Theorem nvrcan

Description: Right cancellation law for vector addition. (Contributed by NM, 4-Dec-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nvgcl.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	nvgcl.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
Assertion	nvrcan	⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐶 ) = ( 𝐵 𝐺 𝐶 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		nvgcl.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		nvgcl.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
3	2	nvgrp	⊢ ( 𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp )
4	1 2	bafval	⊢ 𝑋 = ran 𝐺
5	4	grporcan	⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐶 ) = ( 𝐵 𝐺 𝐶 ) ↔ 𝐴 = 𝐵 ) )
6	3 5	sylan	⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐶 ) = ( 𝐵 𝐺 𝐶 ) ↔ 𝐴 = 𝐵 ) )