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Metamath Proof Explorer

Theorem dihord3

Description: The isomorphism H for a lattice K is order-preserving in the region under co-atom W . (Contributed by NM, 6-Mar-2014)

Ref Expression
Hypotheses dihord3.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihord3.l ⊢ ≤ = ( le ‘ 𝐾 )
dihord3.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihord3.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion dihord3 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ 𝑋 ≤ 𝑌 ) )

Proof

Step Hyp Ref Expression
1 dihord3.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihord3.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihord3.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dihord3.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
5 eqid ⊢ ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
6 1 2 3 4 5 dihvalb ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) )
7 6 3adant3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) )
8 1 2 3 4 5 dihvalb ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑌 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑌 ) )
9 8 3adant2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑌 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑌 ) )
10 7 9 sseq12d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ⊆ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑌 ) ) )
11 1 2 3 5 dibord ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ⊆ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑌 ) ↔ 𝑋 ≤ 𝑌 ) )
12 10 11 bitrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ 𝑋 ≤ 𝑌 ) )