Metamath Proof Explorer

Theorem erng0g

Description: The division ring zero of an endomorphism ring. (Contributed by NM, 5-Nov-2013) (Revised by Mario Carneiro, 23-Jun-2014)

Ref Expression
Hypotheses erng0g.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
erng0g.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
erng0g.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
erng0g.d ⊢ 𝐷 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
erng0g.o ⊢ 𝑂 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
erng0g.z ⊢ 0 = ( 0g ‘ 𝐷 )
Assertion erng0g ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 0 = 𝑂 )

Proof

Step Hyp Ref Expression
1 erng0g.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 erng0g.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 erng0g.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4 erng0g.d ⊢ 𝐷 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
5 erng0g.o ⊢ 𝑂 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
6 erng0g.z ⊢ 0 = ( 0g ‘ 𝐷 )
7 eqid ⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
8 eqid ⊢ ( +g ‘ 𝐷 ) = ( +g ‘ 𝐷 )
9 2 3 7 4 8 erngfplus ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( +g ‘ 𝐷 ) = ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) )
10 9 oveqd ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑂 ( +g ‘ 𝐷 ) 𝑂 ) = ( 𝑂 ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) 𝑂 ) )
11 1 2 3 7 5 tendo0cl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑂 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) )
12 eqid ⊢ ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) = ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) )
13 1 2 3 7 5 12 tendo0pl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑂 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) → ( 𝑂 ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) 𝑂 ) = 𝑂 )
14 11 13 mpdan ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑂 ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) 𝑂 ) = 𝑂 )
15 10 14 eqtrd ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑂 ( +g ‘ 𝐷 ) 𝑂 ) = 𝑂 )
16 2 4 eringring ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐷 ∈ Ring )
17 ringgrp ⊢ ( 𝐷 ∈ Ring → 𝐷 ∈ Grp )
18 16 17 syl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐷 ∈ Grp )
19 eqid ⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 )
20 2 3 7 4 19 erngbase ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( Base ‘ 𝐷 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) )
21 11 20 eleqtrrd ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑂 ∈ ( Base ‘ 𝐷 ) )
22 19 8 6 grpid ⊢ ( ( 𝐷 ∈ Grp ∧ 𝑂 ∈ ( Base ‘ 𝐷 ) ) → ( ( 𝑂 ( +g ‘ 𝐷 ) 𝑂 ) = 𝑂 ↔ 0 = 𝑂 ) )
23 18 21 22 syl2anc ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( 𝑂 ( +g ‘ 𝐷 ) 𝑂 ) = 𝑂 ↔ 0 = 𝑂 ) )
24 15 23 mpbid ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 0 = 𝑂 )