| Step |
Hyp |
Ref |
Expression |
| 1 |
|
reofld |
|- RRfld e. oField |
| 2 |
|
rebase |
|- RR = ( Base ` RRfld ) |
| 3 |
|
eqid |
|- ( ZRHom ` RRfld ) = ( ZRHom ` RRfld ) |
| 4 |
|
relt |
|- < = ( lt ` RRfld ) |
| 5 |
2 3 4
|
isarchiofld |
|- ( RRfld e. oField -> ( RRfld e. Archi <-> A. x e. RR E. n e. NN x < ( ( ZRHom ` RRfld ) ` n ) ) ) |
| 6 |
1 5
|
ax-mp |
|- ( RRfld e. Archi <-> A. x e. RR E. n e. NN x < ( ( ZRHom ` RRfld ) ` n ) ) |
| 7 |
|
arch |
|- ( x e. RR -> E. n e. NN x < n ) |
| 8 |
|
nnz |
|- ( n e. NN -> n e. ZZ ) |
| 9 |
|
refld |
|- RRfld e. Field |
| 10 |
|
isfld |
|- ( RRfld e. Field <-> ( RRfld e. DivRing /\ RRfld e. CRing ) ) |
| 11 |
10
|
simplbi |
|- ( RRfld e. Field -> RRfld e. DivRing ) |
| 12 |
|
drngring |
|- ( RRfld e. DivRing -> RRfld e. Ring ) |
| 13 |
9 11 12
|
mp2b |
|- RRfld e. Ring |
| 14 |
|
eqid |
|- ( .g ` RRfld ) = ( .g ` RRfld ) |
| 15 |
|
re1r |
|- 1 = ( 1r ` RRfld ) |
| 16 |
3 14 15
|
zrhmulg |
|- ( ( RRfld e. Ring /\ n e. ZZ ) -> ( ( ZRHom ` RRfld ) ` n ) = ( n ( .g ` RRfld ) 1 ) ) |
| 17 |
13 16
|
mpan |
|- ( n e. ZZ -> ( ( ZRHom ` RRfld ) ` n ) = ( n ( .g ` RRfld ) 1 ) ) |
| 18 |
|
1re |
|- 1 e. RR |
| 19 |
|
remulg |
|- ( ( n e. ZZ /\ 1 e. RR ) -> ( n ( .g ` RRfld ) 1 ) = ( n x. 1 ) ) |
| 20 |
18 19
|
mpan2 |
|- ( n e. ZZ -> ( n ( .g ` RRfld ) 1 ) = ( n x. 1 ) ) |
| 21 |
|
zcn |
|- ( n e. ZZ -> n e. CC ) |
| 22 |
21
|
mulridd |
|- ( n e. ZZ -> ( n x. 1 ) = n ) |
| 23 |
17 20 22
|
3eqtrd |
|- ( n e. ZZ -> ( ( ZRHom ` RRfld ) ` n ) = n ) |
| 24 |
23
|
breq2d |
|- ( n e. ZZ -> ( x < ( ( ZRHom ` RRfld ) ` n ) <-> x < n ) ) |
| 25 |
8 24
|
syl |
|- ( n e. NN -> ( x < ( ( ZRHom ` RRfld ) ` n ) <-> x < n ) ) |
| 26 |
25
|
rexbiia |
|- ( E. n e. NN x < ( ( ZRHom ` RRfld ) ` n ) <-> E. n e. NN x < n ) |
| 27 |
7 26
|
sylibr |
|- ( x e. RR -> E. n e. NN x < ( ( ZRHom ` RRfld ) ` n ) ) |
| 28 |
6 27
|
mprgbir |
|- RRfld e. Archi |