Metamath Proof Explorer


Theorem rearchi

Description: The field of the real numbers is Archimedean. See also arch . (Contributed by Thierry Arnoux, 9-Apr-2018)

Ref Expression
Assertion rearchi
|- RRfld e. Archi

Proof

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 mulid1d
 |-  ( 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