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Metamath Proof Explorer

Theorem primefld1cl

Description: The prime field contains the unity element of the division ring. (Contributed by Thierry Arnoux, 22-Aug-2023)

Ref Expression
Hypothesis primefld1cl.1 ⊢ 1 = ( 1r ‘ 𝑅 )
Assertion primefld1cl ( 𝑅 ∈ DivRing → 1 ∈ ∩ ( SubDRing ‘ 𝑅 ) )

Proof

Step Hyp Ref Expression
1 primefld1cl.1 ⊢ 1 = ( 1r ‘ 𝑅 )
2 issdrg ⊢ ( 𝑠 ∈ ( SubDRing ‘ 𝑅 ) ↔ ( 𝑅 ∈ DivRing ∧ 𝑠 ∈ ( SubRing ‘ 𝑅 ) ∧ ( 𝑅 ↾s 𝑠 ) ∈ DivRing ) )
3 2 simp2bi ⊢ ( 𝑠 ∈ ( SubDRing ‘ 𝑅 ) → 𝑠 ∈ ( SubRing ‘ 𝑅 ) )
4 3 a1i ⊢ ( 𝑅 ∈ DivRing → ( 𝑠 ∈ ( SubDRing ‘ 𝑅 ) → 𝑠 ∈ ( SubRing ‘ 𝑅 ) ) )
5 4 ssrdv ⊢ ( 𝑅 ∈ DivRing → ( SubDRing ‘ 𝑅 ) ⊆ ( SubRing ‘ 𝑅 ) )
6 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
7 6 sdrgid ⊢ ( 𝑅 ∈ DivRing → ( Base ‘ 𝑅 ) ∈ ( SubDRing ‘ 𝑅 ) )
8 7 ne0d ⊢ ( 𝑅 ∈ DivRing → ( SubDRing ‘ 𝑅 ) ≠ ∅ )
9 subrgint ⊢ ( ( ( SubDRing ‘ 𝑅 ) ⊆ ( SubRing ‘ 𝑅 ) ∧ ( SubDRing ‘ 𝑅 ) ≠ ∅ ) → ∩ ( SubDRing ‘ 𝑅 ) ∈ ( SubRing ‘ 𝑅 ) )
10 5 8 9 syl2anc ⊢ ( 𝑅 ∈ DivRing → ∩ ( SubDRing ‘ 𝑅 ) ∈ ( SubRing ‘ 𝑅 ) )
11 1 subrg1cl ⊢ ( ∩ ( SubDRing ‘ 𝑅 ) ∈ ( SubRing ‘ 𝑅 ) → 1 ∈ ∩ ( SubDRing ‘ 𝑅 ) )
12 10 11 syl ⊢ ( 𝑅 ∈ DivRing → 1 ∈ ∩ ( SubDRing ‘ 𝑅 ) )