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Metamath Proof Explorer

Theorem dibn0

Description: The value of the partial isomorphism B is not empty. (Contributed by NM, 18-Jan-2014)

Ref Expression
Hypotheses dibn0.b ⊢ B = Base K
dibn0.l ⊢ ≤ ˙ = ≤ K
dibn0.h ⊢ H = LHyp ⁡ K
dibn0.i ⊢ I = DIsoB ⁡ K ⁡ W
Assertion dibn0 ⊢ K ∈ HL ∧ W ∈ H ∧ X ∈ B ∧ X ≤ ˙ W → I ⁡ X ≠ ∅

Proof

Step Hyp Ref Expression
1 dibn0.b ⊢ B = Base K
2 dibn0.l ⊢ ≤ ˙ = ≤ K
3 dibn0.h ⊢ H = LHyp ⁡ K
4 dibn0.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 ∧ W ∈ H ∧ X ∈ B ∧ X ≤ ˙ W → I ⁡ X = DIsoA ⁡ K ⁡ W ⁡ X × f ∈ LTrn ⁡ K ⁡ W ⟼ I ↾ B
9 1 2 3 7 dian0 ⊢ K ∈ HL ∧ W ∈ H ∧ X ∈ B ∧ X ≤ ˙ W → DIsoA ⁡ K ⁡ W ⁡ X ≠ ∅
10 fvex ⊢ LTrn ⁡ K ⁡ W ∈ V
11 10 mptex ⊢ f ∈ LTrn ⁡ K ⁡ W ⟼ I ↾ B ∈ V
12 11 snnz ⊢ f ∈ LTrn ⁡ K ⁡ W ⟼ I ↾ B ≠ ∅
13 9 12 jctir ⊢ K ∈ HL ∧ W ∈ H ∧ X ∈ B ∧ X ≤ ˙ W → DIsoA ⁡ K ⁡ W ⁡ X ≠ ∅ ∧ f ∈ LTrn ⁡ K ⁡ W ⟼ I ↾ B ≠ ∅
14 xpnz ⊢ DIsoA ⁡ K ⁡ W ⁡ X ≠ ∅ ∧ f ∈ LTrn ⁡ K ⁡ W ⟼ I ↾ B ≠ ∅ ↔ DIsoA ⁡ K ⁡ W ⁡ X × f ∈ LTrn ⁡ K ⁡ W ⟼ I ↾ B ≠ ∅
15 13 14 sylib ⊢ K ∈ HL ∧ W ∈ H ∧ X ∈ B ∧ X ≤ ˙ W → DIsoA ⁡ K ⁡ W ⁡ X × f ∈ LTrn ⁡ K ⁡ W ⟼ I ↾ B ≠ ∅
16 8 15 eqnetrd ⊢ K ∈ HL ∧ W ∈ H ∧ X ∈ B ∧ X ≤ ˙ W → I ⁡ X ≠ ∅