dalemkehl - Mathbox for Norm Megill

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Theorem dalemkehl 37564

Description: Lemma for dath 37677. 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 1282	. 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 216	1 ⊢ (𝜑 → 𝐾 ∈ HL)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 ∈ wcel 2108 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 HLchlt 37291

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 206 df-an 396 df-3an 1087

This theorem is referenced by: dalemkelat 37565 dalemkeop 37566 dalempjqeb 37586 dalemsjteb 37587 dalemtjueb 37588 dalemqrprot 37589 dalempnes 37592 dalemqnet 37593 dalempjsen 37594 dalemply 37595 dalemsly 37596 dalemswapyz 37597 dalemrot 37598 dalemrotyz 37599 dalem1 37600 dalemcea 37601 dalem2 37602 dalemdea 37603 dalem3 37605 dalem4 37606 dalem5 37608 dalem-cly 37612 dalem9 37613 dalem11 37615 dalem12 37616 dalem13 37617 dalem15 37619 dalem16 37620 dalem17 37621 dalem18 37622 dalem19 37623 dalemswapyzps 37631 dalemcjden 37633 dalem21 37635 dalem22 37636 dalem23 37637 dalem24 37638 dalem25 37639 dalem27 37640 dalem28 37641 dalem38 37651 dalem39 37652 dalem41 37654 dalem42 37655 dalem43 37656 dalem44 37657 dalem45 37658 dalem51 37664 dalem52 37665 dalem54 37667 dalem55 37668 dalem56 37669 dalem57 37670 dalem58 37671 dalem59 37672 dalem60 37673

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