Metamath Proof Explorer

Theorem ltrniotacl

Description: Version of cdleme50ltrn with simpler hypotheses. TODO: Fix comment. (Contributed by NM, 17-Apr-2013)

Ref Expression
Hypotheses ltrniotaval.l ⊢ ≤ ˙ = ≤ K
ltrniotaval.a ⊢ A = Atoms ⁡ K
ltrniotaval.h ⊢ H = LHyp ⁡ K
ltrniotaval.t ⊢ T = LTrn ⁡ K ⁡ W
ltrniotaval.f ⊢ F = ι f ∈ T | f ⁡ P = Q
Assertion ltrniotacl ⊢ K ∈ HL ∧ W ∈ H ∧ P ∈ A ∧ ¬ P ≤ ˙ W ∧ Q ∈ A ∧ ¬ Q ≤ ˙ W → F ∈ T

Proof

Step Hyp Ref Expression
1 ltrniotaval.l ⊢ ≤ ˙ = ≤ K
2 ltrniotaval.a ⊢ A = Atoms ⁡ K
3 ltrniotaval.h ⊢ H = LHyp ⁡ K
4 ltrniotaval.t ⊢ T = LTrn ⁡ K ⁡ W
5 ltrniotaval.f ⊢ F = ι f ∈ T | f ⁡ P = Q
6 eqid ⊢ Base K = Base K
7 eqid ⊢ join ⁡ K = join ⁡ K
8 eqid ⊢ meet ⁡ K = meet ⁡ K
9 eqid ⊢ P join ⁡ K Q meet ⁡ K W = P join ⁡ K Q meet ⁡ K W
10 eqid ⊢ t join ⁡ K P join ⁡ K Q meet ⁡ K W meet ⁡ K Q join ⁡ K P join ⁡ K t meet ⁡ K W = t join ⁡ K P join ⁡ K Q meet ⁡ K W meet ⁡ K Q join ⁡ K P join ⁡ K t meet ⁡ K W
11 eqid ⊢ P join ⁡ K Q meet ⁡ K t join ⁡ K P join ⁡ K Q meet ⁡ K W meet ⁡ K Q join ⁡ K P join ⁡ K t meet ⁡ K W join ⁡ K s join ⁡ K t meet ⁡ K W = P join ⁡ K Q meet ⁡ K t join ⁡ K P join ⁡ K Q meet ⁡ K W meet ⁡ K Q join ⁡ K P join ⁡ K t meet ⁡ K W join ⁡ K s join ⁡ K t meet ⁡ K W
12 eqid ⊢ x ∈ Base K ⟼ if P ≠ Q ∧ ¬ x ≤ ˙ W ι z ∈ Base K | ∀ s ∈ A ¬ s ≤ ˙ W ∧ s join ⁡ K x meet ⁡ K W = x → z = if s ≤ ˙ P join ⁡ K Q ι y ∈ Base K | ∀ t ∈ A ¬ t ≤ ˙ W ∧ ¬ t ≤ ˙ P join ⁡ K Q → y = P join ⁡ K Q meet ⁡ K t join ⁡ K P join ⁡ K Q meet ⁡ K W meet ⁡ K Q join ⁡ K P join ⁡ K t meet ⁡ K W join ⁡ K s join ⁡ K t meet ⁡ K W ⦋ s / t⦌ t join ⁡ K P join ⁡ K Q meet ⁡ K W meet ⁡ K Q join ⁡ K P join ⁡ K t meet ⁡ K W join ⁡ K x meet ⁡ K W x = x ∈ Base K ⟼ if P ≠ Q ∧ ¬ x ≤ ˙ W ι z ∈ Base K | ∀ s ∈ A ¬ s ≤ ˙ W ∧ s join ⁡ K x meet ⁡ K W = x → z = if s ≤ ˙ P join ⁡ K Q ι y ∈ Base K | ∀ t ∈ A ¬ t ≤ ˙ W ∧ ¬ t ≤ ˙ P join ⁡ K Q → y = P join ⁡ K Q meet ⁡ K t join ⁡ K P join ⁡ K Q meet ⁡ K W meet ⁡ K Q join ⁡ K P join ⁡ K t meet ⁡ K W join ⁡ K s join ⁡ K t meet ⁡ K W ⦋ s / t⦌ t join ⁡ K P join ⁡ K Q meet ⁡ K W meet ⁡ K Q join ⁡ K P join ⁡ K t meet ⁡ K W join ⁡ K x meet ⁡ K W x
13 6 1 7 8 2 3 9 10 11 12 4 5 cdlemg1ltrnlem ⊢ K ∈ HL ∧ W ∈ H ∧ P ∈ A ∧ ¬ P ≤ ˙ W ∧ Q ∈ A ∧ ¬ Q ≤ ˙ W → F ∈ T