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Metamath Proof Explorer

Theorem pidlnz

Description: A principal ideal generated by a nonzero element is not the zero ideal. (Contributed by Thierry Arnoux, 11-Apr-2024)

Ref Expression
Hypotheses pidlnz.1 ⊢ 𝐵 = ( Base ‘ 𝑅 )
pidlnz.2 ⊢ 0 = ( 0g ‘ 𝑅 )
pidlnz.3 ⊢ 𝐾 = ( RSpan ‘ 𝑅 )
Assertion pidlnz ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( 𝐾 ‘ { 𝑋 } ) ≠ { 0 } )

Proof

Step Hyp Ref Expression
1 pidlnz.1 ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 pidlnz.2 ⊢ 0 = ( 0g ‘ 𝑅 )
3 pidlnz.3 ⊢ 𝐾 = ( RSpan ‘ 𝑅 )
4 simpl1 ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ ( 𝐾 ‘ { 𝑋 } ) = { 0 } ) → 𝑅 ∈ Ring )
5 simpl2 ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ ( 𝐾 ‘ { 𝑋 } ) = { 0 } ) → 𝑋 ∈ 𝐵 )
6 1 3 rspsnid ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ ( 𝐾 ‘ { 𝑋 } ) )
7 4 5 6 syl2anc ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ ( 𝐾 ‘ { 𝑋 } ) = { 0 } ) → 𝑋 ∈ ( 𝐾 ‘ { 𝑋 } ) )
8 simpr ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ ( 𝐾 ‘ { 𝑋 } ) = { 0 } ) → ( 𝐾 ‘ { 𝑋 } ) = { 0 } )
9 7 8 eleqtrd ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ ( 𝐾 ‘ { 𝑋 } ) = { 0 } ) → 𝑋 ∈ { 0 } )
10 elsni ⊢ ( 𝑋 ∈ { 0 } → 𝑋 = 0 )
11 9 10 syl ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ ( 𝐾 ‘ { 𝑋 } ) = { 0 } ) → 𝑋 = 0 )
12 simpl3 ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ ( 𝐾 ‘ { 𝑋 } ) = { 0 } ) → 𝑋 ≠ 0 )
13 12 neneqd ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ ( 𝐾 ‘ { 𝑋 } ) = { 0 } ) → ¬ 𝑋 = 0 )
14 11 13 pm2.65da ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ¬ ( 𝐾 ‘ { 𝑋 } ) = { 0 } )
15 14 neqned ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( 𝐾 ‘ { 𝑋 } ) ≠ { 0 } )