Theorem 0oval

Description: Value of the zero operator. (Contributed by NM, 28-Nov-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	0oval.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	0oval.6	⊢ 𝑍 = ( 0vec ‘ 𝑊 )
	0oval.0	⊢ 𝑂 = ( 𝑈 0op 𝑊 )
Assertion	0oval	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 𝑂 ‘ 𝐴 ) = 𝑍 )

Proof

Step	Hyp	Ref	Expression
1		0oval.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		0oval.6	⊢ 𝑍 = ( 0vec ‘ 𝑊 )
3		0oval.0	⊢ 𝑂 = ( 𝑈 0op 𝑊 )
4	1 2 3	0ofval	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ) → 𝑂 = ( 𝑋 × { 𝑍 } ) )
5	4	fveq1d	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ) → ( 𝑂 ‘ 𝐴 ) = ( ( 𝑋 × { 𝑍 } ) ‘ 𝐴 ) )
6	5	3adant3	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 𝑂 ‘ 𝐴 ) = ( ( 𝑋 × { 𝑍 } ) ‘ 𝐴 ) )
7	2	fvexi	⊢ 𝑍 ∈ V
8	7	fvconst2	⊢ ( 𝐴 ∈ 𝑋 → ( ( 𝑋 × { 𝑍 } ) ‘ 𝐴 ) = 𝑍 )
9	8	3ad2ant3	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑋 × { 𝑍 } ) ‘ 𝐴 ) = 𝑍 )
10	6 9	eqtrd	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 𝑂 ‘ 𝐴 ) = 𝑍 )