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