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Metamath Proof Explorer

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 ) )