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Metamath Proof Explorer

Theorem dibelval1st2N

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

Ref Expression
Hypotheses dibelval1st2.b ⊢ B = Base K
dibelval1st2.l ⊢ ≤ ˙ = ≤ K
dibelval1st2.h ⊢ H = LHyp ⁡ K
dibelval1st2.t ⊢ T = LTrn ⁡ K ⁡ W
dibelval1st2.r ⊢ R = trL ⁡ K ⁡ W
dibelval1st2.i ⊢ I = DIsoB ⁡ K ⁡ W
Assertion dibelval1st2N ⊢ K ∈ V ∧ W ∈ H ∧ X ∈ B ∧ X ≤ ˙ W ∧ Y ∈ I ⁡ X → R ⁡ 1 st ⁡ Y ≤ ˙ X

Proof

Step Hyp Ref Expression
1 dibelval1st2.b ⊢ B = Base K
2 dibelval1st2.l ⊢ ≤ ˙ = ≤ K
3 dibelval1st2.h ⊢ H = LHyp ⁡ K
4 dibelval1st2.t ⊢ T = LTrn ⁡ K ⁡ W
5 dibelval1st2.r ⊢ R = trL ⁡ K ⁡ W
6 dibelval1st2.i ⊢ I = DIsoB ⁡ K ⁡ W
7 eqid ⊢ DIsoA ⁡ K ⁡ W = DIsoA ⁡ K ⁡ W
8 1 2 3 7 6 dibelval1st ⊢ K ∈ V ∧ W ∈ H ∧ X ∈ B ∧ X ≤ ˙ W ∧ Y ∈ I ⁡ X → 1 st ⁡ Y ∈ DIsoA ⁡ K ⁡ W ⁡ X
9 1 2 3 4 5 7 diatrl ⊢ K ∈ V ∧ W ∈ H ∧ X ∈ B ∧ X ≤ ˙ W ∧ 1 st ⁡ Y ∈ DIsoA ⁡ K ⁡ W ⁡ X → R ⁡ 1 st ⁡ Y ≤ ˙ X
10 8 9 syld3an3 ⊢ K ∈ V ∧ W ∈ H ∧ X ∈ B ∧ X ≤ ˙ W ∧ Y ∈ I ⁡ X → R ⁡ 1 st ⁡ Y ≤ ˙ X