Theorem vclcan

Description: Left cancellation law for vector addition. (Contributed by NM, 4-Nov-2006) (New usage is discouraged.)

	Ref	Expression
Hypotheses	vclcan.1	⊢ 𝐺 = ( 1st ‘ 𝑊 )
	vclcan.2	⊢ 𝑋 = ran 𝐺
Assertion	vclcan	⊢ ( ( 𝑊 ∈ CVecOLD ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐶 𝐺 𝐴 ) = ( 𝐶 𝐺 𝐵 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		vclcan.1	⊢ 𝐺 = ( 1st ‘ 𝑊 )
2		vclcan.2	⊢ 𝑋 = ran 𝐺
3	1	vcgrp	⊢ ( 𝑊 ∈ CVecOLD → 𝐺 ∈ GrpOp )
4	2	grpolcan	⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐶 𝐺 𝐴 ) = ( 𝐶 𝐺 𝐵 ) ↔ 𝐴 = 𝐵 ) )
5	3 4	sylan	⊢ ( ( 𝑊 ∈ CVecOLD ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐶 𝐺 𝐴 ) = ( 𝐶 𝐺 𝐵 ) ↔ 𝐴 = 𝐵 ) )