Theorem nvpncan

Description: Cancellation law for vector subtraction. (Contributed by NM, 24-Jan-2008) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nvpncan2.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	nvpncan2.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
	nvpncan2.3	⊢ 𝑀 = ( −𝑣 ‘ 𝑈 )
Assertion	nvpncan	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝐺 𝐵 ) 𝑀 𝐵 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		nvpncan2.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		nvpncan2.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
3		nvpncan2.3	⊢ 𝑀 = ( −𝑣 ‘ 𝑈 )
4	1 2	nvcom	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐵 𝐺 𝐴 ) = ( 𝐴 𝐺 𝐵 ) )
5	4	oveq1d	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝐵 𝐺 𝐴 ) 𝑀 𝐵 ) = ( ( 𝐴 𝐺 𝐵 ) 𝑀 𝐵 ) )
6	1 2 3	nvpncan2	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝐵 𝐺 𝐴 ) 𝑀 𝐵 ) = 𝐴 )
7	5 6	eqtr3d	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝐴 𝐺 𝐵 ) 𝑀 𝐵 ) = 𝐴 )
8	7	3com23	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝐺 𝐵 ) 𝑀 𝐵 ) = 𝐴 )