Metamath Proof Explorer


Theorem erngdvlem2N

Description: Lemma for eringring . (Contributed by NM, 6-Aug-2013) (New usage is discouraged.)

Ref Expression
Hypotheses ernggrp.h 𝐻 = ( LHyp ‘ 𝐾 )
ernggrp.d 𝐷 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
erngdv.b 𝐵 = ( Base ‘ 𝐾 )
erngdv.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
erngdv.e 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
erngdv.p 𝑃 = ( 𝑎𝐸 , 𝑏𝐸 ↦ ( 𝑓𝑇 ↦ ( ( 𝑎𝑓 ) ∘ ( 𝑏𝑓 ) ) ) )
erngdv.o 0 = ( 𝑓𝑇 ↦ ( I ↾ 𝐵 ) )
erngdv.i 𝐼 = ( 𝑎𝐸 ↦ ( 𝑓𝑇 ( 𝑎𝑓 ) ) )
Assertion erngdvlem2N ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → 𝐷 ∈ Abel )

Proof

Step Hyp Ref Expression
1 ernggrp.h 𝐻 = ( LHyp ‘ 𝐾 )
2 ernggrp.d 𝐷 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
3 erngdv.b 𝐵 = ( Base ‘ 𝐾 )
4 erngdv.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 erngdv.e 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
6 erngdv.p 𝑃 = ( 𝑎𝐸 , 𝑏𝐸 ↦ ( 𝑓𝑇 ↦ ( ( 𝑎𝑓 ) ∘ ( 𝑏𝑓 ) ) ) )
7 erngdv.o 0 = ( 𝑓𝑇 ↦ ( I ↾ 𝐵 ) )
8 erngdv.i 𝐼 = ( 𝑎𝐸 ↦ ( 𝑓𝑇 ( 𝑎𝑓 ) ) )
9 eqid ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 )
10 1 4 5 2 9 erngbase ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( Base ‘ 𝐷 ) = 𝐸 )
11 10 eqcomd ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → 𝐸 = ( Base ‘ 𝐷 ) )
12 eqid ( +g𝐷 ) = ( +g𝐷 )
13 1 4 5 2 12 erngfplus ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( +g𝐷 ) = ( 𝑎𝐸 , 𝑏𝐸 ↦ ( 𝑓𝑇 ↦ ( ( 𝑎𝑓 ) ∘ ( 𝑏𝑓 ) ) ) ) )
14 6 13 eqtr4id ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → 𝑃 = ( +g𝐷 ) )
15 1 2 3 4 5 6 7 8 erngdvlem1 ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → 𝐷 ∈ Grp )
16 1 4 5 6 tendoplcom ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑠𝐸𝑡𝐸 ) → ( 𝑠 𝑃 𝑡 ) = ( 𝑡 𝑃 𝑠 ) )
17 11 14 15 16 isabld ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → 𝐷 ∈ Abel )