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Metamath Proof Explorer

Theorem rspcl

Description: The span of a set of ring elements is an ideal. (Contributed by Stefan O'Rear, 3-Jan-2015) (Revised by Mario Carneiro, 2-Oct-2015)

Ref Expression
Hypotheses rspcl.k ⊢ 𝐾 = ( RSpan ‘ 𝑅 )
rspcl.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
rspcl.u ⊢ 𝑈 = ( LIdeal ‘ 𝑅 )
Assertion rspcl ( ( 𝑅 ∈ Ring ∧ 𝐺 ⊆ 𝐵 ) → ( 𝐾 ‘ 𝐺 ) ∈ 𝑈 )

Proof

Step Hyp Ref Expression
1 rspcl.k ⊢ 𝐾 = ( RSpan ‘ 𝑅 )
2 rspcl.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
3 rspcl.u ⊢ 𝑈 = ( LIdeal ‘ 𝑅 )
4 rlmlmod ⊢ ( 𝑅 ∈ Ring → ( ringLMod ‘ 𝑅 ) ∈ LMod )
5 rlmbas ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ ( ringLMod ‘ 𝑅 ) )
6 2 5 eqtri ⊢ 𝐵 = ( Base ‘ ( ringLMod ‘ 𝑅 ) )
7 lidlval ⊢ ( LIdeal ‘ 𝑅 ) = ( LSubSp ‘ ( ringLMod ‘ 𝑅 ) )
8 3 7 eqtri ⊢ 𝑈 = ( LSubSp ‘ ( ringLMod ‘ 𝑅 ) )
9 rspval ⊢ ( RSpan ‘ 𝑅 ) = ( LSpan ‘ ( ringLMod ‘ 𝑅 ) )
10 1 9 eqtri ⊢ 𝐾 = ( LSpan ‘ ( ringLMod ‘ 𝑅 ) )
11 6 8 10 lspcl ⊢ ( ( ( ringLMod ‘ 𝑅 ) ∈ LMod ∧ 𝐺 ⊆ 𝐵 ) → ( 𝐾 ‘ 𝐺 ) ∈ 𝑈 )
12 4 11 sylan ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐺 ⊆ 𝐵 ) → ( 𝐾 ‘ 𝐺 ) ∈ 𝑈 )