Metamath Proof Explorer


Theorem lmodvsubadd

Description: Relationship between vector subtraction and addition. ( hvsubadd analog.) (Contributed by NM, 31-Mar-2014) (Revised by Mario Carneiro, 19-Jun-2014)

Ref Expression
Hypotheses lmod4.v 𝑉 = ( Base ‘ 𝑊 )
lmod4.p + = ( +g𝑊 )
lmodvaddsub4.m = ( -g𝑊 )
Assertion lmodvsubadd ( ( 𝑊 ∈ LMod ∧ ( 𝐴𝑉𝐵𝑉𝐶𝑉 ) ) → ( ( 𝐴 𝐵 ) = 𝐶 ↔ ( 𝐵 + 𝐶 ) = 𝐴 ) )

Proof

Step Hyp Ref Expression
1 lmod4.v 𝑉 = ( Base ‘ 𝑊 )
2 lmod4.p + = ( +g𝑊 )
3 lmodvaddsub4.m = ( -g𝑊 )
4 lmodabl ( 𝑊 ∈ LMod → 𝑊 ∈ Abel )
5 1 2 3 ablsubadd ( ( 𝑊 ∈ Abel ∧ ( 𝐴𝑉𝐵𝑉𝐶𝑉 ) ) → ( ( 𝐴 𝐵 ) = 𝐶 ↔ ( 𝐵 + 𝐶 ) = 𝐴 ) )
6 4 5 sylan ( ( 𝑊 ∈ LMod ∧ ( 𝐴𝑉𝐵𝑉𝐶𝑉 ) ) → ( ( 𝐴 𝐵 ) = 𝐶 ↔ ( 𝐵 + 𝐶 ) = 𝐴 ) )