Metamath Proof Explorer

Theorem dibelval1st

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

Ref Expression
Hypotheses dibelval1.b ⊢ B = Base K
dibelval1.l ⊢ ≤ ˙ = ≤ K
dibelval1.h ⊢ H = LHyp ⁡ K
dibelval1.j ⊢ J = DIsoA ⁡ K ⁡ W
dibelval1.i ⊢ I = DIsoB ⁡ K ⁡ W
Assertion dibelval1st ⊢ K ∈ V ∧ W ∈ H ∧ X ∈ B ∧ X ≤ ˙ W ∧ Y ∈ I ⁡ X → 1 st ⁡ Y ∈ J ⁡ X

Proof

Step Hyp Ref Expression
1 dibelval1.b ⊢ B = Base K
2 dibelval1.l ⊢ ≤ ˙ = ≤ K
3 dibelval1.h ⊢ H = LHyp ⁡ K
4 dibelval1.j ⊢ J = DIsoA ⁡ K ⁡ W
5 dibelval1.i ⊢ I = DIsoB ⁡ K ⁡ W
6 eqid ⊢ LTrn ⁡ K ⁡ W = LTrn ⁡ K ⁡ W
7 eqid ⊢ f ∈ LTrn ⁡ K ⁡ W ⟼ I ↾ B = f ∈ LTrn ⁡ K ⁡ W ⟼ I ↾ B
8 1 2 3 6 7 4 5 dibval2 ⊢ K ∈ V ∧ W ∈ H ∧ X ∈ B ∧ X ≤ ˙ W → I ⁡ X = J ⁡ X × f ∈ LTrn ⁡ K ⁡ W ⟼ I ↾ B
9 8 eleq2d ⊢ K ∈ V ∧ W ∈ H ∧ X ∈ B ∧ X ≤ ˙ W → Y ∈ I ⁡ X ↔ Y ∈ J ⁡ X × f ∈ LTrn ⁡ K ⁡ W ⟼ I ↾ B
10 9 biimp3a ⊢ K ∈ V ∧ W ∈ H ∧ X ∈ B ∧ X ≤ ˙ W ∧ Y ∈ I ⁡ X → Y ∈ J ⁡ X × f ∈ LTrn ⁡ K ⁡ W ⟼ I ↾ B
11 xp1st ⊢ Y ∈ J ⁡ X × f ∈ LTrn ⁡ K ⁡ W ⟼ I ↾ B → 1 st ⁡ Y ∈ J ⁡ X
12 10 11 syl ⊢ K ∈ V ∧ W ∈ H ∧ X ∈ B ∧ X ≤ ˙ W ∧ Y ∈ I ⁡ X → 1 st ⁡ Y ∈ J ⁡ X