Step |
Hyp |
Ref |
Expression |
1 |
|
lpival.p |
⊢ 𝑃 = ( LPIdeal ‘ 𝑅 ) |
2 |
|
lpi0.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
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 ) |