Step |
Hyp |
Ref |
Expression |
1 |
|
reofld |
⊢ ℝfld ∈ 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 |
⊢ ℝfld ∈ 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 |
⊢ ℝfld ∈ 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 |
⊢ ℝfld ∈ Archi |