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 · = ( .r𝐷 )
Assertion erngmul ( ( ( 𝐾𝑋𝑊𝐻 ) ∧ ( 𝑈𝐸𝑉𝐸 ) ) → ( 𝑈 · 𝑉 ) = ( 𝑈𝑉 ) )

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 · = ( .r𝐷 )
6 1 2 3 4 5 erngfmul ( ( 𝐾𝑋𝑊𝐻 ) → · = ( 𝑠𝐸 , 𝑡𝐸 ↦ ( 𝑠𝑡 ) ) )
7 6 oveqd ( ( 𝐾𝑋𝑊𝐻 ) → ( 𝑈 · 𝑉 ) = ( 𝑈 ( 𝑠𝐸 , 𝑡𝐸 ↦ ( 𝑠𝑡 ) ) 𝑉 ) )
8 coexg ( ( 𝑈𝐸𝑉𝐸 ) → ( 𝑈𝑉 ) ∈ V )
9 coeq1 ( 𝑠 = 𝑈 → ( 𝑠𝑡 ) = ( 𝑈𝑡 ) )
10 coeq2 ( 𝑡 = 𝑉 → ( 𝑈𝑡 ) = ( 𝑈𝑉 ) )
11 eqid ( 𝑠𝐸 , 𝑡𝐸 ↦ ( 𝑠𝑡 ) ) = ( 𝑠𝐸 , 𝑡𝐸 ↦ ( 𝑠𝑡 ) )
12 9 10 11 ovmpog ( ( 𝑈𝐸𝑉𝐸 ∧ ( 𝑈𝑉 ) ∈ V ) → ( 𝑈 ( 𝑠𝐸 , 𝑡𝐸 ↦ ( 𝑠𝑡 ) ) 𝑉 ) = ( 𝑈𝑉 ) )
13 8 12 mpd3an3 ( ( 𝑈𝐸𝑉𝐸 ) → ( 𝑈 ( 𝑠𝐸 , 𝑡𝐸 ↦ ( 𝑠𝑡 ) ) 𝑉 ) = ( 𝑈𝑉 ) )
14 7 13 sylan9eq ( ( ( 𝐾𝑋𝑊𝐻 ) ∧ ( 𝑈𝐸𝑉𝐸 ) ) → ( 𝑈 · 𝑉 ) = ( 𝑈𝑉 ) )