Theorem lindepsnlininds

Description: A linearly dependent subset is not a linearly independent subset. (Contributed by AV, 26-Apr-2019)

	Ref	Expression
Assertion	lindepsnlininds	⊢ ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ 𝑊 ) → ( 𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀 ) )

Proof

Step	Hyp	Ref	Expression
1		breq12	⊢ ( ( 𝑠 = 𝑆 ∧ 𝑚 = 𝑀 ) → ( 𝑠 linIndS 𝑚 ↔ 𝑆 linIndS 𝑀 ) )
2	1	notbid	⊢ ( ( 𝑠 = 𝑆 ∧ 𝑚 = 𝑀 ) → ( ¬ 𝑠 linIndS 𝑚 ↔ ¬ 𝑆 linIndS 𝑀 ) )
3		df-lindeps	⊢ linDepS = { ⟨ 𝑠 , 𝑚 ⟩ ∣ ¬ 𝑠 linIndS 𝑚 }
4	2 3	brabga	⊢ ( ( 𝑆 ∈ 𝑉 ∧ 𝑀 ∈ 𝑊 ) → ( 𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀 ) )