dalemkehl - Mathbox for Norm Megill

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Theorem dalemkehl 40247

Description: Lemma for dath 40360. 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 1298	. 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 219	1 ⊢ (𝜑 → 𝐾 ∈ HL)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1098 ∈ wcel 2142 class class class wbr 5100 ‘cfv 6521 (class class class)co 7396 Basecbs 17245 HLchlt 39974

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 209 df-an 400 df-3an 1100

This theorem is referenced by: dalemkelat 40248 dalemkeop 40249 dalempjqeb 40269 dalemsjteb 40270 dalemtjueb 40271 dalemqrprot 40272 dalempnes 40275 dalemqnet 40276 dalempjsen 40277 dalemply 40278 dalemsly 40279 dalemswapyz 40280 dalemrot 40281 dalemrotyz 40282 dalem1 40283 dalemcea 40284 dalem2 40285 dalemdea 40286 dalem3 40288 dalem4 40289 dalem5 40291 dalem-cly 40295 dalem9 40296 dalem11 40298 dalem12 40299 dalem13 40300 dalem15 40302 dalem16 40303 dalem17 40304 dalem18 40305 dalem19 40306 dalemswapyzps 40314 dalemcjden 40316 dalem21 40318 dalem22 40319 dalem23 40320 dalem24 40321 dalem25 40322 dalem27 40323 dalem28 40324 dalem38 40334 dalem39 40335 dalem41 40337 dalem42 40338 dalem43 40339 dalem44 40340 dalem45 40341 dalem51 40347 dalem52 40348 dalem54 40350 dalem55 40351 dalem56 40352 dalem57 40353 dalem58 40354 dalem59 40355 dalem60 40356

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