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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | cdlemednpq 40301 | Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. 𝐷 represents s2. (Contributed by NM, 18-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝐷 ≤ (𝑃 ∨ 𝑄)) | ||
| Theorem | cdlemednuN 40302 | Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. 𝐷 represents s2. (Contributed by NM, 18-Nov-2012.) (New usage is discouraged.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐷 ≠ 𝑈) | ||
| Theorem | cdleme20zN 40303 | Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. (Contributed by NM, 17-Nov-2012.) (New usage is discouraged.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → ((𝑆 ∨ 𝑅) ∧ 𝑇) = (0.‘𝐾)) | ||
| Theorem | cdleme20y 40304 | Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. (Contributed by NM, 17-Nov-2012.) (Proof shortened by OpenAI, 25-Mar-2020.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑆 ≠ 𝑇 ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → ((𝑆 ∨ 𝑅) ∧ (𝑇 ∨ 𝑅)) = 𝑅) | ||
| Theorem | cdleme19a 40305 | Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, 1st line. 𝐷 represents s2. In their notation, we prove that if r ≤ s ∨ t, then s2=(s ∨ t) ∧ w. (Contributed by NM, 13-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝐷 = ((𝑆 ∨ 𝑇) ∧ 𝑊)) | ||
| Theorem | cdleme19b 40306 | Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, 1st line. 𝐷, 𝐹, 𝐺 represent s2, f(s), f(t). In their notation, we prove that if r ≤ s ∨ t, then s2 ≤ f(s) ∨ f(t). (Contributed by NM, 13-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑆 ∨ 𝑇)))) → 𝐷 ≤ (𝐹 ∨ 𝐺)) | ||
| Theorem | cdleme19c 40307 | Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, 1st line. 𝐷, 𝐹 represent s2, f(s). We prove f(s) ≠ s2. (Contributed by NM, 13-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐹 ≠ 𝐷) | ||
| Theorem | cdleme19d 40308 | Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114. 𝐷, 𝐹, 𝐺 represent s2, f(s), f(t). We prove f(s) ∨ s2 = f(s) ∨ f(t). (Contributed by NM, 14-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑆 ∨ 𝑇)))) → (𝐹 ∨ 𝐷) = (𝐹 ∨ 𝐺)) | ||
| Theorem | cdleme19e 40309 | Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, line 2. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). We prove f(s) ∨ s2=f(t) ∨ t2. (Contributed by NM, 14-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑆 ∨ 𝑇)))) → (𝐹 ∨ 𝐷) = (𝐺 ∨ 𝑌)) | ||
| Theorem | cdleme19f 40310 | Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, line 3. 𝐷, 𝐹, 𝑁, 𝑌, 𝐺, 𝑂 represent s2, f(s), fs(r), t2, f(t), ft(r). We prove that if r ≤ s ∨ t, then ft(r) = ft(r). (Contributed by NM, 14-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝐷)) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ 𝑌)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑆 ∨ 𝑇)))) → 𝑁 = 𝑂) | ||
| Theorem | cdleme20aN 40311 | Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). (Contributed by NM, 14-Nov-2012.) (New usage is discouraged.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑉 ∨ 𝐷) = (((𝑆 ∨ 𝑅) ∨ 𝑇) ∧ 𝑊)) | ||
| Theorem | cdleme20bN 40312 | Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, second line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). We show v ∨ s2 = v ∨ t2. (Contributed by NM, 15-Nov-2012.) (New usage is discouraged.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑉 ∨ 𝐷) = (𝑉 ∨ 𝑌)) | ||
| Theorem | cdleme20c 40313 | Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, second line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). (Contributed by NM, 15-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝐷 ∨ 𝑌) = (((𝑅 ∨ 𝑆) ∨ 𝑇) ∧ 𝑊)) | ||
| Theorem | cdleme20d 40314 | Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, second line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). (Contributed by NM, 17-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝐹 ∨ 𝐺) ∧ (𝐷 ∨ 𝑌)) = 𝑉) | ||
| Theorem | cdleme20e 40315 | Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, 4th line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). We show <f(s),s2,s> and <f(t),t2,t> are centrally perspective. (Contributed by NM, 17-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝐹 ∨ 𝐺) ∧ (𝐷 ∨ 𝑌)) ≤ (𝑆 ∨ 𝑇)) | ||
| Theorem | cdleme20f 40316 | Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, 4th line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). We show <f(s),s2,s> and <f(t),t2,t> are axially perspective. (Contributed by NM, 17-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝐹 ∨ 𝐷) ∧ (𝐺 ∨ 𝑌)) ≤ (((𝐷 ∨ 𝑆) ∧ (𝑌 ∨ 𝑇)) ∨ ((𝑆 ∨ 𝐹) ∧ (𝑇 ∨ 𝐺)))) | ||
| Theorem | cdleme20g 40317 | Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, antepenultimate line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). (Contributed by NM, 18-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (((𝐷 ∨ 𝑆) ∧ (𝑌 ∨ 𝑇)) ∨ ((𝑆 ∨ 𝐹) ∧ (𝑇 ∨ 𝐺))) = (((𝑆 ∨ 𝑅) ∧ (𝑇 ∨ 𝑅)) ∨ ((𝑆 ∨ 𝑈) ∧ (𝑇 ∨ 𝑈)))) | ||
| Theorem | cdleme20h 40318 | Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, antepenultimate line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). (Contributed by NM, 18-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝑅 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → (((𝑆 ∨ 𝑅) ∧ (𝑇 ∨ 𝑅)) ∨ ((𝑆 ∨ 𝑈) ∧ (𝑇 ∨ 𝑈))) = (𝑅 ∨ 𝑈)) | ||
| Theorem | cdleme20i 40319 | Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, antepenultimate line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). We show (f(s) ∨ s2) ∧ (f(t) ∨ t2) ≤ p ∨ q. (Contributed by NM, 18-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝑅 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → ((𝐹 ∨ 𝐷) ∧ (𝐺 ∨ 𝑌)) ≤ (𝑃 ∨ 𝑄)) | ||
| Theorem | cdleme20j 40320 | Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). We show s2 ≠ t2. (Contributed by NM, 18-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝐷 ≠ 𝑌) | ||
| Theorem | cdleme20k 40321 | Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, antepenultimate line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). (Contributed by NM, 20-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝐹 ∨ 𝐷) ≠ (𝑃 ∨ 𝑄)) | ||
| Theorem | cdleme20l1 40322 | Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, penultimate line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t) respectively. (Contributed by NM, 20-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝐹 ∨ 𝐷) ∈ (LLines‘𝐾)) | ||
| Theorem | cdleme20l2 40323 | Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, penultimate line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t) respectively. (Contributed by NM, 20-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝑅 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → ((𝐹 ∨ 𝐷) ∧ (𝐺 ∨ 𝑌)) ∈ 𝐴) | ||
| Theorem | cdleme20l 40324 | Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, penultimate line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t) respectively. (Contributed by NM, 20-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝑅 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → ((𝐹 ∨ 𝐷) ∧ (𝐺 ∨ 𝑌)) = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝐷))) | ||
| Theorem | cdleme20m 40325 | Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, penultimate line. 𝐷, 𝐹, 𝑁, 𝑌, 𝐺, 𝑂 represent s2, f(s), fs(r), t2, f(t), ft(r) respectively. We prove that if ¬ r ≤ s ∨ t and ¬ u ≤ s ∨ t, then fs(r) = ft(r). (Contributed by NM, 20-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝐷)) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ 𝑌)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝑅 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → 𝑁 = 𝑂) | ||
| Theorem | cdleme20 40326 | Combine cdleme19f 40310 and cdleme20m 40325 to eliminate ¬ 𝑅 ≤ (𝑆 ∨ 𝑇) condition. (Contributed by NM, 28-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝐷)) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ 𝑌)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝑁 = 𝑂) | ||
| Theorem | cdleme21a 40327 | Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 28-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → 𝑆 ≠ 𝑧) | ||
| Theorem | cdleme21b 40328 | Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 28-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) | ||
| Theorem | cdleme21c 40329 | Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 28-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → ¬ 𝑈 ≤ (𝑆 ∨ 𝑧)) | ||
| Theorem | cdleme21at 40330 | Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → 𝑇 ≠ 𝑧) | ||
| Theorem | cdleme21ct 40331 | Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇))) ∧ ((𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊) ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → ¬ 𝑈 ≤ (𝑇 ∨ 𝑧)) | ||
| Theorem | cdleme21d 40332 | Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 115, 3rd line. 𝐷, 𝐹, 𝑁, 𝐸, 𝐵, 𝑍 represent s2, f(s), fs(r), z2, f(z), fz(r) respectively. We prove fs(r) = fz(r). (Contributed by NM, 29-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐵 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝐸 = ((𝑅 ∨ 𝑧) ∧ 𝑊) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝐷)) & ⊢ 𝑍 = ((𝑃 ∨ 𝑄) ∧ (𝐵 ∨ 𝐸)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊) ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))) → 𝑁 = 𝑍) | ||
| Theorem | cdleme21e 40333 | Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 115, 3rd line. 𝑌, 𝐺, 𝑂, 𝐸, 𝐵, 𝑍 represent s2, f(s), fs(r), z2, f(z), fz(r) respectively. We prove that if u ≤ s ∨ z, then ft(r) = fz(r). (Contributed by NM, 29-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐵 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝐸 = ((𝑅 ∨ 𝑧) ∧ 𝑊) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝐷)) & ⊢ 𝑍 = ((𝑃 ∨ 𝑄) ∧ (𝐵 ∨ 𝐸)) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ 𝑌)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) ∧ ((𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊) ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))) → 𝑂 = 𝑍) | ||
| Theorem | cdleme21f 40334 | Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐵 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝐸 = ((𝑅 ∨ 𝑧) ∧ 𝑊) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝐷)) & ⊢ 𝑍 = ((𝑃 ∨ 𝑄) ∧ (𝐵 ∨ 𝐸)) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ 𝑌)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) ∧ ((𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊) ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))) → 𝑁 = 𝑂) | ||
| Theorem | cdleme21g 40335 | Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝐷)) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ 𝑌)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) ∧ ((𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊) ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))) → 𝑁 = 𝑂) | ||
| Theorem | cdleme21h 40336* | Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝐷)) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ 𝑌)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → (∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)) → 𝑁 = 𝑂)) | ||
| Theorem | cdleme21i 40337* | Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝐷)) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ 𝑌)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)))) → (∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) → 𝑁 = 𝑂)) | ||
| Theorem | cdleme21j 40338* | Combine cdleme20 40326 and cdleme21i 40337 to eliminate 𝑈 ≤ (𝑆 ∨ 𝑇) condition. (Contributed by NM, 29-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝐷)) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ 𝑌)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑁 = 𝑂) | ||
| Theorem | cdleme21 40339 | Part of proof of Lemma E in [Crawley] p. 113, 3rd line on p. 115. 𝐷, 𝐹, 𝑁, 𝑌, 𝐺, 𝑂 represent s2, f(s), fs(r), t2, f(t), ft(r) respectively. Combine cdleme18d 40297 and cdleme21j 40338 to eliminate existence condition, proving fs(r) = ft(r) with fewer conditions. (Contributed by NM, 29-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝐷)) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ 𝑌)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)))) → 𝑁 = 𝑂) | ||
| Theorem | cdleme21k 40340 | Eliminate 𝑆 ≠ 𝑇 condition in cdleme21 40339. (Contributed by NM, 26-Dec-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) & ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝐷)) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ 𝑌)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)))) → 𝑁 = 𝑂) | ||
| Theorem | cdleme22aa 40341 | Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 3rd line on p. 115. Show that t ∨ v = p ∨ q implies v = u. (Contributed by NM, 2-Dec-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ 𝑉 ≤ (𝑃 ∨ 𝑄))) → 𝑉 = 𝑈) | ||
| Theorem | cdleme22a 40342 | Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 3rd line on p. 115. Show that t ∨ v = p ∨ q implies v = u. (Contributed by NM, 30-Nov-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) → 𝑉 = 𝑈) | ||
| Theorem | cdleme22b 40343 | Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 5th line on p. 115. Show that t ∨ v =/= p ∨ q and s ≤ p ∨ q implies ¬ t ≤ p ∨ q. (Contributed by NM, 2-Dec-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) | ||
| Theorem | cdleme22cN 40344 | Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 5th line on p. 115. Show that t ∨ v =/= p ∨ q and s ≤ p ∨ q implies ¬ v ≤ p ∨ q. (Contributed by NM, 3-Dec-2012.) (New usage is discouraged.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄))) → ¬ 𝑉 ≤ (𝑃 ∨ 𝑄)) | ||
| Theorem | cdleme22d 40345 | Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 9th line on p. 115. (Contributed by NM, 4-Dec-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊)) | ||
| Theorem | cdleme22e 40346 | Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. 𝐹, 𝑁, 𝑂 represent f(z), fz(s), fz(t) respectively. When t ∨ v = p ∨ q, fz(s) ≤ fz(t) ∨ v. (Contributed by NM, 6-Dec-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊))) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑁 ≤ (𝑂 ∨ 𝑉)) | ||
| Theorem | cdleme22eALTN 40347 | Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. 𝐹, 𝑁, 𝑂 represent f(z), fz(s), fz(t) respectively. When t ∨ v = p ∨ q, fz(s) ≤ fz(t) ∨ v. (Contributed by NM, 6-Dec-2012.) (New usage is discouraged.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑦 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑦) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑦) ∧ 𝑊))) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊))) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑁 ≤ (𝑂 ∨ 𝑉)) | ||
| Theorem | cdleme22f 40348 | Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 6th and 7th lines on p. 115. 𝐹, 𝑁 represent f(t), ft(s) respectively. If s ≤ t ∨ v, then ft(s) ≤ f(t) ∨ v. (Contributed by NM, 6-Dec-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑇) ∧ 𝑊))) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝑁 ≤ (𝐹 ∨ 𝑉)) | ||
| Theorem | cdleme22f2 40349 | Part of proof of Lemma E in [Crawley] p. 113. cdleme22f 40348 with s and t swapped (this case is not mentioned by them). If s ≤ t ∨ v, then f(s) ≤ fs(t) ∨ v. (Contributed by NM, 7-Dec-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑇 ∨ 𝑆) ∧ 𝑊))) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐹 ≤ (𝑁 ∨ 𝑉)) | ||
| Theorem | cdleme22g 40350 | Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 6th and 7th lines on p. 115. 𝐹, 𝐺 represent f(s), f(t) respectively. If s ≤ t ∨ v and ¬ s ≤ p ∨ q, then f(s) ≤ f(t) ∨ v. (Contributed by NM, 6-Dec-2012.) |
| ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐹 ≤ (𝐺 ∨ 𝑉)) | ||
| Theorem | cdleme23a 40351 | Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 8-Dec-2012.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ (𝑋 ∧ 𝑊)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑉 ≤ 𝑊) | ||
| Theorem | cdleme23b 40352 | Part of proof of Lemma E in [Crawley] p. 113, 4th paragraph, 6th line on p. 115. (Contributed by NM, 8-Dec-2012.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ (𝑋 ∧ 𝑊)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑉 ∈ 𝐴) | ||
| Theorem | cdleme23c 40353 | Part of proof of Lemma E in [Crawley] p. 113, 4th paragraph, 6th line on p. 115. (Contributed by NM, 8-Dec-2012.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ (𝑋 ∧ 𝑊)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑆 ≤ (𝑇 ∨ 𝑉)) | ||
| Theorem | cdleme24 40354* | Quantified version of cdleme21k 40340. (Contributed by NM, 26-Dec-2012.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ∀𝑠 ∈ 𝐴 ∀𝑡 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄))) → 𝑁 = 𝑂)) | ||
| Theorem | cdleme25a 40355* | Lemma for cdleme25b 40356. (Contributed by NM, 1-Jan-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑠) ∧ 𝑊))) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ∃𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑁 ∈ 𝐵)) | ||
| Theorem | cdleme25b 40356* | Transform cdleme24 40354. TODO get rid of $d's on 𝑈, 𝑁 (Contributed by NM, 1-Jan-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑠) ∧ 𝑊))) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ∃𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) | ||
| Theorem | cdleme25c 40357* | Transform cdleme25b 40356. (Contributed by NM, 1-Jan-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑠) ∧ 𝑊))) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ∃!𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) | ||
| Theorem | cdleme25dN 40358* | Transform cdleme25c 40357. (Contributed by NM, 19-Jan-2013.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑠) ∧ 𝑊))) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ∃!𝑢 ∈ 𝐵 ∃𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑢 = 𝑁)) | ||
| Theorem | cdleme25cl 40359* | Show closure of the unique element in cdleme25c 40357. (Contributed by NM, 2-Feb-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝐼 = (℩𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝐼 ∈ 𝐵) | ||
| Theorem | cdleme25cv 40360* | Change bound variables in cdleme25c 40357. (Contributed by NM, 2-Feb-2013.) |
| ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ ((𝑅 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐼 = (℩𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) & ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂)) ⇒ ⊢ 𝐼 = 𝐸 | ||
| Theorem | cdleme26e 40361* | Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. 𝐹, 𝑁, 𝑂 represent f(z), fz(s), fz(t) respectively. When t ∨ v = p ∨ q, fz(s) ≤ fz(t) ∨ v. TODO: FIX COMMENT. (Contributed by NM, 2-Feb-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐼 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) & ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄) ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝐼 ≤ (𝐸 ∨ 𝑉)) | ||
| Theorem | cdleme26ee 40362* | Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. 𝐹, 𝑁, 𝑂 represent f(z), fz(s), fz(t) respectively. When t ∨ v = p ∨ q, fz(s) ≤ fz(t) ∨ v. TODO: FIX COMMENT. (Contributed by NM, 2-Feb-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐼 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) & ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) → 𝐼 ≤ (𝐸 ∨ 𝑉)) | ||
| Theorem | cdleme26eALTN 40363* | Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. 𝐹, 𝑁, 𝑂 represent f(z), fz(s), fz(t) respectively. When t ∨ v = p ∨ q, fz(s) ≤ fz(t) ∨ v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑦 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑦) ∧ 𝑊))) & ⊢ 𝐺 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑦) ∧ 𝑊))) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐼 = (℩𝑢 ∈ 𝐵 ∀𝑦 ∈ 𝐴 ((¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) & ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊 ∧ ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))) → 𝐼 ≤ (𝐸 ∨ 𝑉)) | ||
| Theorem | cdleme26fALTN 40364* | Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 6th and 7th lines on p. 115. 𝐹, 𝑁 represent f(t), ft(s) respectively. If t ≤ t ∨ v, then ft(s) ≤ f(t) ∨ v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑡) ∧ 𝑊))) & ⊢ 𝐼 = (℩𝑢 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑆 ≠ 𝑡 ∧ 𝑆 ≤ (𝑡 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐼 ≤ (𝐹 ∨ 𝑉)) | ||
| Theorem | cdleme26f 40365* | Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 6th and 7th lines on p. 115. 𝐹, 𝑁 represent f(t), ft(s) respectively. If t ≤ t ∨ v, then ft(s) ≤ f(t) ∨ v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑡) ∧ 𝑊))) & ⊢ 𝐼 = (℩𝑢 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (¬ 𝑡 ≤ (𝑃 ∨ 𝑄) ∧ (𝑆 ≠ 𝑡 ∧ 𝑆 ≤ (𝑡 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐼 ≤ (𝐹 ∨ 𝑉)) | ||
| Theorem | cdleme26f2ALTN 40366* | Part of proof of Lemma E in [Crawley] p. 113. cdleme26fALTN 40364 with s and t swapped (this case is not mentioned by them). If s ≤ t ∨ v, then f(s) ≤ fs(t) ∨ v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐺 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ ((𝑇 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐺 ≤ (𝐸 ∨ 𝑉)) | ||
| Theorem | cdleme26f2 40367* | Part of proof of Lemma E in [Crawley] p. 113. cdleme26fALTN 40364 with s and t swapped (this case is not mentioned by them). If s ≤ t ∨ v, then f(s) ≤ fs(t) ∨ v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐺 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ ((𝑇 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (¬ 𝑠 ≤ (𝑃 ∨ 𝑄) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐺 ≤ (𝐸 ∨ 𝑉)) | ||
| Theorem | cdleme27cl 40368* | Part of proof of Lemma E in [Crawley] p. 113. Closure of 𝐶. (Contributed by NM, 6-Feb-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑍 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐷 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) & ⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄)) → 𝐶 ∈ 𝐵) | ||
| Theorem | cdleme27a 40369* | Part of proof of Lemma E in [Crawley] p. 113. cdleme26f 40365 with s and t swapped (this case is not mentioned by them). If s ≤ t ∨ v, then f(s) ≤ fs(t) ∨ v. TODO: FIX COMMENT. (Contributed by NM, 3-Feb-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑍 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐷 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) & ⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹) & ⊢ 𝐺 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑡 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂)) & ⊢ 𝑌 = if(𝑡 ≤ (𝑃 ∨ 𝑄), 𝐸, 𝐺) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ ((𝑠 ≠ 𝑡 ∧ 𝑠 ≤ (𝑡 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐶 ≤ (𝑌 ∨ 𝑉)) | ||
| Theorem | cdleme27b 40370* | Lemma for cdleme27N 40371. (Contributed by NM, 3-Feb-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑍 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐷 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) & ⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹) & ⊢ 𝐺 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑡 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂)) & ⊢ 𝑌 = if(𝑡 ≤ (𝑃 ∨ 𝑄), 𝐸, 𝐺) ⇒ ⊢ (𝑠 = 𝑡 → 𝐶 = 𝑌) | ||
| Theorem | cdleme27N 40371* | Part of proof of Lemma E in [Crawley] p. 113. Eliminate the 𝑠 ≠ 𝑡 antecedent in cdleme27a 40369. (Contributed by NM, 3-Feb-2013.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑍 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐷 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) & ⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹) & ⊢ 𝐺 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑡 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂)) & ⊢ 𝑌 = if(𝑡 ≤ (𝑃 ∨ 𝑄), 𝐸, 𝐺) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐶 ≤ (𝑌 ∨ 𝑉)) | ||
| Theorem | cdleme28a 40372* | Lemma for cdleme25b 40356. TODO: FIX COMMENT. (Contributed by NM, 4-Feb-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑍 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐷 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) & ⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹) & ⊢ 𝐺 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑡 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂)) & ⊢ 𝑌 = if(𝑡 ≤ (𝑃 ∨ 𝑄), 𝐸, 𝐺) & ⊢ 𝑉 = ((𝑠 ∨ 𝑡) ∧ (𝑋 ∧ 𝑊)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≠ 𝑡 ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑡 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊))) → (𝐶 ∨ (𝑋 ∧ 𝑊)) ≤ (𝑌 ∨ (𝑋 ∧ 𝑊))) | ||
| Theorem | cdleme28b 40373* | Lemma for cdleme25b 40356. TODO: FIX COMMENT. (Contributed by NM, 6-Feb-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑍 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐷 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) & ⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹) & ⊢ 𝐺 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑡 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂)) & ⊢ 𝑌 = if(𝑡 ≤ (𝑃 ∨ 𝑄), 𝐸, 𝐺) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≠ 𝑡 ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑡 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊))) → (𝐶 ∨ (𝑋 ∧ 𝑊)) = (𝑌 ∨ (𝑋 ∧ 𝑊))) | ||
| Theorem | cdleme28c 40374* | Part of proof of Lemma E in [Crawley] p. 113. Eliminate the 𝑠 ≠ 𝑡 antecedent in cdleme28b 40373. TODO: FIX COMMENT. (Contributed by NM, 6-Feb-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑍 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐷 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) & ⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹) & ⊢ 𝐺 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑡 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂)) & ⊢ 𝑌 = if(𝑡 ≤ (𝑃 ∨ 𝑄), 𝐸, 𝐺) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑡 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊))) → (𝐶 ∨ (𝑋 ∧ 𝑊)) = (𝑌 ∨ (𝑋 ∧ 𝑊))) | ||
| Theorem | cdleme28 40375* | Quantified version of cdleme28c 40374. (Compare cdleme24 40354.) (Contributed by NM, 7-Feb-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑍 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐷 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) & ⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹) & ⊢ 𝐺 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) & ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑡 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂)) & ⊢ 𝑌 = if(𝑡 ≤ (𝑃 ∨ 𝑄), 𝐸, 𝐺) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∀𝑠 ∈ 𝐴 ∀𝑡 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (¬ 𝑡 ≤ 𝑊 ∧ (𝑡 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐶 ∨ (𝑋 ∧ 𝑊)) = (𝑌 ∨ (𝑋 ∧ 𝑊)))) | ||
| Theorem | cdleme29ex 40376* | Lemma for cdleme29b 40377. (Compare cdleme25a 40355.) TODO: FIX COMMENT. (Contributed by NM, 7-Feb-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑍 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐷 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) & ⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (𝐶 ∨ (𝑋 ∧ 𝑊)) ∈ 𝐵)) | ||
| Theorem | cdleme29b 40377* | Transform cdleme28 40375. (Compare cdleme25b 40356.) TODO: FIX COMMENT. (Contributed by NM, 7-Feb-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑍 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐷 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) & ⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃𝑣 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑣 = (𝐶 ∨ (𝑋 ∧ 𝑊)))) | ||
| Theorem | cdleme29c 40378* | Transform cdleme28b 40373. (Compare cdleme25c 40357.) TODO: FIX COMMENT. (Contributed by NM, 8-Feb-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑍 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐷 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) & ⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃!𝑣 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑣 = (𝐶 ∨ (𝑋 ∧ 𝑊)))) | ||
| Theorem | cdleme29cl 40379* | Show closure of the unique element in cdleme28c 40374. (Contributed by NM, 8-Feb-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑍 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊))) & ⊢ 𝐷 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) & ⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹) & ⊢ 𝐼 = (℩𝑣 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑣 = (𝐶 ∨ (𝑋 ∧ 𝑊)))) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → 𝐼 ∈ 𝐵) | ||
| Theorem | cdleme30a 40380 | Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Feb-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑠 ∨ (𝑌 ∧ 𝑊)) = 𝑌) | ||
| Theorem | cdleme31so 40381* | Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Feb-2013.) |
| ⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊)))) & ⊢ 𝐶 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊)))) ⇒ ⊢ (𝑋 ∈ 𝐵 → ⦋𝑋 / 𝑥⦌𝑂 = 𝐶) | ||
| Theorem | cdleme31sn 40382* | Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.) |
| ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) & ⊢ 𝐶 = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷) ⇒ ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶) | ||
| Theorem | cdleme31sn1 40383* | Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.) |
| ⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺)) & ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) & ⊢ 𝐶 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺)) ⇒ ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶) | ||
| Theorem | cdleme31se 40384* | Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.) |
| ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑇) ∧ 𝑊))) & ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑅 ∨ 𝑇) ∧ 𝑊))) ⇒ ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝐸 = 𝑌) | ||
| Theorem | cdleme31se2 40385* | Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 3-Apr-2013.) |
| ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))) & ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∧ (⦋𝑆 / 𝑡⦌𝐷 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))) ⇒ ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑡⦌𝐸 = 𝑌) | ||
| Theorem | cdleme31sc 40386* | Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.) |
| ⊢ 𝐶 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑋 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))) ⇒ ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝐶 = 𝑋) | ||
| Theorem | cdleme31sde 40387* | Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.) |
| ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) & ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) & ⊢ 𝑌 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) & ⊢ 𝑍 = ((𝑃 ∨ 𝑄) ∧ (𝑌 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))) ⇒ ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → ⦋𝑅 / 𝑠⦌⦋𝑆 / 𝑡⦌𝐸 = 𝑍) | ||
| Theorem | cdleme31snd 40388* | Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Apr-2013.) |
| ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) & ⊢ 𝑁 = ((𝑣 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊))) & ⊢ 𝐸 = ((𝑂 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑂) ∧ 𝑊))) & ⊢ 𝑂 = ((𝑆 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑆) ∧ 𝑊))) ⇒ ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑣⦌⦋𝑁 / 𝑡⦌𝐷 = 𝐸) | ||
| Theorem | cdleme31sdnN 40389* | Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.) (New usage is discouraged.) |
| ⊢ 𝐶 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) & ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐶) ⇒ ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, ⦋𝑠 / 𝑡⦌𝐷) | ||
| Theorem | cdleme31sn1c 40390* | Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 1-Mar-2013.) |
| ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) & ⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺)) & ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) & ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))) & ⊢ 𝐶 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝑌)) ⇒ ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶) | ||
| Theorem | cdleme31sn2 40391* | Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.) |
| ⊢ 𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) & ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) & ⊢ 𝐶 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))) ⇒ ⊢ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶) | ||
| Theorem | cdleme31fv 40392* | Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 10-Feb-2013.) |
| ⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊)))) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥)) & ⊢ 𝐶 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊)))) ⇒ ⊢ (𝑋 ∈ 𝐵 → (𝐹‘𝑋) = if((𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊), 𝐶, 𝑋)) | ||
| Theorem | cdleme31fv1 40393* | Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 10-Feb-2013.) |
| ⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊)))) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥)) & ⊢ 𝐶 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊)))) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝐶) | ||
| Theorem | cdleme31fv1s 40394* | Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Feb-2013.) |
| ⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊)))) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥)) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = ⦋𝑋 / 𝑥⦌𝑂) | ||
| Theorem | cdleme31fv2 40395* | Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 23-Feb-2013.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥)) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋) | ||
| Theorem | cdleme31id 40396* | Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 18-Apr-2013.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥)) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑃 = 𝑄) → (𝐹‘𝑋) = 𝑋) | ||
| Theorem | cdlemefrs29pre00 40397 | ***START OF VALUE AT ATOM STUFF TO REPLACE ONES BELOW*** FIX COMMENT. TODO: see if this is the optimal utility theorem using lhpmat 40032. (Contributed by NM, 29-Mar-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ (𝑠 = 𝑅 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝜓) ∧ 𝑠 ∈ 𝐴) → (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) ↔ (¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅))) | ||
| Theorem | cdlemefrs29bpre0 40398* | TODO fix comment. (Contributed by NM, 29-Mar-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ (𝑠 = 𝑅 → (𝜑 ↔ 𝜓)) & ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))) → 𝑁 ∈ 𝐵) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → (∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ 𝑧 = ⦋𝑅 / 𝑠⦌𝑁)) | ||
| Theorem | cdlemefrs29bpre1 40399* | TODO: FIX COMMENT. (Contributed by NM, 29-Mar-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ (𝑠 = 𝑅 → (𝜑 ↔ 𝜓)) & ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))) → 𝑁 ∈ 𝐵) & ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ⦋𝑅 / 𝑠⦌𝑁 ∈ 𝐵) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ∃𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))) | ||
| Theorem | cdlemefrs29cpre1 40400* | TODO: FIX COMMENT. (Contributed by NM, 29-Mar-2013.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ (𝑠 = 𝑅 → (𝜑 ↔ 𝜓)) & ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))) → 𝑁 ∈ 𝐵) & ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ⦋𝑅 / 𝑠⦌𝑁 ∈ 𝐵) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ∃!𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))) | ||
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