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Metamath Proof Explorer

Theorem lpi1

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

Ref Expression
Hypotheses lpival.p ⊢ 𝑃 = ( LPIdeal ‘ 𝑅 )
lpi1.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
Assertion lpi1 ( 𝑅 ∈ Ring → 𝐵 ∈ 𝑃 )

Proof

Step Hyp Ref Expression
1 lpival.p ⊢ 𝑃 = ( LPIdeal ‘ 𝑅 )
2 lpi1.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
3 eqid ⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
4 2 3 ringidcl ⊢ ( 𝑅 ∈ Ring → ( 1r ‘ 𝑅 ) ∈ 𝐵 )
5 eqid ⊢ ( RSpan ‘ 𝑅 ) = ( RSpan ‘ 𝑅 )
6 5 2 3 rsp1 ⊢ ( 𝑅 ∈ Ring → ( ( RSpan ‘ 𝑅 ) ‘ { ( 1r ‘ 𝑅 ) } ) = 𝐵 )
7 6 eqcomd ⊢ ( 𝑅 ∈ Ring → 𝐵 = ( ( RSpan ‘ 𝑅 ) ‘ { ( 1r ‘ 𝑅 ) } ) )
8 sneq ⊢ ( 𝑔 = ( 1r ‘ 𝑅 ) → { 𝑔 } = { ( 1r ‘ 𝑅 ) } )
9 8 fveq2d ⊢ ( 𝑔 = ( 1r ‘ 𝑅 ) → ( ( RSpan ‘ 𝑅 ) ‘ { 𝑔 } ) = ( ( RSpan ‘ 𝑅 ) ‘ { ( 1r ‘ 𝑅 ) } ) )
10 9 rspceeqv ⊢ ( ( ( 1r ‘ 𝑅 ) ∈ 𝐵 ∧ 𝐵 = ( ( RSpan ‘ 𝑅 ) ‘ { ( 1r ‘ 𝑅 ) } ) ) → ∃ 𝑔 ∈ 𝐵 𝐵 = ( ( RSpan ‘ 𝑅 ) ‘ { 𝑔 } ) )
11 4 7 10 syl2anc ⊢ ( 𝑅 ∈ Ring → ∃ 𝑔 ∈ 𝐵 𝐵 = ( ( RSpan ‘ 𝑅 ) ‘ { 𝑔 } ) )
12 1 5 2 islpidl ⊢ ( 𝑅 ∈ Ring → ( 𝐵 ∈ 𝑃 ↔ ∃ 𝑔 ∈ 𝐵 𝐵 = ( ( RSpan ‘ 𝑅 ) ‘ { 𝑔 } ) ) )
13 11 12 mpbird ⊢ ( 𝑅 ∈ Ring → 𝐵 ∈ 𝑃 )