Step |
Hyp |
Ref |
Expression |
1 |
|
ernggrp.h-r |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
ernggrp.d-r |
⊢ 𝐷 = ( ( EDRingR ‘ 𝐾 ) ‘ 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 = ( 𝑎 ∈ 𝐸 , 𝑏 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑎 ‘ 𝑓 ) ∘ ( 𝑏 ‘ 𝑓 ) ) ) ) |
7 |
|
ernggrplem.o-r |
⊢ 𝑂 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) |
8 |
|
ernggrplem.i-r |
⊢ 𝐼 = ( 𝑎 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ◡ ( 𝑎 ‘ 𝑓 ) ) ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 ) |
10 |
1 4 5 2 9
|
erngbase-rN |
⊢ ( ( 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-rN |
⊢ ( ( 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 4 5 6
|
tendoplcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ) → ( 𝑠 𝑃 𝑡 ) ∈ 𝐸 ) |
16 |
1 4 5 6
|
tendoplass |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸 ) ) → ( ( 𝑠 𝑃 𝑡 ) 𝑃 𝑢 ) = ( 𝑠 𝑃 ( 𝑡 𝑃 𝑢 ) ) ) |
17 |
3 1 4 5 7
|
tendo0cl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑂 ∈ 𝐸 ) |
18 |
3 1 4 5 7 6
|
tendo0pl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑠 ∈ 𝐸 ) → ( 𝑂 𝑃 𝑠 ) = 𝑠 ) |
19 |
1 4 5 8
|
tendoicl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑠 ∈ 𝐸 ) → ( 𝐼 ‘ 𝑠 ) ∈ 𝐸 ) |
20 |
1 4 5 8 3 6 7
|
tendoipl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑠 ∈ 𝐸 ) → ( ( 𝐼 ‘ 𝑠 ) 𝑃 𝑠 ) = 𝑂 ) |
21 |
11 14 15 16 17 18 19 20
|
isgrpd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐷 ∈ Grp ) |