Theorem lmodlcan

Description: Left cancellation law for vector sum. (Contributed by NM, 12-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmodvacl.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lmodvacl.a	⊢ + = ( +g ‘ 𝑊 )
Assertion	lmodlcan	⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉 ) ) → ( ( 𝑍 + 𝑋 ) = ( 𝑍 + 𝑌 ) ↔ 𝑋 = 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		lmodvacl.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		lmodvacl.a	⊢ + = ( +g ‘ 𝑊 )
3		lmodgrp	⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ Grp )
4	1 2	grplcan	⊢ ( ( 𝑊 ∈ Grp ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉 ) ) → ( ( 𝑍 + 𝑋 ) = ( 𝑍 + 𝑌 ) ↔ 𝑋 = 𝑌 ) )
5	3 4	sylan	⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉 ) ) → ( ( 𝑍 + 𝑋 ) = ( 𝑍 + 𝑌 ) ↔ 𝑋 = 𝑌 ) )