Step |
Hyp |
Ref |
Expression |
1 |
|
resvsca.r |
|- R = ( W |`v A ) |
2 |
|
resvsca.f |
|- F = ( Scalar ` W ) |
3 |
|
resvsca.b |
|- B = ( Base ` F ) |
4 |
|
elex |
|- ( W e. X -> W e. _V ) |
5 |
|
elex |
|- ( A e. Y -> A e. _V ) |
6 |
|
ovex |
|- ( W sSet <. ( Scalar ` ndx ) , ( F |`s A ) >. ) e. _V |
7 |
|
ifcl |
|- ( ( W e. _V /\ ( W sSet <. ( Scalar ` ndx ) , ( F |`s A ) >. ) e. _V ) -> if ( B C_ A , W , ( W sSet <. ( Scalar ` ndx ) , ( F |`s A ) >. ) ) e. _V ) |
8 |
6 7
|
mpan2 |
|- ( W e. _V -> if ( B C_ A , W , ( W sSet <. ( Scalar ` ndx ) , ( F |`s A ) >. ) ) e. _V ) |
9 |
8
|
adantr |
|- ( ( W e. _V /\ A e. _V ) -> if ( B C_ A , W , ( W sSet <. ( Scalar ` ndx ) , ( F |`s A ) >. ) ) e. _V ) |
10 |
|
simpl |
|- ( ( w = W /\ x = A ) -> w = W ) |
11 |
10
|
fveq2d |
|- ( ( w = W /\ x = A ) -> ( Scalar ` w ) = ( Scalar ` W ) ) |
12 |
11 2
|
eqtr4di |
|- ( ( w = W /\ x = A ) -> ( Scalar ` w ) = F ) |
13 |
12
|
fveq2d |
|- ( ( w = W /\ x = A ) -> ( Base ` ( Scalar ` w ) ) = ( Base ` F ) ) |
14 |
13 3
|
eqtr4di |
|- ( ( w = W /\ x = A ) -> ( Base ` ( Scalar ` w ) ) = B ) |
15 |
|
simpr |
|- ( ( w = W /\ x = A ) -> x = A ) |
16 |
14 15
|
sseq12d |
|- ( ( w = W /\ x = A ) -> ( ( Base ` ( Scalar ` w ) ) C_ x <-> B C_ A ) ) |
17 |
12 15
|
oveq12d |
|- ( ( w = W /\ x = A ) -> ( ( Scalar ` w ) |`s x ) = ( F |`s A ) ) |
18 |
17
|
opeq2d |
|- ( ( w = W /\ x = A ) -> <. ( Scalar ` ndx ) , ( ( Scalar ` w ) |`s x ) >. = <. ( Scalar ` ndx ) , ( F |`s A ) >. ) |
19 |
10 18
|
oveq12d |
|- ( ( w = W /\ x = A ) -> ( w sSet <. ( Scalar ` ndx ) , ( ( Scalar ` w ) |`s x ) >. ) = ( W sSet <. ( Scalar ` ndx ) , ( F |`s A ) >. ) ) |
20 |
16 10 19
|
ifbieq12d |
|- ( ( w = W /\ x = A ) -> if ( ( Base ` ( Scalar ` w ) ) C_ x , w , ( w sSet <. ( Scalar ` ndx ) , ( ( Scalar ` w ) |`s x ) >. ) ) = if ( B C_ A , W , ( W sSet <. ( Scalar ` ndx ) , ( F |`s A ) >. ) ) ) |
21 |
|
df-resv |
|- |`v = ( w e. _V , x e. _V |-> if ( ( Base ` ( Scalar ` w ) ) C_ x , w , ( w sSet <. ( Scalar ` ndx ) , ( ( Scalar ` w ) |`s x ) >. ) ) ) |
22 |
20 21
|
ovmpoga |
|- ( ( W e. _V /\ A e. _V /\ if ( B C_ A , W , ( W sSet <. ( Scalar ` ndx ) , ( F |`s A ) >. ) ) e. _V ) -> ( W |`v A ) = if ( B C_ A , W , ( W sSet <. ( Scalar ` ndx ) , ( F |`s A ) >. ) ) ) |
23 |
9 22
|
mpd3an3 |
|- ( ( W e. _V /\ A e. _V ) -> ( W |`v A ) = if ( B C_ A , W , ( W sSet <. ( Scalar ` ndx ) , ( F |`s A ) >. ) ) ) |
24 |
4 5 23
|
syl2an |
|- ( ( W e. X /\ A e. Y ) -> ( W |`v A ) = if ( B C_ A , W , ( W sSet <. ( Scalar ` ndx ) , ( F |`s A ) >. ) ) ) |
25 |
1 24
|
syl5eq |
|- ( ( W e. X /\ A e. Y ) -> R = if ( B C_ A , W , ( W sSet <. ( Scalar ` ndx ) , ( F |`s A ) >. ) ) ) |