Theorem frlmbasmap

Description: Elements of the free module are set functions. (Contributed by Stefan O'Rear, 3-Feb-2015) (Proof shortened by AV, 21-Jul-2019)

	Ref	Expression
Hypotheses	frlmval.f	⊢ 𝐹 = ( 𝑅 freeLMod 𝐼 )
	frlmbasmap.n	⊢ 𝑁 = ( Base ‘ 𝑅 )
	frlmbasmap.b	⊢ 𝐵 = ( Base ‘ 𝐹 )
Assertion	frlmbasmap	⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ ( 𝑁 ↑m 𝐼 ) )

Proof

Step	Hyp	Ref	Expression
1		frlmval.f	⊢ 𝐹 = ( 𝑅 freeLMod 𝐼 )
2		frlmbasmap.n	⊢ 𝑁 = ( Base ‘ 𝑅 )
3		frlmbasmap.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	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
8	1 2 7 3	frlmelbas	⊢ ( ( 𝑅 ∈ V ∧ 𝐼 ∈ 𝑊 ) → ( 𝑋 ∈ 𝐵 ↔ ( 𝑋 ∈ ( 𝑁 ↑m 𝐼 ) ∧ 𝑋 finSupp ( 0g ‘ 𝑅 ) ) ) )
9	5 6 8	syl2an2	⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ 𝐵 ↔ ( 𝑋 ∈ ( 𝑁 ↑m 𝐼 ) ∧ 𝑋 finSupp ( 0g ‘ 𝑅 ) ) ) )
10	4 9	mpbid	⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ ( 𝑁 ↑m 𝐼 ) ∧ 𝑋 finSupp ( 0g ‘ 𝑅 ) ) )
11	10	simpld	⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ ( 𝑁 ↑m 𝐼 ) )