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 = ( 0g𝑅 )
frlmbasfsupp.b 𝐵 = ( Base ‘ 𝐹 )
Assertion frlmbasfsupp ( ( 𝐼𝑊𝑋𝐵 ) → 𝑋 finSupp 0 )

Proof

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