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 )