Metamath Proof Explorer


Theorem rspsnid

Description: A principal ideal contains the element that generates it. (Contributed by Thierry Arnoux, 15-Jan-2024)

Ref Expression
Hypotheses rspsnid.b 𝐵 = ( Base ‘ 𝑅 )
rspsnid.k 𝐾 = ( RSpan ‘ 𝑅 )
Assertion rspsnid ( ( 𝑅 ∈ Ring ∧ 𝐺𝐵 ) → 𝐺 ∈ ( 𝐾 ‘ { 𝐺 } ) )

Proof

Step Hyp Ref Expression
1 rspsnid.b 𝐵 = ( Base ‘ 𝑅 )
2 rspsnid.k 𝐾 = ( RSpan ‘ 𝑅 )
3 snssi ( 𝐺𝐵 → { 𝐺 } ⊆ 𝐵 )
4 2 1 rspssid ( ( 𝑅 ∈ Ring ∧ { 𝐺 } ⊆ 𝐵 ) → { 𝐺 } ⊆ ( 𝐾 ‘ { 𝐺 } ) )
5 3 4 sylan2 ( ( 𝑅 ∈ Ring ∧ 𝐺𝐵 ) → { 𝐺 } ⊆ ( 𝐾 ‘ { 𝐺 } ) )
6 snssg ( 𝐺𝐵 → ( 𝐺 ∈ ( 𝐾 ‘ { 𝐺 } ) ↔ { 𝐺 } ⊆ ( 𝐾 ‘ { 𝐺 } ) ) )
7 6 adantl ( ( 𝑅 ∈ Ring ∧ 𝐺𝐵 ) → ( 𝐺 ∈ ( 𝐾 ‘ { 𝐺 } ) ↔ { 𝐺 } ⊆ ( 𝐾 ‘ { 𝐺 } ) ) )
8 5 7 mpbird ( ( 𝑅 ∈ Ring ∧ 𝐺𝐵 ) → 𝐺 ∈ ( 𝐾 ‘ { 𝐺 } ) )