Metamath Proof Explorer

Theorem dia0

Description: The value of the partial isomorphism A at the lattice zero is the singleton of the identity translation i.e. the zero subspace. (Contributed by NM, 26-Nov-2013)

Ref Expression
Hypotheses dia0.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dia0.z ⊢ 0 = ( 0. ‘ 𝐾 )
dia0.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dia0.i ⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
Assertion dia0 ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐼 ‘ 0 ) = { ( I ↾ 𝐵 ) } )

Proof

Step Hyp Ref Expression
1 dia0.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dia0.z ⊢ 0 = ( 0. ‘ 𝐾 )
3 dia0.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dia0.i ⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
5 id ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6 hlatl ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
7 1 2 atl0cl ⊢ ( 𝐾 ∈ AtLat → 0 ∈ 𝐵 )
8 6 7 syl ⊢ ( 𝐾 ∈ HL → 0 ∈ 𝐵 )
9 8 adantr ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 0 ∈ 𝐵 )
10 1 3 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
11 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
12 1 11 2 atl0le ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑊 ∈ 𝐵 ) → 0 ( le ‘ 𝐾 ) 𝑊 )
13 6 10 12 syl2an ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 0 ( le ‘ 𝐾 ) 𝑊 )
14 eqid ⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
15 eqid ⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
16 1 11 3 14 15 4 diaval ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 0 ∈ 𝐵 ∧ 0 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ 0 ) = { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) ( le ‘ 𝐾 ) 0 } )
17 5 9 13 16 syl12anc ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐼 ‘ 0 ) = { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) ( le ‘ 𝐾 ) 0 } )
18 6 ad2antrr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) → 𝐾 ∈ AtLat )
19 1 3 14 15 trlcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) → ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) ∈ 𝐵 )
20 1 11 2 atlle0 ⊢ ( ( 𝐾 ∈ AtLat ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) ∈ 𝐵 ) → ( ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) ( le ‘ 𝐾 ) 0 ↔ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) = 0 ) )
21 18 19 20 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) → ( ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) ( le ‘ 𝐾 ) 0 ↔ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) = 0 ) )
22 1 2 3 14 15 trlid0b ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) → ( 𝑓 = ( I ↾ 𝐵 ) ↔ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) = 0 ) )
23 21 22 bitr4d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) → ( ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) ( le ‘ 𝐾 ) 0 ↔ 𝑓 = ( I ↾ 𝐵 ) ) )
24 23 rabbidva ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) ( le ‘ 𝐾 ) 0 } = { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∣ 𝑓 = ( I ↾ 𝐵 ) } )
25 1 3 14 idltrn ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( I ↾ 𝐵 ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
26 rabsn ⊢ ( ( I ↾ 𝐵 ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) → { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∣ 𝑓 = ( I ↾ 𝐵 ) } = { ( I ↾ 𝐵 ) } )
27 25 26 syl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∣ 𝑓 = ( I ↾ 𝐵 ) } = { ( I ↾ 𝐵 ) } )
28 17 24 27 3eqtrd ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐼 ‘ 0 ) = { ( I ↾ 𝐵 ) } )