Step |
Hyp |
Ref |
Expression |
1 |
|
ernggrp.h-r |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
ernggrp.d-r |
⊢ 𝐷 = ( ( EDRingR ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
4 |
|
eqid |
⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
3 1 4
|
cdlemftr0 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) 𝑓 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ) |
6 |
|
eqid |
⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
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 ) ) ) ) |
8 |
|
eqid |
⊢ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) = ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) |
9 |
|
eqid |
⊢ ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ◡ ( 𝑎 ‘ 𝑓 ) ) ) = ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ◡ ( 𝑎 ‘ 𝑓 ) ) ) |
10 |
|
eqid |
⊢ ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑏 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑏 ∘ 𝑎 ) ) = ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑏 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑏 ∘ 𝑎 ) ) |
11 |
|
eqid |
⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 ) |
12 |
|
eqid |
⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 ) |
13 |
|
eqid |
⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
14 |
|
eqid |
⊢ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) |
15 |
|
eqid |
⊢ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑏 ∘ ◡ ( 𝑠 ‘ 𝑓 ) ) ) ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑏 ∘ ◡ ( 𝑠 ‘ 𝑓 ) ) ) ) ) |
16 |
|
eqid |
⊢ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑔 ) ) ( meet ‘ 𝐾 ) ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑏 ∘ ◡ ( 𝑠 ‘ 𝑓 ) ) ) ) ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑔 ∘ ◡ 𝑏 ) ) ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑔 ) ) ( meet ‘ 𝐾 ) ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑏 ∘ ◡ ( 𝑠 ‘ 𝑓 ) ) ) ) ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑔 ∘ ◡ 𝑏 ) ) ) ) |
17 |
|
eqid |
⊢ ( ℩ 𝑧 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∀ 𝑏 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ( 𝑏 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑏 ) ≠ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑏 ) ≠ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑔 ) ) → ( 𝑧 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑔 ) ) ( meet ‘ 𝐾 ) ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑏 ∘ ◡ ( 𝑠 ‘ 𝑓 ) ) ) ) ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑔 ∘ ◡ 𝑏 ) ) ) ) ) ) = ( ℩ 𝑧 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∀ 𝑏 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ( 𝑏 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑏 ) ≠ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑏 ) ≠ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑔 ) ) → ( 𝑧 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑔 ) ) ( meet ‘ 𝐾 ) ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑏 ∘ ◡ ( 𝑠 ‘ 𝑓 ) ) ) ) ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑔 ∘ ◡ 𝑏 ) ) ) ) ) ) |
18 |
|
eqid |
⊢ ( 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ if ( ( 𝑠 ‘ 𝑓 ) = 𝑓 , 𝑔 , ( ℩ 𝑧 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∀ 𝑏 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ( 𝑏 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑏 ) ≠ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑏 ) ≠ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑔 ) ) → ( 𝑧 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑔 ) ) ( meet ‘ 𝐾 ) ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑏 ∘ ◡ ( 𝑠 ‘ 𝑓 ) ) ) ) ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑔 ∘ ◡ 𝑏 ) ) ) ) ) ) ) ) = ( 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ if ( ( 𝑠 ‘ 𝑓 ) = 𝑓 , 𝑔 , ( ℩ 𝑧 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∀ 𝑏 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ( 𝑏 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑏 ) ≠ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑏 ) ≠ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑔 ) ) → ( 𝑧 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑔 ) ) ( meet ‘ 𝐾 ) ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑏 ) ) ( meet ‘ 𝐾 ) ( ( 𝑓 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑏 ∘ ◡ ( 𝑠 ‘ 𝑓 ) ) ) ) ) ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑔 ∘ ◡ 𝑏 ) ) ) ) ) ) ) ) |
19 |
1 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18
|
erngdvlem4-rN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑓 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ) ) → 𝐷 ∈ DivRing ) |
20 |
5 19
|
rexlimddv |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐷 ∈ DivRing ) |