Theorem erngplus-rN

Description: Ring addition operation. (Contributed by NM, 10-Jun-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	erngset.h-r	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	erngset.t-r	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	erngset.e-r	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
	erngset.d-r	⊢ 𝐷 = ( ( EDRingR ‘ 𝐾 ) ‘ 𝑊 )
	erng.p-r	⊢ + = ( +g ‘ 𝐷 )
Assertion	erngplus-rN	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ) → ( 𝑈 + 𝑉 ) = ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑈 ‘ 𝑓 ) ∘ ( 𝑉 ‘ 𝑓 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		erngset.h-r	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		erngset.t-r	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3		erngset.e-r	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4		erngset.d-r	⊢ 𝐷 = ( ( EDRingR ‘ 𝐾 ) ‘ 𝑊 )
5		erng.p-r	⊢ + = ( +g ‘ 𝐷 )
6	1 2 3 4 5	erngfplus-rN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → + = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑔 ) ∘ ( 𝑡 ‘ 𝑔 ) ) ) ) )
7	6	oveqd	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑈 + 𝑉 ) = ( 𝑈 ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑔 ) ∘ ( 𝑡 ‘ 𝑔 ) ) ) ) 𝑉 ) )
8		eqid	⊢ ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑔 ) ∘ ( 𝑡 ‘ 𝑔 ) ) ) ) = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑔 ) ∘ ( 𝑡 ‘ 𝑔 ) ) ) )
9	8 2	tendopl	⊢ ( ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) → ( 𝑈 ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑔 ) ∘ ( 𝑡 ‘ 𝑔 ) ) ) ) 𝑉 ) = ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑈 ‘ 𝑓 ) ∘ ( 𝑉 ‘ 𝑓 ) ) ) )
10	7 9	sylan9eq	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ) → ( 𝑈 + 𝑉 ) = ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑈 ‘ 𝑓 ) ∘ ( 𝑉 ‘ 𝑓 ) ) ) )