dalemkehl - Mathbox for Norm Megill

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Theorem dalemkehl 39624

Description: Lemma for dath 39737. 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 1285	. 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 217	1 ⊢ (𝜑 → 𝐾 ∈ HL)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2109 class class class wbr 5110 ‘cfv 6514 (class class class)co 7390 Basecbs 17186 HLchlt 39350

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 207 df-an 396 df-3an 1088

This theorem is referenced by: dalemkelat 39625 dalemkeop 39626 dalempjqeb 39646 dalemsjteb 39647 dalemtjueb 39648 dalemqrprot 39649 dalempnes 39652 dalemqnet 39653 dalempjsen 39654 dalemply 39655 dalemsly 39656 dalemswapyz 39657 dalemrot 39658 dalemrotyz 39659 dalem1 39660 dalemcea 39661 dalem2 39662 dalemdea 39663 dalem3 39665 dalem4 39666 dalem5 39668 dalem-cly 39672 dalem9 39673 dalem11 39675 dalem12 39676 dalem13 39677 dalem15 39679 dalem16 39680 dalem17 39681 dalem18 39682 dalem19 39683 dalemswapyzps 39691 dalemcjden 39693 dalem21 39695 dalem22 39696 dalem23 39697 dalem24 39698 dalem25 39699 dalem27 39700 dalem28 39701 dalem38 39711 dalem39 39712 dalem41 39714 dalem42 39715 dalem43 39716 dalem44 39717 dalem45 39718 dalem51 39724 dalem52 39725 dalem54 39727 dalem55 39728 dalem56 39729 dalem57 39730 dalem58 39731 dalem59 39732 dalem60 39733

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