Theorem lnof

Description: A linear operator is a mapping. (Contributed by NM, 4-Dec-2007) (Revised by Mario Carneiro, 18-Nov-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	lnof.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	lnof.2	⊢ 𝑌 = ( BaseSet ‘ 𝑊 )
	lnof.7	⊢ 𝐿 = ( 𝑈 LnOp 𝑊 )
Assertion	lnof	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿 ) → 𝑇 : 𝑋 ⟶ 𝑌 )

Proof

Step	Hyp	Ref	Expression
1		lnof.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		lnof.2	⊢ 𝑌 = ( BaseSet ‘ 𝑊 )
3		lnof.7	⊢ 𝐿 = ( 𝑈 LnOp 𝑊 )
4		eqid	⊢ ( +𝑣 ‘ 𝑈 ) = ( +𝑣 ‘ 𝑈 )
5		eqid	⊢ ( +𝑣 ‘ 𝑊 ) = ( +𝑣 ‘ 𝑊 )
6		eqid	⊢ ( ·𝑠OLD ‘ 𝑈 ) = ( ·𝑠OLD ‘ 𝑈 )
7		eqid	⊢ ( ·𝑠OLD ‘ 𝑊 ) = ( ·𝑠OLD ‘ 𝑊 )
8	1 2 4 5 6 7 3	islno	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ) → ( 𝑇 ∈ 𝐿 ↔ ( 𝑇 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( 𝑇 ‘ ( ( 𝑥 ( ·𝑠OLD ‘ 𝑈 ) 𝑦 ) ( +𝑣 ‘ 𝑈 ) 𝑧 ) ) = ( ( 𝑥 ( ·𝑠OLD ‘ 𝑊 ) ( 𝑇 ‘ 𝑦 ) ) ( +𝑣 ‘ 𝑊 ) ( 𝑇 ‘ 𝑧 ) ) ) ) )
9	8	simprbda	⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ) ∧ 𝑇 ∈ 𝐿 ) → 𝑇 : 𝑋 ⟶ 𝑌 )
10	9	3impa	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿 ) → 𝑇 : 𝑋 ⟶ 𝑌 )