Proof of Theorem cdlemc5
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simp1l 1198 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝐾 ∈ HL) | 
| 2 |  | simp23l 1295 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑄 ∈ 𝐴) | 
| 3 |  | simp1 1137 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | 
| 4 |  | simp21 1207 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝐹 ∈ 𝑇) | 
| 5 |  | cdlemc3.l | . . . . . . 7
⊢  ≤ =
(le‘𝐾) | 
| 6 |  | cdlemc3.a | . . . . . . 7
⊢ 𝐴 = (Atoms‘𝐾) | 
| 7 |  | cdlemc3.h | . . . . . . 7
⊢ 𝐻 = (LHyp‘𝐾) | 
| 8 |  | cdlemc3.t | . . . . . . 7
⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | 
| 9 | 5, 6, 7, 8 | ltrnat 40142 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐴) → (𝐹‘𝑄) ∈ 𝐴) | 
| 10 | 3, 4, 2, 9 | syl3anc 1373 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑄) ∈ 𝐴) | 
| 11 |  | cdlemc3.j | . . . . . 6
⊢  ∨ =
(join‘𝐾) | 
| 12 | 5, 11, 6 | hlatlej2 39377 | . . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ (𝐹‘𝑄) ∈ 𝐴) → (𝐹‘𝑄) ≤ (𝑄 ∨ (𝐹‘𝑄))) | 
| 13 | 1, 2, 10, 12 | syl3anc 1373 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑄) ≤ (𝑄 ∨ (𝐹‘𝑄))) | 
| 14 |  | simp23 1209 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) | 
| 15 |  | cdlemc3.r | . . . . . 6
⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | 
| 16 | 5, 11, 6, 7, 8, 15 | trljat1 40168 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑄 ∨ (𝑅‘𝐹)) = (𝑄 ∨ (𝐹‘𝑄))) | 
| 17 | 3, 4, 14, 16 | syl3anc 1373 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑄 ∨ (𝑅‘𝐹)) = (𝑄 ∨ (𝐹‘𝑄))) | 
| 18 | 13, 17 | breqtrrd 5171 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑄) ≤ (𝑄 ∨ (𝑅‘𝐹))) | 
| 19 |  | simp22 1208 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) | 
| 20 |  | cdlemc3.m | . . . . 5
⊢  ∧ =
(meet‘𝐾) | 
| 21 | 5, 11, 20, 6, 7, 8 | cdlemc2 40194 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝐹‘𝑄) ≤ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) | 
| 22 | 3, 4, 19, 14, 21 | syl112anc 1376 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑄) ≤ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) | 
| 23 | 1 | hllatd 39365 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝐾 ∈ Lat) | 
| 24 |  | eqid 2737 | . . . . . . 7
⊢
(Base‘𝐾) =
(Base‘𝐾) | 
| 25 | 24, 6 | atbase 39290 | . . . . . 6
⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) | 
| 26 | 2, 25 | syl 17 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑄 ∈ (Base‘𝐾)) | 
| 27 | 24, 7, 8 | ltrncl 40127 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑄 ∈ (Base‘𝐾)) → (𝐹‘𝑄) ∈ (Base‘𝐾)) | 
| 28 | 3, 4, 26, 27 | syl3anc 1373 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑄) ∈ (Base‘𝐾)) | 
| 29 | 24, 7, 8, 15 | trlcl 40166 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ (Base‘𝐾)) | 
| 30 | 3, 4, 29 | syl2anc 584 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ (Base‘𝐾)) | 
| 31 | 24, 11 | latjcl 18484 | . . . . 5
⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑅‘𝐹) ∈ (Base‘𝐾)) → (𝑄 ∨ (𝑅‘𝐹)) ∈ (Base‘𝐾)) | 
| 32 | 23, 26, 30, 31 | syl3anc 1373 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑄 ∨ (𝑅‘𝐹)) ∈ (Base‘𝐾)) | 
| 33 |  | simp22l 1293 | . . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑃 ∈ 𝐴) | 
| 34 | 24, 6 | atbase 39290 | . . . . . . 7
⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) | 
| 35 | 33, 34 | syl 17 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑃 ∈ (Base‘𝐾)) | 
| 36 | 24, 7, 8 | ltrncl 40127 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ (Base‘𝐾)) → (𝐹‘𝑃) ∈ (Base‘𝐾)) | 
| 37 | 3, 4, 35, 36 | syl3anc 1373 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑃) ∈ (Base‘𝐾)) | 
| 38 | 24, 11, 6 | hlatjcl 39368 | . . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) | 
| 39 | 1, 33, 2, 38 | syl3anc 1373 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) | 
| 40 |  | simp1r 1199 | . . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑊 ∈ 𝐻) | 
| 41 | 24, 7 | lhpbase 40000 | . . . . . . 7
⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) | 
| 42 | 40, 41 | syl 17 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑊 ∈ (Base‘𝐾)) | 
| 43 | 24, 20 | latmcl 18485 | . . . . . 6
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ (Base‘𝐾)) | 
| 44 | 23, 39, 42, 43 | syl3anc 1373 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ (Base‘𝐾)) | 
| 45 | 24, 11 | latjcl 18484 | . . . . 5
⊢ ((𝐾 ∈ Lat ∧ (𝐹‘𝑃) ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ (Base‘𝐾)) → ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (Base‘𝐾)) | 
| 46 | 23, 37, 44, 45 | syl3anc 1373 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (Base‘𝐾)) | 
| 47 | 24, 5, 20 | latlem12 18511 | . . . 4
⊢ ((𝐾 ∈ Lat ∧ ((𝐹‘𝑄) ∈ (Base‘𝐾) ∧ (𝑄 ∨ (𝑅‘𝐹)) ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (Base‘𝐾))) → (((𝐹‘𝑄) ≤ (𝑄 ∨ (𝑅‘𝐹)) ∧ (𝐹‘𝑄) ≤ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ↔ (𝐹‘𝑄) ≤ ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))))) | 
| 48 | 23, 28, 32, 46, 47 | syl13anc 1374 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (((𝐹‘𝑄) ≤ (𝑄 ∨ (𝑅‘𝐹)) ∧ (𝐹‘𝑄) ≤ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ↔ (𝐹‘𝑄) ≤ ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))))) | 
| 49 | 18, 22, 48 | mpbi2and 712 | . 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑄) ≤ ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))) | 
| 50 |  | hlatl 39361 | . . . 4
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | 
| 51 | 1, 50 | syl 17 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝐾 ∈ AtLat) | 
| 52 |  | simp3r 1203 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑃) ≠ 𝑃) | 
| 53 | 5, 6, 7, 8, 15 | trlat 40171 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴) | 
| 54 | 3, 19, 4, 52, 53 | syl112anc 1376 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴) | 
| 55 | 5, 7, 8, 15 | trlle 40186 | . . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ≤ 𝑊) | 
| 56 | 3, 4, 55 | syl2anc 584 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ≤ 𝑊) | 
| 57 |  | simp23r 1296 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ¬ 𝑄 ≤ 𝑊) | 
| 58 |  | nbrne2 5163 | . . . . . . 7
⊢ (((𝑅‘𝐹) ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊) → (𝑅‘𝐹) ≠ 𝑄) | 
| 59 | 58 | necomd 2996 | . . . . . 6
⊢ (((𝑅‘𝐹) ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊) → 𝑄 ≠ (𝑅‘𝐹)) | 
| 60 | 56, 57, 59 | syl2anc 584 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑄 ≠ (𝑅‘𝐹)) | 
| 61 |  | eqid 2737 | . . . . . 6
⊢
(LLines‘𝐾) =
(LLines‘𝐾) | 
| 62 | 11, 6, 61 | llni2 39514 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ (𝑅‘𝐹) ∈ 𝐴) ∧ 𝑄 ≠ (𝑅‘𝐹)) → (𝑄 ∨ (𝑅‘𝐹)) ∈ (LLines‘𝐾)) | 
| 63 | 1, 2, 54, 60, 62 | syl31anc 1375 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑄 ∨ (𝑅‘𝐹)) ∈ (LLines‘𝐾)) | 
| 64 | 5, 6, 7, 8 | ltrnat 40142 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐹‘𝑃) ∈ 𝐴) | 
| 65 | 3, 4, 33, 64 | syl3anc 1373 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑃) ∈ 𝐴) | 
| 66 | 5, 11, 6 | hlatlej1 39376 | . . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃))) | 
| 67 | 1, 33, 65, 66 | syl3anc 1373 | . . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃))) | 
| 68 |  | simp3l 1202 | . . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) | 
| 69 |  | nbrne2 5163 | . . . . . . 7
⊢ ((𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) → 𝑃 ≠ 𝑄) | 
| 70 | 67, 68, 69 | syl2anc 584 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑃 ≠ 𝑄) | 
| 71 | 5, 11, 20, 6, 7 | lhpat 40045 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐴) | 
| 72 | 3, 19, 2, 70, 71 | syl112anc 1376 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐴) | 
| 73 | 24, 5, 20 | latmle2 18510 | . . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊) | 
| 74 | 23, 39, 42, 73 | syl3anc 1373 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊) | 
| 75 | 5, 6, 7, 8 | ltrnel 40141 | . . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) | 
| 76 | 75 | simprd 495 | . . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ¬ (𝐹‘𝑃) ≤ 𝑊) | 
| 77 | 3, 4, 19, 76 | syl3anc 1373 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ¬ (𝐹‘𝑃) ≤ 𝑊) | 
| 78 |  | nbrne2 5163 | . . . . . . 7
⊢ ((((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≠ (𝐹‘𝑃)) | 
| 79 | 78 | necomd 2996 | . . . . . 6
⊢ ((((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊) → (𝐹‘𝑃) ≠ ((𝑃 ∨ 𝑄) ∧ 𝑊)) | 
| 80 | 74, 77, 79 | syl2anc 584 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑃) ≠ ((𝑃 ∨ 𝑄) ∧ 𝑊)) | 
| 81 | 11, 6, 61 | llni2 39514 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐴) ∧ (𝐹‘𝑃) ≠ ((𝑃 ∨ 𝑄) ∧ 𝑊)) → ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (LLines‘𝐾)) | 
| 82 | 1, 65, 72, 80, 81 | syl31anc 1375 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (LLines‘𝐾)) | 
| 83 | 5, 11, 20, 6, 7, 8,
15 | cdlemc4 40196 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) → (𝑄 ∨ (𝑅‘𝐹)) ≠ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) | 
| 84 | 83 | 3adant3r 1182 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑄 ∨ (𝑅‘𝐹)) ≠ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) | 
| 85 | 24, 20 | latmcl 18485 | . . . . . 6
⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ (𝑅‘𝐹)) ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (Base‘𝐾)) → ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ∈ (Base‘𝐾)) | 
| 86 | 23, 32, 46, 85 | syl3anc 1373 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ∈ (Base‘𝐾)) | 
| 87 |  | eqid 2737 | . . . . . 6
⊢
(0.‘𝐾) =
(0.‘𝐾) | 
| 88 | 24, 5, 87, 6 | atlen0 39311 | . . . . 5
⊢ (((𝐾 ∈ AtLat ∧ ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ∈ (Base‘𝐾) ∧ (𝐹‘𝑄) ∈ 𝐴) ∧ (𝐹‘𝑄) ≤ ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))) → ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ≠ (0.‘𝐾)) | 
| 89 | 51, 86, 10, 49, 88 | syl31anc 1375 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ≠ (0.‘𝐾)) | 
| 90 | 20, 87, 6, 61 | 2llnmat 39526 | . . . 4
⊢ (((𝐾 ∈ HL ∧ (𝑄 ∨ (𝑅‘𝐹)) ∈ (LLines‘𝐾) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (LLines‘𝐾)) ∧ ((𝑄 ∨ (𝑅‘𝐹)) ≠ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∧ ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ≠ (0.‘𝐾))) → ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ∈ 𝐴) | 
| 91 | 1, 63, 82, 84, 89, 90 | syl32anc 1380 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ∈ 𝐴) | 
| 92 | 5, 6 | atcmp 39312 | . . 3
⊢ ((𝐾 ∈ AtLat ∧ (𝐹‘𝑄) ∈ 𝐴 ∧ ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ∈ 𝐴) → ((𝐹‘𝑄) ≤ ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ↔ (𝐹‘𝑄) = ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))))) | 
| 93 | 51, 10, 91, 92 | syl3anc 1373 | . 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ((𝐹‘𝑄) ≤ ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ↔ (𝐹‘𝑄) = ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))))) | 
| 94 | 49, 93 | mpbid 232 | 1
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑄) = ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))) |