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Metamath Proof Explorer

Theorem erngdv

Description: An endomorphism ring is a division ring. TODO: fix comment. (Contributed by NM, 11-Aug-2013)

Ref Expression
Hypotheses ernggrp.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
ernggrp.d ⊢ 𝐷 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
Assertion erngdv ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐷 ∈ DivRing )

Proof

Step Hyp Ref Expression
1 ernggrp.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 ernggrp.d ⊢ 𝐷 = ( ( EDRing ‘ 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 ) ↦ ( 𝑎 ∘ 𝑏 ) ) = ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑏 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑎 ∘ 𝑏 ) )
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 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑓 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ) ) → 𝐷 ∈ DivRing )
20 5 19 rexlimddv ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐷 ∈ DivRing )