Theorem nvnpcan

Description: Cancellation law for a normed complex vector space. (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	nvnpcan	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝑀 𝐵 ) 𝐺 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		simprl	⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → 𝐴 ∈ 𝑋 )
5		simprr	⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → 𝐵 ∈ 𝑋 )
6	4 5 5	3jca	⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) )
7	1 2 3	nvaddsub	⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝑀 𝐵 ) = ( ( 𝐴 𝑀 𝐵 ) 𝐺 𝐵 ) )
8	6 7	syldan	⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝑀 𝐵 ) = ( ( 𝐴 𝑀 𝐵 ) 𝐺 𝐵 ) )
9	8	3impb	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝐺 𝐵 ) 𝑀 𝐵 ) = ( ( 𝐴 𝑀 𝐵 ) 𝐺 𝐵 ) )
10	1 2 3	nvpncan	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝐺 𝐵 ) 𝑀 𝐵 ) = 𝐴 )
11	9 10	eqtr3d	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝑀 𝐵 ) 𝐺 𝐵 ) = 𝐴 )