Theorem erngmul

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

	Ref	Expression
Hypotheses	erngset.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	erngset.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	erngset.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
	erngset.d	⊢ 𝐷 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
	erng.m	⊢ · = ( .r ‘ 𝐷 )
Assertion	erngmul	⊢ ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ) → ( 𝑈 · 𝑉 ) = ( 𝑈 ∘ 𝑉 ) )

Proof

Step	Hyp	Ref	Expression
1		erngset.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		erngset.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3		erngset.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4		erngset.d	⊢ 𝐷 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
5		erng.m	⊢ · = ( .r ‘ 𝐷 )
6	1 2 3 4 5	erngfmul	⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → · = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) )
7	6	oveqd	⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → ( 𝑈 · 𝑉 ) = ( 𝑈 ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) 𝑉 ) )
8		coexg	⊢ ( ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) → ( 𝑈 ∘ 𝑉 ) ∈ V )
9		coeq1	⊢ ( 𝑠 = 𝑈 → ( 𝑠 ∘ 𝑡 ) = ( 𝑈 ∘ 𝑡 ) )
10		coeq2	⊢ ( 𝑡 = 𝑉 → ( 𝑈 ∘ 𝑡 ) = ( 𝑈 ∘ 𝑉 ) )
11		eqid	⊢ ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) )
12	9 10 11	ovmpog	⊢ ( ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ∘ 𝑉 ) ∈ V ) → ( 𝑈 ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) 𝑉 ) = ( 𝑈 ∘ 𝑉 ) )
13	8 12	mpd3an3	⊢ ( ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) → ( 𝑈 ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) 𝑉 ) = ( 𝑈 ∘ 𝑉 ) )
14	7 13	sylan9eq	⊢ ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ) → ( 𝑈 · 𝑉 ) = ( 𝑈 ∘ 𝑉 ) )