Theorem lmodvaddsub4

Description: Vector addition/subtraction law. (Contributed by NM, 31-Mar-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmod4.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lmod4.p	⊢ + = ( +g ‘ 𝑊 )
	lmodvaddsub4.m	⊢ − = ( -g ‘ 𝑊 )
Assertion	lmodvaddsub4	⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) → ( ( 𝐴 + 𝐵 ) = ( 𝐶 + 𝐷 ) ↔ ( 𝐴 − 𝐶 ) = ( 𝐷 − 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lmod4.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		lmod4.p	⊢ + = ( +g ‘ 𝑊 )
3		lmodvaddsub4.m	⊢ − = ( -g ‘ 𝑊 )
4		lmodabl	⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ Abel )
5	1 2 3	abladdsub4	⊢ ( ( 𝑊 ∈ Abel ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) → ( ( 𝐴 + 𝐵 ) = ( 𝐶 + 𝐷 ) ↔ ( 𝐴 − 𝐶 ) = ( 𝐷 − 𝐵 ) ) )
6	4 5	syl3an1	⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) → ( ( 𝐴 + 𝐵 ) = ( 𝐶 + 𝐷 ) ↔ ( 𝐴 − 𝐶 ) = ( 𝐷 − 𝐵 ) ) )