Metamath Proof Explorer


Theorem erngdvlem1-rN

Description: Lemma for eringring . (Contributed by NM, 4-Aug-2013) (New usage is discouraged.)

Ref Expression
Hypotheses ernggrp.h-r 𝐻 = ( LHyp ‘ 𝐾 )
ernggrp.d-r 𝐷 = ( ( EDRingR𝐾 ) ‘ 𝑊 )
ernggrplem.b-r 𝐵 = ( Base ‘ 𝐾 )
ernggrplem.t-r 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
ernggrplem.e-r 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
ernggrplem.p-r 𝑃 = ( 𝑎𝐸 , 𝑏𝐸 ↦ ( 𝑓𝑇 ↦ ( ( 𝑎𝑓 ) ∘ ( 𝑏𝑓 ) ) ) )
ernggrplem.o-r 𝑂 = ( 𝑓𝑇 ↦ ( I ↾ 𝐵 ) )
ernggrplem.i-r 𝐼 = ( 𝑎𝐸 ↦ ( 𝑓𝑇 ( 𝑎𝑓 ) ) )
Assertion erngdvlem1-rN ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → 𝐷 ∈ Grp )

Proof

Step Hyp Ref Expression
1 ernggrp.h-r 𝐻 = ( LHyp ‘ 𝐾 )
2 ernggrp.d-r 𝐷 = ( ( EDRingR𝐾 ) ‘ 𝑊 )
3 ernggrplem.b-r 𝐵 = ( Base ‘ 𝐾 )
4 ernggrplem.t-r 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 ernggrplem.e-r 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
6 ernggrplem.p-r 𝑃 = ( 𝑎𝐸 , 𝑏𝐸 ↦ ( 𝑓𝑇 ↦ ( ( 𝑎𝑓 ) ∘ ( 𝑏𝑓 ) ) ) )
7 ernggrplem.o-r 𝑂 = ( 𝑓𝑇 ↦ ( I ↾ 𝐵 ) )
8 ernggrplem.i-r 𝐼 = ( 𝑎𝐸 ↦ ( 𝑓𝑇 ( 𝑎𝑓 ) ) )
9 eqid ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 )
10 1 4 5 2 9 erngbase-rN ( ( 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-rN ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( +g𝐷 ) = ( 𝑎𝐸 , 𝑏𝐸 ↦ ( 𝑓𝑇 ↦ ( ( 𝑎𝑓 ) ∘ ( 𝑏𝑓 ) ) ) ) )
14 6 13 eqtr4id ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → 𝑃 = ( +g𝐷 ) )
15 1 4 5 6 tendoplcl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑠𝐸𝑡𝐸 ) → ( 𝑠 𝑃 𝑡 ) ∈ 𝐸 )
16 1 4 5 6 tendoplass ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑠𝐸𝑡𝐸𝑢𝐸 ) ) → ( ( 𝑠 𝑃 𝑡 ) 𝑃 𝑢 ) = ( 𝑠 𝑃 ( 𝑡 𝑃 𝑢 ) ) )
17 3 1 4 5 7 tendo0cl ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → 𝑂𝐸 )
18 3 1 4 5 7 6 tendo0pl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑠𝐸 ) → ( 𝑂 𝑃 𝑠 ) = 𝑠 )
19 1 4 5 8 tendoicl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑠𝐸 ) → ( 𝐼𝑠 ) ∈ 𝐸 )
20 1 4 5 8 3 6 7 tendoipl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑠𝐸 ) → ( ( 𝐼𝑠 ) 𝑃 𝑠 ) = 𝑂 )
21 11 14 15 16 17 18 19 20 isgrpd ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → 𝐷 ∈ Grp )