| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ldualvaddval.v |
|- V = ( Base ` W ) |
| 2 |
|
ldualvaddval.r |
|- R = ( Scalar ` W ) |
| 3 |
|
ldualvaddval.a |
|- .+ = ( +g ` R ) |
| 4 |
|
ldualvaddval.f |
|- F = ( LFnl ` W ) |
| 5 |
|
ldualvaddval.d |
|- D = ( LDual ` W ) |
| 6 |
|
ldualvaddval.p |
|- .+b = ( +g ` D ) |
| 7 |
|
ldualvaddval.w |
|- ( ph -> W e. LMod ) |
| 8 |
|
ldualvaddval.g |
|- ( ph -> G e. F ) |
| 9 |
|
ldualvaddval.h |
|- ( ph -> H e. F ) |
| 10 |
|
ldualvaddval.x |
|- ( ph -> X e. V ) |
| 11 |
4 2 3 5 6 7 8 9
|
ldualvadd |
|- ( ph -> ( G .+b H ) = ( G oF .+ H ) ) |
| 12 |
11
|
fveq1d |
|- ( ph -> ( ( G .+b H ) ` X ) = ( ( G oF .+ H ) ` X ) ) |
| 13 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
| 14 |
2 13 1 4
|
lflf |
|- ( ( W e. LMod /\ G e. F ) -> G : V --> ( Base ` R ) ) |
| 15 |
14
|
ffnd |
|- ( ( W e. LMod /\ G e. F ) -> G Fn V ) |
| 16 |
7 8 15
|
syl2anc |
|- ( ph -> G Fn V ) |
| 17 |
2 13 1 4
|
lflf |
|- ( ( W e. LMod /\ H e. F ) -> H : V --> ( Base ` R ) ) |
| 18 |
17
|
ffnd |
|- ( ( W e. LMod /\ H e. F ) -> H Fn V ) |
| 19 |
7 9 18
|
syl2anc |
|- ( ph -> H Fn V ) |
| 20 |
1
|
fvexi |
|- V e. _V |
| 21 |
20
|
a1i |
|- ( ph -> V e. _V ) |
| 22 |
|
inidm |
|- ( V i^i V ) = V |
| 23 |
|
eqidd |
|- ( ( ph /\ X e. V ) -> ( G ` X ) = ( G ` X ) ) |
| 24 |
|
eqidd |
|- ( ( ph /\ X e. V ) -> ( H ` X ) = ( H ` X ) ) |
| 25 |
16 19 21 21 22 23 24
|
ofval |
|- ( ( ph /\ X e. V ) -> ( ( G oF .+ H ) ` X ) = ( ( G ` X ) .+ ( H ` X ) ) ) |
| 26 |
10 25
|
mpdan |
|- ( ph -> ( ( G oF .+ H ) ` X ) = ( ( G ` X ) .+ ( H ` X ) ) ) |
| 27 |
12 26
|
eqtrd |
|- ( ph -> ( ( G .+b H ) ` X ) = ( ( G ` X ) .+ ( H ` X ) ) ) |