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Metamath Proof Explorer

Theorem ltrniotaidvalN

Description: Value of the unique translation specified by identity value. (Contributed by NM, 25-Aug-2014) (New usage is discouraged.)

Ref Expression
Hypotheses ltrniotaidval.b ⊢ B = Base K
ltrniotaidval.l ⊢ ≤ ˙ = ≤ K
ltrniotaidval.a ⊢ A = Atoms ⁡ K
ltrniotaidval.h ⊢ H = LHyp ⁡ K
ltrniotaidval.t ⊢ T = LTrn ⁡ K ⁡ W
ltrniotaidval.f ⊢ F = ι f ∈ T | f ⁡ P = P
Assertion ltrniotaidvalN ⊢ K ∈ HL ∧ W ∈ H ∧ P ∈ A ∧ ¬ P ≤ ˙ W → F = I ↾ B

Proof

Step Hyp Ref Expression
1 ltrniotaidval.b ⊢ B = Base K
2 ltrniotaidval.l ⊢ ≤ ˙ = ≤ K
3 ltrniotaidval.a ⊢ A = Atoms ⁡ K
4 ltrniotaidval.h ⊢ H = LHyp ⁡ K
5 ltrniotaidval.t ⊢ T = LTrn ⁡ K ⁡ W
6 ltrniotaidval.f ⊢ F = ι f ∈ T | f ⁡ P = P
7 2 3 4 5 6 ltrniotaval ⊢ K ∈ HL ∧ W ∈ H ∧ P ∈ A ∧ ¬ P ≤ ˙ W ∧ P ∈ A ∧ ¬ P ≤ ˙ W → F ⁡ P = P
8 7 3anidm23 ⊢ K ∈ HL ∧ W ∈ H ∧ P ∈ A ∧ ¬ P ≤ ˙ W → F ⁡ P = P
9 simpl ⊢ K ∈ HL ∧ W ∈ H ∧ P ∈ A ∧ ¬ P ≤ ˙ W → K ∈ HL ∧ W ∈ H
10 2 3 4 5 6 ltrniotacl ⊢ K ∈ HL ∧ W ∈ H ∧ P ∈ A ∧ ¬ P ≤ ˙ W ∧ P ∈ A ∧ ¬ P ≤ ˙ W → F ∈ T
11 10 3anidm23 ⊢ K ∈ HL ∧ W ∈ H ∧ P ∈ A ∧ ¬ P ≤ ˙ W → F ∈ T
12 simpr ⊢ K ∈ HL ∧ W ∈ H ∧ P ∈ A ∧ ¬ P ≤ ˙ W → P ∈ A ∧ ¬ P ≤ ˙ W
13 1 2 3 4 5 ltrnideq ⊢ K ∈ HL ∧ W ∈ H ∧ F ∈ T ∧ P ∈ A ∧ ¬ P ≤ ˙ W → F = I ↾ B ↔ F ⁡ P = P
14 9 11 12 13 syl3anc ⊢ K ∈ HL ∧ W ∈ H ∧ P ∈ A ∧ ¬ P ≤ ˙ W → F = I ↾ B ↔ F ⁡ P = P
15 8 14 mpbird ⊢ K ∈ HL ∧ W ∈ H ∧ P ∈ A ∧ ¬ P ≤ ˙ W → F = I ↾ B