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 )