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 = ( 𝑎 ∈ 𝐸 , 𝑏 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑎 ‘ 𝑓 ) ∘ ( 𝑏 ‘ 𝑓 ) ) ) ) |
7 |
|
erngdv.o |
⊢ 0 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) |
8 |
|
erngdv.i |
⊢ 𝐼 = ( 𝑎 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ◡ ( 𝑎 ‘ 𝑓 ) ) ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 ) |
10 |
1 4 5 2 9
|
erngbase |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( Base ‘ 𝐷 ) = 𝐸 ) |
11 |
10
|
eqcomd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐸 = ( Base ‘ 𝐷 ) ) |
12 |
|
eqid |
⊢ ( +g ‘ 𝐷 ) = ( +g ‘ 𝐷 ) |
13 |
1 4 5 2 12
|
erngfplus |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( +g ‘ 𝐷 ) = ( 𝑎 ∈ 𝐸 , 𝑏 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑎 ‘ 𝑓 ) ∘ ( 𝑏 ‘ 𝑓 ) ) ) ) ) |
14 |
6 13
|
eqtr4id |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑃 = ( +g ‘ 𝐷 ) ) |
15 |
1 2 3 4 5 6 7 8
|
erngdvlem1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐷 ∈ Grp ) |
16 |
1 4 5 6
|
tendoplcom |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ) → ( 𝑠 𝑃 𝑡 ) = ( 𝑡 𝑃 𝑠 ) ) |
17 |
11 14 15 16
|
isabld |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐷 ∈ Abel ) |