Theorem rmsuppfi

Description: The support of a mapping of a multiplication of a constant with a function into a ring is finite if the support of the function is finite. (Contributed by AV, 11-Apr-2019)

	Ref	Expression
Hypothesis	rmsuppfi.r	⊢ 𝑅 = ( Base ‘ 𝑀 )
Assertion	rmsuppfi	⊢ ( ( ( 𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅 ) ∧ 𝐴 ∈ ( 𝑅 ↑m 𝑉 ) ∧ ( 𝐴 supp ( 0g ‘ 𝑀 ) ) ∈ Fin ) → ( ( 𝑣 ∈ 𝑉 ↦ ( 𝐶 ( .r ‘ 𝑀 ) ( 𝐴 ‘ 𝑣 ) ) ) supp ( 0g ‘ 𝑀 ) ) ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		rmsuppfi.r	⊢ 𝑅 = ( Base ‘ 𝑀 )
2		simp3	⊢ ( ( ( 𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅 ) ∧ 𝐴 ∈ ( 𝑅 ↑m 𝑉 ) ∧ ( 𝐴 supp ( 0g ‘ 𝑀 ) ) ∈ Fin ) → ( 𝐴 supp ( 0g ‘ 𝑀 ) ) ∈ Fin )
3	1	rmsuppss	⊢ ( ( ( 𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅 ) ∧ 𝐴 ∈ ( 𝑅 ↑m 𝑉 ) ) → ( ( 𝑣 ∈ 𝑉 ↦ ( 𝐶 ( .r ‘ 𝑀 ) ( 𝐴 ‘ 𝑣 ) ) ) supp ( 0g ‘ 𝑀 ) ) ⊆ ( 𝐴 supp ( 0g ‘ 𝑀 ) ) )
4	3	3adant3	⊢ ( ( ( 𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅 ) ∧ 𝐴 ∈ ( 𝑅 ↑m 𝑉 ) ∧ ( 𝐴 supp ( 0g ‘ 𝑀 ) ) ∈ Fin ) → ( ( 𝑣 ∈ 𝑉 ↦ ( 𝐶 ( .r ‘ 𝑀 ) ( 𝐴 ‘ 𝑣 ) ) ) supp ( 0g ‘ 𝑀 ) ) ⊆ ( 𝐴 supp ( 0g ‘ 𝑀 ) ) )
5		ssfi	⊢ ( ( ( 𝐴 supp ( 0g ‘ 𝑀 ) ) ∈ Fin ∧ ( ( 𝑣 ∈ 𝑉 ↦ ( 𝐶 ( .r ‘ 𝑀 ) ( 𝐴 ‘ 𝑣 ) ) ) supp ( 0g ‘ 𝑀 ) ) ⊆ ( 𝐴 supp ( 0g ‘ 𝑀 ) ) ) → ( ( 𝑣 ∈ 𝑉 ↦ ( 𝐶 ( .r ‘ 𝑀 ) ( 𝐴 ‘ 𝑣 ) ) ) supp ( 0g ‘ 𝑀 ) ) ∈ Fin )
6	2 4 5	syl2anc	⊢ ( ( ( 𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅 ) ∧ 𝐴 ∈ ( 𝑅 ↑m 𝑉 ) ∧ ( 𝐴 supp ( 0g ‘ 𝑀 ) ) ∈ Fin ) → ( ( 𝑣 ∈ 𝑉 ↦ ( 𝐶 ( .r ‘ 𝑀 ) ( 𝐴 ‘ 𝑣 ) ) ) supp ( 0g ‘ 𝑀 ) ) ∈ Fin )