Step |
Hyp |
Ref |
Expression |
1 |
|
dipdir.1 |
|- X = ( BaseSet ` U ) |
2 |
|
dipdir.2 |
|- G = ( +v ` U ) |
3 |
|
dipdir.7 |
|- P = ( .iOLD ` U ) |
4 |
|
fveq2 |
|- ( U = if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) -> ( BaseSet ` U ) = ( BaseSet ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) ) |
5 |
1 4
|
syl5eq |
|- ( U = if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) -> X = ( BaseSet ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) ) |
6 |
5
|
eleq2d |
|- ( U = if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) -> ( A e. X <-> A e. ( BaseSet ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) ) ) |
7 |
5
|
eleq2d |
|- ( U = if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) -> ( B e. X <-> B e. ( BaseSet ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) ) ) |
8 |
5
|
eleq2d |
|- ( U = if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) -> ( C e. X <-> C e. ( BaseSet ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) ) ) |
9 |
6 7 8
|
3anbi123d |
|- ( U = if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) -> ( ( A e. X /\ B e. X /\ C e. X ) <-> ( A e. ( BaseSet ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) /\ B e. ( BaseSet ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) /\ C e. ( BaseSet ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) ) ) ) |
10 |
|
fveq2 |
|- ( U = if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) -> ( +v ` U ) = ( +v ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) ) |
11 |
2 10
|
syl5eq |
|- ( U = if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) -> G = ( +v ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) ) |
12 |
11
|
oveqd |
|- ( U = if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) -> ( A G B ) = ( A ( +v ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) B ) ) |
13 |
12
|
oveq1d |
|- ( U = if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) -> ( ( A G B ) P C ) = ( ( A ( +v ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) B ) P C ) ) |
14 |
|
fveq2 |
|- ( U = if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) -> ( .iOLD ` U ) = ( .iOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) ) |
15 |
3 14
|
syl5eq |
|- ( U = if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) -> P = ( .iOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) ) |
16 |
15
|
oveqd |
|- ( U = if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) -> ( ( A ( +v ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) B ) P C ) = ( ( A ( +v ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) B ) ( .iOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) C ) ) |
17 |
13 16
|
eqtrd |
|- ( U = if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) -> ( ( A G B ) P C ) = ( ( A ( +v ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) B ) ( .iOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) C ) ) |
18 |
15
|
oveqd |
|- ( U = if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) -> ( A P C ) = ( A ( .iOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) C ) ) |
19 |
15
|
oveqd |
|- ( U = if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) -> ( B P C ) = ( B ( .iOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) C ) ) |
20 |
18 19
|
oveq12d |
|- ( U = if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) -> ( ( A P C ) + ( B P C ) ) = ( ( A ( .iOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) C ) + ( B ( .iOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) C ) ) ) |
21 |
17 20
|
eqeq12d |
|- ( U = if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) -> ( ( ( A G B ) P C ) = ( ( A P C ) + ( B P C ) ) <-> ( ( A ( +v ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) B ) ( .iOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) C ) = ( ( A ( .iOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) C ) + ( B ( .iOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) C ) ) ) ) |
22 |
9 21
|
imbi12d |
|- ( U = if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) -> ( ( ( A e. X /\ B e. X /\ C e. X ) -> ( ( A G B ) P C ) = ( ( A P C ) + ( B P C ) ) ) <-> ( ( A e. ( BaseSet ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) /\ B e. ( BaseSet ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) /\ C e. ( BaseSet ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) ) -> ( ( A ( +v ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) B ) ( .iOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) C ) = ( ( A ( .iOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) C ) + ( B ( .iOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) C ) ) ) ) ) |
23 |
|
eqid |
|- ( BaseSet ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) = ( BaseSet ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) |
24 |
|
eqid |
|- ( +v ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) = ( +v ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) |
25 |
|
eqid |
|- ( .sOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) = ( .sOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) |
26 |
|
eqid |
|- ( .iOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) = ( .iOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) |
27 |
|
elimphu |
|- if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) e. CPreHilOLD |
28 |
23 24 25 26 27
|
ipdiri |
|- ( ( A e. ( BaseSet ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) /\ B e. ( BaseSet ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) /\ C e. ( BaseSet ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) ) -> ( ( A ( +v ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) B ) ( .iOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) C ) = ( ( A ( .iOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) C ) + ( B ( .iOLD ` if ( U e. CPreHilOLD , U , <. <. + , x. >. , abs >. ) ) C ) ) ) |
29 |
22 28
|
dedth |
|- ( U e. CPreHilOLD -> ( ( A e. X /\ B e. X /\ C e. X ) -> ( ( A G B ) P C ) = ( ( A P C ) + ( B P C ) ) ) ) |
30 |
29
|
imp |
|- ( ( U e. CPreHilOLD /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) P C ) = ( ( A P C ) + ( B P C ) ) ) |