| Step |
Hyp |
Ref |
Expression |
| 1 |
|
rpnnen1lem.1 |
|- T = { n e. ZZ | ( n / k ) < x } |
| 2 |
|
rpnnen1lem.2 |
|- F = ( x e. RR |-> ( k e. NN |-> ( sup ( T , RR , < ) / k ) ) ) |
| 3 |
|
rpnnen1lem.n |
|- NN e. _V |
| 4 |
|
rpnnen1lem.q |
|- QQ e. _V |
| 5 |
|
ovex |
|- ( QQ ^m NN ) e. _V |
| 6 |
1 2 3 4
|
rpnnen1lem1 |
|- ( x e. RR -> ( F ` x ) e. ( QQ ^m NN ) ) |
| 7 |
|
rneq |
|- ( ( F ` x ) = ( F ` y ) -> ran ( F ` x ) = ran ( F ` y ) ) |
| 8 |
7
|
supeq1d |
|- ( ( F ` x ) = ( F ` y ) -> sup ( ran ( F ` x ) , RR , < ) = sup ( ran ( F ` y ) , RR , < ) ) |
| 9 |
1 2 3 4
|
rpnnen1lem5 |
|- ( x e. RR -> sup ( ran ( F ` x ) , RR , < ) = x ) |
| 10 |
|
fveq2 |
|- ( x = y -> ( F ` x ) = ( F ` y ) ) |
| 11 |
10
|
rneqd |
|- ( x = y -> ran ( F ` x ) = ran ( F ` y ) ) |
| 12 |
11
|
supeq1d |
|- ( x = y -> sup ( ran ( F ` x ) , RR , < ) = sup ( ran ( F ` y ) , RR , < ) ) |
| 13 |
|
id |
|- ( x = y -> x = y ) |
| 14 |
12 13
|
eqeq12d |
|- ( x = y -> ( sup ( ran ( F ` x ) , RR , < ) = x <-> sup ( ran ( F ` y ) , RR , < ) = y ) ) |
| 15 |
14 9
|
vtoclga |
|- ( y e. RR -> sup ( ran ( F ` y ) , RR , < ) = y ) |
| 16 |
9 15
|
eqeqan12d |
|- ( ( x e. RR /\ y e. RR ) -> ( sup ( ran ( F ` x ) , RR , < ) = sup ( ran ( F ` y ) , RR , < ) <-> x = y ) ) |
| 17 |
8 16
|
imbitrid |
|- ( ( x e. RR /\ y e. RR ) -> ( ( F ` x ) = ( F ` y ) -> x = y ) ) |
| 18 |
17 10
|
impbid1 |
|- ( ( x e. RR /\ y e. RR ) -> ( ( F ` x ) = ( F ` y ) <-> x = y ) ) |
| 19 |
6 18
|
dom2 |
|- ( ( QQ ^m NN ) e. _V -> RR ~<_ ( QQ ^m NN ) ) |
| 20 |
5 19
|
ax-mp |
|- RR ~<_ ( QQ ^m NN ) |