Step |
Hyp |
Ref |
Expression |
1 |
|
carsgval.1 |
|- ( ph -> O e. V ) |
2 |
|
carsgval.2 |
|- ( ph -> M : ~P O --> ( 0 [,] +oo ) ) |
3 |
|
baselcarsg.1 |
|- ( ph -> ( M ` (/) ) = 0 ) |
4 |
|
0ss |
|- (/) C_ O |
5 |
4
|
a1i |
|- ( ph -> (/) C_ O ) |
6 |
|
in0 |
|- ( e i^i (/) ) = (/) |
7 |
6
|
fveq2i |
|- ( M ` ( e i^i (/) ) ) = ( M ` (/) ) |
8 |
7 3
|
syl5eq |
|- ( ph -> ( M ` ( e i^i (/) ) ) = 0 ) |
9 |
|
dif0 |
|- ( e \ (/) ) = e |
10 |
9
|
fveq2i |
|- ( M ` ( e \ (/) ) ) = ( M ` e ) |
11 |
10
|
a1i |
|- ( ph -> ( M ` ( e \ (/) ) ) = ( M ` e ) ) |
12 |
8 11
|
oveq12d |
|- ( ph -> ( ( M ` ( e i^i (/) ) ) +e ( M ` ( e \ (/) ) ) ) = ( 0 +e ( M ` e ) ) ) |
13 |
12
|
adantr |
|- ( ( ph /\ e e. ~P O ) -> ( ( M ` ( e i^i (/) ) ) +e ( M ` ( e \ (/) ) ) ) = ( 0 +e ( M ` e ) ) ) |
14 |
|
iccssxr |
|- ( 0 [,] +oo ) C_ RR* |
15 |
2
|
ffvelrnda |
|- ( ( ph /\ e e. ~P O ) -> ( M ` e ) e. ( 0 [,] +oo ) ) |
16 |
14 15
|
sselid |
|- ( ( ph /\ e e. ~P O ) -> ( M ` e ) e. RR* ) |
17 |
|
xaddid2 |
|- ( ( M ` e ) e. RR* -> ( 0 +e ( M ` e ) ) = ( M ` e ) ) |
18 |
16 17
|
syl |
|- ( ( ph /\ e e. ~P O ) -> ( 0 +e ( M ` e ) ) = ( M ` e ) ) |
19 |
13 18
|
eqtrd |
|- ( ( ph /\ e e. ~P O ) -> ( ( M ` ( e i^i (/) ) ) +e ( M ` ( e \ (/) ) ) ) = ( M ` e ) ) |
20 |
19
|
ralrimiva |
|- ( ph -> A. e e. ~P O ( ( M ` ( e i^i (/) ) ) +e ( M ` ( e \ (/) ) ) ) = ( M ` e ) ) |
21 |
1 2
|
elcarsg |
|- ( ph -> ( (/) e. ( toCaraSiga ` M ) <-> ( (/) C_ O /\ A. e e. ~P O ( ( M ` ( e i^i (/) ) ) +e ( M ` ( e \ (/) ) ) ) = ( M ` e ) ) ) ) |
22 |
5 20 21
|
mpbir2and |
|- ( ph -> (/) e. ( toCaraSiga ` M ) ) |