( I e. W /\ J e. I ) ) -> ( ( X oF .+ Y ) ` J ) = ( ( X ` J ) .+ ( Y ` J ) ) ) |
| 25 |
18 23 4 7 24
|
syl22anc |
|- ( ph -> ( ( X oF .+ Y ) ` J ) = ( ( X ` J ) .+ ( Y ` J ) ) ) |
| 26 |
11 25
|
eqtrd |
|- ( ph -> ( ( X .+b Y ) ` J ) = ( ( X ` J ) .+ ( Y ` J ) ) ) |