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 ) ) ) |