dalemkehl - Mathbox for Norm Megill

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Theorem dalemkehl 39588

Description: Lemma for dath 39701. 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 2108 class class class wbr 5119 ‘cfv 6530 (class class class)co 7403 Basecbs 17226 HLchlt 39314

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 39589 dalemkeop 39590 dalempjqeb 39610 dalemsjteb 39611 dalemtjueb 39612 dalemqrprot 39613 dalempnes 39616 dalemqnet 39617 dalempjsen 39618 dalemply 39619 dalemsly 39620 dalemswapyz 39621 dalemrot 39622 dalemrotyz 39623 dalem1 39624 dalemcea 39625 dalem2 39626 dalemdea 39627 dalem3 39629 dalem4 39630 dalem5 39632 dalem-cly 39636 dalem9 39637 dalem11 39639 dalem12 39640 dalem13 39641 dalem15 39643 dalem16 39644 dalem17 39645 dalem18 39646 dalem19 39647 dalemswapyzps 39655 dalemcjden 39657 dalem21 39659 dalem22 39660 dalem23 39661 dalem24 39662 dalem25 39663 dalem27 39664 dalem28 39665 dalem38 39675 dalem39 39676 dalem41 39678 dalem42 39679 dalem43 39680 dalem44 39681 dalem45 39682 dalem51 39688 dalem52 39689 dalem54 39691 dalem55 39692 dalem56 39693 dalem57 39694 dalem58 39695 dalem59 39696 dalem60 39697

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