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Metamath Proof Explorer

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 ) )