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Metamath Proof Explorer

Theorem dip0l

Description: Inner product with a zero first argument. Part of proof of Theorem 6.44 of Ponnusamy p. 361. (Contributed by NM, 5-Feb-2007) (New usage is discouraged.)

Ref Expression
Hypotheses dip0r.1 ⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
dip0r.5 ⊢ 𝑍 = ( 0vec ‘ 𝑈 )
dip0r.7 ⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
Assertion dip0l ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 𝑍 𝑃 𝐴 ) = 0 )

Proof

Step Hyp Ref Expression
1 dip0r.1 ⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2 dip0r.5 ⊢ 𝑍 = ( 0vec ‘ 𝑈 )
3 dip0r.7 ⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
4 1 2 nvzcl ⊢ ( 𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋 )
5 4 adantr ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → 𝑍 ∈ 𝑋 )
6 1 3 dipcj ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ) → ( ∗ ‘ ( 𝐴 𝑃 𝑍 ) ) = ( 𝑍 𝑃 𝐴 ) )
7 5 6 mpd3an3 ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( ∗ ‘ ( 𝐴 𝑃 𝑍 ) ) = ( 𝑍 𝑃 𝐴 ) )
8 1 2 3 dip0r ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝑃 𝑍 ) = 0 )
9 8 fveq2d ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( ∗ ‘ ( 𝐴 𝑃 𝑍 ) ) = ( ∗ ‘ 0 ) )
10 cj0 ⊢ ( ∗ ‘ 0 ) = 0
11 9 10 eqtrdi ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( ∗ ‘ ( 𝐴 𝑃 𝑍 ) ) = 0 )
12 7 11 eqtr3d ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 𝑍 𝑃 𝐴 ) = 0 )