Metamath Proof Explorer


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 = ( +𝑣𝑈 )
nvpncan2.3 𝑀 = ( −𝑣𝑈 )
Assertion nvnpcan ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) → ( ( 𝐴 𝑀 𝐵 ) 𝐺 𝐵 ) = 𝐴 )

Proof

Step Hyp Ref Expression
1 nvpncan2.1 𝑋 = ( BaseSet ‘ 𝑈 )
2 nvpncan2.2 𝐺 = ( +𝑣𝑈 )
3 nvpncan2.3 𝑀 = ( −𝑣𝑈 )
4 simprl ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴𝑋𝐵𝑋 ) ) → 𝐴𝑋 )
5 simprr ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴𝑋𝐵𝑋 ) ) → 𝐵𝑋 )
6 4 5 5 3jca ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴𝑋𝐵𝑋 ) ) → ( 𝐴𝑋𝐵𝑋𝐵𝑋 ) )
7 1 2 3 nvaddsub ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴𝑋𝐵𝑋𝐵𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝑀 𝐵 ) = ( ( 𝐴 𝑀 𝐵 ) 𝐺 𝐵 ) )
8 6 7 syldan ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴𝑋𝐵𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝑀 𝐵 ) = ( ( 𝐴 𝑀 𝐵 ) 𝐺 𝐵 ) )
9 8 3impb ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) → ( ( 𝐴 𝐺 𝐵 ) 𝑀 𝐵 ) = ( ( 𝐴 𝑀 𝐵 ) 𝐺 𝐵 ) )
10 1 2 3 nvpncan ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) → ( ( 𝐴 𝐺 𝐵 ) 𝑀 𝐵 ) = 𝐴 )
11 9 10 eqtr3d ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) → ( ( 𝐴 𝑀 𝐵 ) 𝐺 𝐵 ) = 𝐴 )