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Metamath Proof Explorer

Theorem frlmbasfsupp

Description: Elements of the free module are finitely supported. (Contributed by Stefan O'Rear, 3-Feb-2015) (Revised by Thierry Arnoux, 21-Jun-2019) (Proof shortened by AV, 20-Jul-2019)

Ref Expression
Hypotheses frlmval.f ⊢ 𝐹 = ( 𝑅 freeLMod 𝐼 )
frlmbasfsupp.z ⊢ 0 = ( 0g ‘ 𝑅 )
frlmbasfsupp.b ⊢ 𝐵 = ( Base ‘ 𝐹 )
Assertion frlmbasfsupp ( ( 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 finSupp 0 )

Proof

Step Hyp Ref Expression
1 frlmval.f ⊢ 𝐹 = ( 𝑅 freeLMod 𝐼 )
2 frlmbasfsupp.z ⊢ 0 = ( 0g ‘ 𝑅 )
3 frlmbasfsupp.b ⊢ 𝐵 = ( Base ‘ 𝐹 )
4 simpr ⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
5 1 3 frlmrcl ⊢ ( 𝑋 ∈ 𝐵 → 𝑅 ∈ V )
6 simpl ⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵 ) → 𝐼 ∈ 𝑊 )
7 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
8 1 7 2 3 frlmelbas ⊢ ( ( 𝑅 ∈ V ∧ 𝐼 ∈ 𝑊 ) → ( 𝑋 ∈ 𝐵 ↔ ( 𝑋 ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐼 ) ∧ 𝑋 finSupp 0 ) ) )
9 5 6 8 syl2an2 ⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ 𝐵 ↔ ( 𝑋 ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐼 ) ∧ 𝑋 finSupp 0 ) ) )
10 4 9 mpbid ⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐼 ) ∧ 𝑋 finSupp 0 ) )
11 10 simprd ⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 finSupp 0 )