Metamath Proof Explorer


Theorem dibord

Description: The isomorphism B for a lattice K is order-preserving in the region under co-atom W . (Contributed by NM, 24-Feb-2014)

Ref Expression
Hypotheses dib11.b 𝐵 = ( Base ‘ 𝐾 )
dib11.l = ( le ‘ 𝐾 )
dib11.h 𝐻 = ( LHyp ‘ 𝐾 )
dib11.i 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
Assertion dibord ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → ( ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ↔ 𝑋 𝑌 ) )

Proof

Step Hyp Ref Expression
1 dib11.b 𝐵 = ( Base ‘ 𝐾 )
2 dib11.l = ( le ‘ 𝐾 )
3 dib11.h 𝐻 = ( LHyp ‘ 𝐾 )
4 dib11.i 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
5 eqid ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6 eqid ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) = ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) )
7 eqid ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
8 1 2 3 5 6 7 4 dibval2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( 𝐼𝑋 ) = ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) )
9 8 3adant3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → ( 𝐼𝑋 ) = ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) )
10 1 2 3 5 6 7 4 dibval2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → ( 𝐼𝑌 ) = ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑌 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) )
11 10 3adant2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → ( 𝐼𝑌 ) = ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑌 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) )
12 9 11 sseq12d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → ( ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ↔ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) ⊆ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑌 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) ) )
13 1 2 3 4 dibn0 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( 𝐼𝑋 ) ≠ ∅ )
14 13 3adant3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → ( 𝐼𝑋 ) ≠ ∅ )
15 9 14 eqnetrrd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) ≠ ∅ )
16 ssxpb ( ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) ≠ ∅ → ( ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) ⊆ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑌 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) ↔ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ⊆ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑌 ) ∧ { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ⊆ { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) ) )
17 15 16 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → ( ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) ⊆ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑌 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) ↔ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ⊆ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑌 ) ∧ { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ⊆ { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) ) )
18 ssid { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ⊆ { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) }
19 18 biantru ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ⊆ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑌 ) ↔ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ⊆ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑌 ) ∧ { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ⊆ { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) )
20 1 2 3 7 diaord ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ⊆ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑌 ) ↔ 𝑋 𝑌 ) )
21 19 20 bitr3id ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → ( ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ⊆ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑌 ) ∧ { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ⊆ { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) ↔ 𝑋 𝑌 ) )
22 12 17 21 3bitrd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → ( ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ↔ 𝑋 𝑌 ) )