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 fld ∈ Archi

Proof

Step Hyp Ref Expression
1 reofld fld ∈ oField
2 rebase ℝ = ( Base ‘ ℝfld )
3 eqid ( ℤRHom ‘ ℝfld ) = ( ℤRHom ‘ ℝfld )
4 relt < = ( lt ‘ ℝfld )
5 2 3 4 isarchiofld ( ℝfld ∈ oField → ( ℝfld ∈ Archi ↔ ∀ 𝑥 ∈ ℝ ∃ 𝑛 ∈ ℕ 𝑥 < ( ( ℤRHom ‘ ℝfld ) ‘ 𝑛 ) ) )
6 1 5 ax-mp ( ℝfld ∈ Archi ↔ ∀ 𝑥 ∈ ℝ ∃ 𝑛 ∈ ℕ 𝑥 < ( ( ℤRHom ‘ ℝfld ) ‘ 𝑛 ) )
7 arch ( 𝑥 ∈ ℝ → ∃ 𝑛 ∈ ℕ 𝑥 < 𝑛 )
8 nnz ( 𝑛 ∈ ℕ → 𝑛 ∈ ℤ )
9 refld fld ∈ Field
10 isfld ( ℝfld ∈ Field ↔ ( ℝfld ∈ DivRing ∧ ℝfld ∈ CRing ) )
11 10 simplbi ( ℝfld ∈ Field → ℝfld ∈ DivRing )
12 drngring ( ℝfld ∈ DivRing → ℝfld ∈ Ring )
13 9 11 12 mp2b fld ∈ Ring
14 eqid ( .g ‘ ℝfld ) = ( .g ‘ ℝfld )
15 re1r 1 = ( 1r ‘ ℝfld )
16 3 14 15 zrhmulg ( ( ℝfld ∈ Ring ∧ 𝑛 ∈ ℤ ) → ( ( ℤRHom ‘ ℝfld ) ‘ 𝑛 ) = ( 𝑛 ( .g ‘ ℝfld ) 1 ) )
17 13 16 mpan ( 𝑛 ∈ ℤ → ( ( ℤRHom ‘ ℝfld ) ‘ 𝑛 ) = ( 𝑛 ( .g ‘ ℝfld ) 1 ) )
18 1re 1 ∈ ℝ
19 remulg ( ( 𝑛 ∈ ℤ ∧ 1 ∈ ℝ ) → ( 𝑛 ( .g ‘ ℝfld ) 1 ) = ( 𝑛 · 1 ) )
20 18 19 mpan2 ( 𝑛 ∈ ℤ → ( 𝑛 ( .g ‘ ℝfld ) 1 ) = ( 𝑛 · 1 ) )
21 zcn ( 𝑛 ∈ ℤ → 𝑛 ∈ ℂ )
22 21 mulid1d ( 𝑛 ∈ ℤ → ( 𝑛 · 1 ) = 𝑛 )
23 17 20 22 3eqtrd ( 𝑛 ∈ ℤ → ( ( ℤRHom ‘ ℝfld ) ‘ 𝑛 ) = 𝑛 )
24 23 breq2d ( 𝑛 ∈ ℤ → ( 𝑥 < ( ( ℤRHom ‘ ℝfld ) ‘ 𝑛 ) ↔ 𝑥 < 𝑛 ) )
25 8 24 syl ( 𝑛 ∈ ℕ → ( 𝑥 < ( ( ℤRHom ‘ ℝfld ) ‘ 𝑛 ) ↔ 𝑥 < 𝑛 ) )
26 25 rexbiia ( ∃ 𝑛 ∈ ℕ 𝑥 < ( ( ℤRHom ‘ ℝfld ) ‘ 𝑛 ) ↔ ∃ 𝑛 ∈ ℕ 𝑥 < 𝑛 )
27 7 26 sylibr ( 𝑥 ∈ ℝ → ∃ 𝑛 ∈ ℕ 𝑥 < ( ( ℤRHom ‘ ℝfld ) ‘ 𝑛 ) )
28 6 27 mprgbir fld ∈ Archi