Theorem lmodvnpcan

Description: Cancellation law for vector subtraction ( npcan analog). (Contributed by NM, 19-Apr-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	lmodvnpcan	⊢ ( ( 𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝐴 − 𝐵 ) + 𝐵 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		lmod4.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		lmod4.p	⊢ + = ( +g ‘ 𝑊 )
3		lmodvaddsub4.m	⊢ − = ( -g ‘ 𝑊 )
4		lmodgrp	⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ Grp )
5	1 2 3	grpnpcan	⊢ ( ( 𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝐴 − 𝐵 ) + 𝐵 ) = 𝐴 )
6	4 5	syl3an1	⊢ ( ( 𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝐴 − 𝐵 ) + 𝐵 ) = 𝐴 )