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Metamath Proof Explorer

Theorem hvsubassi

Description: Hilbert vector space associative law for subtraction. (Contributed by NM, 7-Oct-1999) (Revised by Mario Carneiro, 15-May-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hvass.1	⊢ 𝐴 ∈ ℋ
	hvass.2	⊢ 𝐵 ∈ ℋ
	hvass.3	⊢ 𝐶 ∈ ℋ
Assertion	hvsubassi	⊢ ( ( 𝐴 −_ℎ 𝐵 ) −_ℎ 𝐶 ) = ( 𝐴 −_ℎ ( 𝐵 +_ℎ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		hvass.1	⊢ 𝐴 ∈ ℋ
2		hvass.2	⊢ 𝐵 ∈ ℋ
3		hvass.3	⊢ 𝐶 ∈ ℋ
4		hvsubass	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 −_ℎ 𝐵 ) −_ℎ 𝐶 ) = ( 𝐴 −_ℎ ( 𝐵 +_ℎ 𝐶 ) ) )
5	1 2 3 4	mp3an	⊢ ( ( 𝐴 −_ℎ 𝐵 ) −_ℎ 𝐶 ) = ( 𝐴 −_ℎ ( 𝐵 +_ℎ 𝐶 ) )