dalemkehl - Mathbox for Norm Megill

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Theorem dalemkehl 40115

Description: Lemma for dath 40228. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

**Hypothesis**
Ref	Expression
dalema.ph	⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))

**Assertion**
Ref	Expression
dalemkehl	⊢ (𝜑 → 𝐾 ∈ HL)

**Proof of Theorem dalemkehl**
Step	Hyp	Ref	Expression
1		dalema.ph	. 2 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))
2		simp11l 1291	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))) → 𝐾 ∈ HL)
3	1, 2	sylbi 218	1 ⊢ (𝜑 → 𝐾 ∈ HL)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 ∈ wcel 2119 class class class wbr 5072 ‘cfv 6485 (class class class)co 7356 Basecbs 17170 HLchlt 39842

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8

This theorem depends on definitions: df-bi 208 df-an 397 df-3an 1094

This theorem is referenced by: dalemkelat 40116 dalemkeop 40117 dalempjqeb 40137 dalemsjteb 40138 dalemtjueb 40139 dalemqrprot 40140 dalempnes 40143 dalemqnet 40144 dalempjsen 40145 dalemply 40146 dalemsly 40147 dalemswapyz 40148 dalemrot 40149 dalemrotyz 40150 dalem1 40151 dalemcea 40152 dalem2 40153 dalemdea 40154 dalem3 40156 dalem4 40157 dalem5 40159 dalem-cly 40163 dalem9 40164 dalem11 40166 dalem12 40167 dalem13 40168 dalem15 40170 dalem16 40171 dalem17 40172 dalem18 40173 dalem19 40174 dalemswapyzps 40182 dalemcjden 40184 dalem21 40186 dalem22 40187 dalem23 40188 dalem24 40189 dalem25 40190 dalem27 40191 dalem28 40192 dalem38 40202 dalem39 40203 dalem41 40205 dalem42 40206 dalem43 40207 dalem44 40208 dalem45 40209 dalem51 40215 dalem52 40216 dalem54 40218 dalem55 40219 dalem56 40220 dalem57 40221 dalem58 40222 dalem59 40223 dalem60 40224

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