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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 = ( +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 )