Metamath Proof Explorer


Theorem lpi0

Description: The zero ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015)

Ref Expression
Hypotheses lpival.p 𝑃 = ( LPIdeal ‘ 𝑅 )
lpi0.z 0 = ( 0g𝑅 )
Assertion lpi0 ( 𝑅 ∈ Ring → { 0 } ∈ 𝑃 )

Proof

Step Hyp Ref Expression
1 lpival.p 𝑃 = ( LPIdeal ‘ 𝑅 )
2 lpi0.z 0 = ( 0g𝑅 )
3 eqid ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
4 3 2 ring0cl ( 𝑅 ∈ Ring → 0 ∈ ( Base ‘ 𝑅 ) )
5 eqid ( RSpan ‘ 𝑅 ) = ( RSpan ‘ 𝑅 )
6 5 2 rsp0 ( 𝑅 ∈ Ring → ( ( RSpan ‘ 𝑅 ) ‘ { 0 } ) = { 0 } )
7 6 eqcomd ( 𝑅 ∈ Ring → { 0 } = ( ( RSpan ‘ 𝑅 ) ‘ { 0 } ) )
8 sneq ( 𝑔 = 0 → { 𝑔 } = { 0 } )
9 8 fveq2d ( 𝑔 = 0 → ( ( RSpan ‘ 𝑅 ) ‘ { 𝑔 } ) = ( ( RSpan ‘ 𝑅 ) ‘ { 0 } ) )
10 9 rspceeqv ( ( 0 ∈ ( Base ‘ 𝑅 ) ∧ { 0 } = ( ( RSpan ‘ 𝑅 ) ‘ { 0 } ) ) → ∃ 𝑔 ∈ ( Base ‘ 𝑅 ) { 0 } = ( ( RSpan ‘ 𝑅 ) ‘ { 𝑔 } ) )
11 4 7 10 syl2anc ( 𝑅 ∈ Ring → ∃ 𝑔 ∈ ( Base ‘ 𝑅 ) { 0 } = ( ( RSpan ‘ 𝑅 ) ‘ { 𝑔 } ) )
12 1 5 3 islpidl ( 𝑅 ∈ Ring → ( { 0 } ∈ 𝑃 ↔ ∃ 𝑔 ∈ ( Base ‘ 𝑅 ) { 0 } = ( ( RSpan ‘ 𝑅 ) ‘ { 𝑔 } ) ) )
13 11 12 mpbird ( 𝑅 ∈ Ring → { 0 } ∈ 𝑃 )