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Metamath Proof Explorer

Theorem ltrneq3

Description: Two translations agree at any atom not under the fiducial co-atom W iff they are equal. (Contributed by NM, 25-Jul-2013)

Ref Expression
Hypotheses cdlemd.l ⊢ ≤ ˙ = ≤ K
cdlemd.a ⊢ A = Atoms ⁡ K
cdlemd.h ⊢ H = LHyp ⁡ K
cdlemd.t ⊢ T = LTrn ⁡ K ⁡ W
Assertion ltrneq3 ⊢ K ∈ HL ∧ W ∈ H ∧ F ∈ T ∧ G ∈ T ∧ P ∈ A ∧ ¬ P ≤ ˙ W → F ⁡ P = G ⁡ P ↔ F = G

Proof

Step Hyp Ref Expression
1 cdlemd.l ⊢ ≤ ˙ = ≤ K
2 cdlemd.a ⊢ A = Atoms ⁡ K
3 cdlemd.h ⊢ H = LHyp ⁡ K
4 cdlemd.t ⊢ T = LTrn ⁡ K ⁡ W
5 simpl1 ⊢ K ∈ HL ∧ W ∈ H ∧ F ∈ T ∧ G ∈ T ∧ P ∈ A ∧ ¬ P ≤ ˙ W ∧ F ⁡ P = G ⁡ P → K ∈ HL ∧ W ∈ H
6 simpl2l ⊢ K ∈ HL ∧ W ∈ H ∧ F ∈ T ∧ G ∈ T ∧ P ∈ A ∧ ¬ P ≤ ˙ W ∧ F ⁡ P = G ⁡ P → F ∈ T
7 simpl2r ⊢ K ∈ HL ∧ W ∈ H ∧ F ∈ T ∧ G ∈ T ∧ P ∈ A ∧ ¬ P ≤ ˙ W ∧ F ⁡ P = G ⁡ P → G ∈ T
8 simpl3 ⊢ K ∈ HL ∧ W ∈ H ∧ F ∈ T ∧ G ∈ T ∧ P ∈ A ∧ ¬ P ≤ ˙ W ∧ F ⁡ P = G ⁡ P → P ∈ A ∧ ¬ P ≤ ˙ W
9 simpr ⊢ K ∈ HL ∧ W ∈ H ∧ F ∈ T ∧ G ∈ T ∧ P ∈ A ∧ ¬ P ≤ ˙ W ∧ F ⁡ P = G ⁡ P → F ⁡ P = G ⁡ P
10 1 2 3 4 cdlemd ⊢ K ∈ HL ∧ W ∈ H ∧ F ∈ T ∧ G ∈ T ∧ P ∈ A ∧ ¬ P ≤ ˙ W ∧ F ⁡ P = G ⁡ P → F = G
11 5 6 7 8 9 10 syl311anc ⊢ K ∈ HL ∧ W ∈ H ∧ F ∈ T ∧ G ∈ T ∧ P ∈ A ∧ ¬ P ≤ ˙ W ∧ F ⁡ P = G ⁡ P → F = G
12 fveq1 ⊢ F = G → F ⁡ P = G ⁡ P
13 12 adantl ⊢ K ∈ HL ∧ W ∈ H ∧ F ∈ T ∧ G ∈ T ∧ P ∈ A ∧ ¬ P ≤ ˙ W ∧ F = G → F ⁡ P = G ⁡ P
14 11 13 impbida ⊢ K ∈ HL ∧ W ∈ H ∧ F ∈ T ∧ G ∈ T ∧ P ∈ A ∧ ¬ P ≤ ˙ W → F ⁡ P = G ⁡ P ↔ F = G