Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
|- ( RR*s |`s ( RR* \ { -oo } ) ) = ( RR*s |`s ( RR* \ { -oo } ) ) |
2 |
1
|
xrs1mnd |
|- ( RR*s |`s ( RR* \ { -oo } ) ) e. Mnd |
3 |
|
xrge0cmn |
|- ( RR*s |`s ( 0 [,] +oo ) ) e. CMnd |
4 |
|
cmnmnd |
|- ( ( RR*s |`s ( 0 [,] +oo ) ) e. CMnd -> ( RR*s |`s ( 0 [,] +oo ) ) e. Mnd ) |
5 |
3 4
|
ax-mp |
|- ( RR*s |`s ( 0 [,] +oo ) ) e. Mnd |
6 |
|
mnflt0 |
|- -oo < 0 |
7 |
|
mnfxr |
|- -oo e. RR* |
8 |
|
0xr |
|- 0 e. RR* |
9 |
|
xrltnle |
|- ( ( -oo e. RR* /\ 0 e. RR* ) -> ( -oo < 0 <-> -. 0 <_ -oo ) ) |
10 |
7 8 9
|
mp2an |
|- ( -oo < 0 <-> -. 0 <_ -oo ) |
11 |
6 10
|
mpbi |
|- -. 0 <_ -oo |
12 |
11
|
intnan |
|- -. ( -oo e. RR* /\ 0 <_ -oo ) |
13 |
|
elxrge0 |
|- ( -oo e. ( 0 [,] +oo ) <-> ( -oo e. RR* /\ 0 <_ -oo ) ) |
14 |
12 13
|
mtbir |
|- -. -oo e. ( 0 [,] +oo ) |
15 |
|
difsn |
|- ( -. -oo e. ( 0 [,] +oo ) -> ( ( 0 [,] +oo ) \ { -oo } ) = ( 0 [,] +oo ) ) |
16 |
14 15
|
ax-mp |
|- ( ( 0 [,] +oo ) \ { -oo } ) = ( 0 [,] +oo ) |
17 |
|
iccssxr |
|- ( 0 [,] +oo ) C_ RR* |
18 |
|
ssdif |
|- ( ( 0 [,] +oo ) C_ RR* -> ( ( 0 [,] +oo ) \ { -oo } ) C_ ( RR* \ { -oo } ) ) |
19 |
17 18
|
ax-mp |
|- ( ( 0 [,] +oo ) \ { -oo } ) C_ ( RR* \ { -oo } ) |
20 |
16 19
|
eqsstrri |
|- ( 0 [,] +oo ) C_ ( RR* \ { -oo } ) |
21 |
|
0e0iccpnf |
|- 0 e. ( 0 [,] +oo ) |
22 |
|
difss |
|- ( RR* \ { -oo } ) C_ RR* |
23 |
|
df-ss |
|- ( ( RR* \ { -oo } ) C_ RR* <-> ( ( RR* \ { -oo } ) i^i RR* ) = ( RR* \ { -oo } ) ) |
24 |
22 23
|
mpbi |
|- ( ( RR* \ { -oo } ) i^i RR* ) = ( RR* \ { -oo } ) |
25 |
|
xrex |
|- RR* e. _V |
26 |
|
difexg |
|- ( RR* e. _V -> ( RR* \ { -oo } ) e. _V ) |
27 |
25 26
|
ax-mp |
|- ( RR* \ { -oo } ) e. _V |
28 |
|
xrsbas |
|- RR* = ( Base ` RR*s ) |
29 |
1 28
|
ressbas |
|- ( ( RR* \ { -oo } ) e. _V -> ( ( RR* \ { -oo } ) i^i RR* ) = ( Base ` ( RR*s |`s ( RR* \ { -oo } ) ) ) ) |
30 |
27 29
|
ax-mp |
|- ( ( RR* \ { -oo } ) i^i RR* ) = ( Base ` ( RR*s |`s ( RR* \ { -oo } ) ) ) |
31 |
24 30
|
eqtr3i |
|- ( RR* \ { -oo } ) = ( Base ` ( RR*s |`s ( RR* \ { -oo } ) ) ) |
32 |
1
|
xrs10 |
|- 0 = ( 0g ` ( RR*s |`s ( RR* \ { -oo } ) ) ) |
33 |
|
ovex |
|- ( 0 [,] +oo ) e. _V |
34 |
|
ressress |
|- ( ( ( RR* \ { -oo } ) e. _V /\ ( 0 [,] +oo ) e. _V ) -> ( ( RR*s |`s ( RR* \ { -oo } ) ) |`s ( 0 [,] +oo ) ) = ( RR*s |`s ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) ) ) |
35 |
27 33 34
|
mp2an |
|- ( ( RR*s |`s ( RR* \ { -oo } ) ) |`s ( 0 [,] +oo ) ) = ( RR*s |`s ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) ) |
36 |
|
dfss |
|- ( ( 0 [,] +oo ) C_ ( RR* \ { -oo } ) <-> ( 0 [,] +oo ) = ( ( 0 [,] +oo ) i^i ( RR* \ { -oo } ) ) ) |
37 |
20 36
|
mpbi |
|- ( 0 [,] +oo ) = ( ( 0 [,] +oo ) i^i ( RR* \ { -oo } ) ) |
38 |
|
incom |
|- ( ( 0 [,] +oo ) i^i ( RR* \ { -oo } ) ) = ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) |
39 |
37 38
|
eqtr2i |
|- ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) = ( 0 [,] +oo ) |
40 |
39
|
oveq2i |
|- ( RR*s |`s ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) ) = ( RR*s |`s ( 0 [,] +oo ) ) |
41 |
35 40
|
eqtr2i |
|- ( RR*s |`s ( 0 [,] +oo ) ) = ( ( RR*s |`s ( RR* \ { -oo } ) ) |`s ( 0 [,] +oo ) ) |
42 |
31 32 41
|
submnd0 |
|- ( ( ( ( RR*s |`s ( RR* \ { -oo } ) ) e. Mnd /\ ( RR*s |`s ( 0 [,] +oo ) ) e. Mnd ) /\ ( ( 0 [,] +oo ) C_ ( RR* \ { -oo } ) /\ 0 e. ( 0 [,] +oo ) ) ) -> 0 = ( 0g ` ( RR*s |`s ( 0 [,] +oo ) ) ) ) |
43 |
2 5 20 21 42
|
mp4an |
|- 0 = ( 0g ` ( RR*s |`s ( 0 [,] +oo ) ) ) |