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Metamath Proof Explorer

Theorem eringring

Description: An endomorphism ring is a ring. TODO: fix comment. (Contributed by NM, 4-Aug-2013)

Ref Expression
Hypotheses ernggrp.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
ernggrp.d ⊢ 𝐷 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
Assertion eringring ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐷 ∈ Ring )

Proof

Step Hyp Ref Expression
1 ernggrp.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 ernggrp.d ⊢ 𝐷 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
3 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
4 eqid ⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 eqid ⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
6 eqid ⊢ ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑏 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( 𝑎 ‘ 𝑓 ) ∘ ( 𝑏 ‘ 𝑓 ) ) ) ) = ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑏 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( 𝑎 ‘ 𝑓 ) ∘ ( 𝑏 ‘ 𝑓 ) ) ) )
7 eqid ⊢ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) = ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) )
8 eqid ⊢ ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ◡ ( 𝑎 ‘ 𝑓 ) ) ) = ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ◡ ( 𝑎 ‘ 𝑓 ) ) )
9 eqid ⊢ ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑏 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑎 ∘ 𝑏 ) ) = ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑏 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑎 ∘ 𝑏 ) )
10 1 2 3 4 5 6 7 8 9 erngdvlem3 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐷 ∈ Ring )