| Step |
Hyp |
Ref |
Expression |
| 1 |
|
1xr |
|- 1 e. RR* |
| 2 |
|
mnfxr |
|- -oo e. RR* |
| 3 |
|
pnfxr |
|- +oo e. RR* |
| 4 |
1 2 3
|
3pm3.2i |
|- ( 1 e. RR* /\ -oo e. RR* /\ +oo e. RR* ) |
| 5 |
|
xaddcom |
|- ( ( 1 e. RR* /\ -oo e. RR* ) -> ( 1 +e -oo ) = ( -oo +e 1 ) ) |
| 6 |
1 2 5
|
mp2an |
|- ( 1 +e -oo ) = ( -oo +e 1 ) |
| 7 |
|
1re |
|- 1 e. RR |
| 8 |
|
renepnf |
|- ( 1 e. RR -> 1 =/= +oo ) |
| 9 |
7 8
|
ax-mp |
|- 1 =/= +oo |
| 10 |
|
xaddmnf2 |
|- ( ( 1 e. RR* /\ 1 =/= +oo ) -> ( -oo +e 1 ) = -oo ) |
| 11 |
1 9 10
|
mp2an |
|- ( -oo +e 1 ) = -oo |
| 12 |
6 11
|
eqtri |
|- ( 1 +e -oo ) = -oo |
| 13 |
12
|
oveq1i |
|- ( ( 1 +e -oo ) +e +oo ) = ( -oo +e +oo ) |
| 14 |
|
mnfaddpnf |
|- ( -oo +e +oo ) = 0 |
| 15 |
13 14
|
eqtri |
|- ( ( 1 +e -oo ) +e +oo ) = 0 |
| 16 |
|
0ne1 |
|- 0 =/= 1 |
| 17 |
15 16
|
eqnetri |
|- ( ( 1 +e -oo ) +e +oo ) =/= 1 |
| 18 |
14
|
oveq2i |
|- ( 1 +e ( -oo +e +oo ) ) = ( 1 +e 0 ) |
| 19 |
|
xaddrid |
|- ( 1 e. RR* -> ( 1 +e 0 ) = 1 ) |
| 20 |
1 19
|
ax-mp |
|- ( 1 +e 0 ) = 1 |
| 21 |
18 20
|
eqtri |
|- ( 1 +e ( -oo +e +oo ) ) = 1 |
| 22 |
17 21
|
neeqtrri |
|- ( ( 1 +e -oo ) +e +oo ) =/= ( 1 +e ( -oo +e +oo ) ) |
| 23 |
|
xrsbas |
|- RR* = ( Base ` RR*s ) |
| 24 |
|
xrsadd |
|- +e = ( +g ` RR*s ) |
| 25 |
23 24
|
isnsgrp |
|- ( ( 1 e. RR* /\ -oo e. RR* /\ +oo e. RR* ) -> ( ( ( 1 +e -oo ) +e +oo ) =/= ( 1 +e ( -oo +e +oo ) ) -> RR*s e/ Smgrp ) ) |
| 26 |
4 22 25
|
mp2 |
|- RR*s e/ Smgrp |