Metamath Proof Explorer


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 = ( +𝑣𝑈 )
Assertion nvrcan ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴𝑋𝐵𝑋𝐶𝑋 ) ) → ( ( 𝐴 𝐺 𝐶 ) = ( 𝐵 𝐺 𝐶 ) ↔ 𝐴 = 𝐵 ) )

Proof

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