Metamath Proof Explorer

Theorem dibelval1st1

Description: Membership in value of the partial isomorphism B for a lattice K . (Contributed by NM, 13-Feb-2014)

Ref Expression
Hypotheses dibelval1st1.b ⊢ B = Base K
dibelval1st1.l ⊢ ≤ ˙ = ≤ K
dibelval1st1.h ⊢ H = LHyp ⁡ K
dibelval1st1.t ⊢ T = LTrn ⁡ K ⁡ W
dibelval1st1.i ⊢ I = DIsoB ⁡ K ⁡ W
Assertion dibelval1st1 ⊢ K ∈ V ∧ W ∈ H ∧ X ∈ B ∧ X ≤ ˙ W ∧ Y ∈ I ⁡ X → 1 st ⁡ Y ∈ T

Proof

Step Hyp Ref Expression
1 dibelval1st1.b ⊢ B = Base K
2 dibelval1st1.l ⊢ ≤ ˙ = ≤ K
3 dibelval1st1.h ⊢ H = LHyp ⁡ K
4 dibelval1st1.t ⊢ T = LTrn ⁡ K ⁡ W
5 dibelval1st1.i ⊢ I = DIsoB ⁡ K ⁡ W
6 eqid ⊢ DIsoA ⁡ K ⁡ W = DIsoA ⁡ K ⁡ W
7 1 2 3 6 5 dibelval1st ⊢ K ∈ V ∧ W ∈ H ∧ X ∈ B ∧ X ≤ ˙ W ∧ Y ∈ I ⁡ X → 1 st ⁡ Y ∈ DIsoA ⁡ K ⁡ W ⁡ X
8 1 2 3 4 6 diael ⊢ K ∈ V ∧ W ∈ H ∧ X ∈ B ∧ X ≤ ˙ W ∧ 1 st ⁡ Y ∈ DIsoA ⁡ K ⁡ W ⁡ X → 1 st ⁡ Y ∈ T
9 7 8 syld3an3 ⊢ K ∈ V ∧ W ∈ H ∧ X ∈ B ∧ X ≤ ˙ W ∧ Y ∈ I ⁡ X → 1 st ⁡ Y ∈ T