Metamath Proof Explorer
Description: The real numbers form a star ring. (Contributed by Thierry Arnoux, 19-Apr-2019) (Proof shortened by Thierry Arnoux, 11-Jan-2025)
|
|
Ref |
Expression |
|
Assertion |
resrng |
⊢ ℝfld ∈ *-Ring |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
rebase |
⊢ ℝ = ( Base ‘ ℝfld ) |
2 |
|
refldcj |
⊢ ∗ = ( *𝑟 ‘ ℝfld ) |
3 |
|
refld |
⊢ ℝfld ∈ Field |
4 |
3
|
a1i |
⊢ ( ⊤ → ℝfld ∈ Field ) |
5 |
4
|
fldcrngd |
⊢ ( ⊤ → ℝfld ∈ CRing ) |
6 |
|
cjre |
⊢ ( 𝑥 ∈ ℝ → ( ∗ ‘ 𝑥 ) = 𝑥 ) |
7 |
6
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ ) → ( ∗ ‘ 𝑥 ) = 𝑥 ) |
8 |
1 2 5 7
|
idsrngd |
⊢ ( ⊤ → ℝfld ∈ *-Ring ) |
9 |
8
|
mptru |
⊢ ℝfld ∈ *-Ring |