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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dalemkelat | Structured version Visualization version GIF version | ||
| Description: Lemma for dath 39738. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
| 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))))) |
| Ref | Expression |
|---|---|
| dalemkelat | ⊢ (𝜑 → 𝐾 ∈ Lat) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dalema.ph | . . 3 ⊢ (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 | 1 | dalemkehl 39625 | . 2 ⊢ (𝜑 → 𝐾 ∈ HL) |
| 3 | 2 | hllatd 39365 | 1 ⊢ (𝜑 → 𝐾 ∈ Lat) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Latclat 18476 HLchlt 39351 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-dm 5695 df-iota 6514 df-fv 6569 df-ov 7434 df-atl 39299 df-cvlat 39323 df-hlat 39352 |
| This theorem is referenced by: dalemcnes 39652 dalempnes 39653 dalemqnet 39654 dalemply 39656 dalemsly 39657 dalem1 39661 dalemcea 39662 dalem3 39666 dalem4 39667 dalem5 39669 dalem8 39672 dalem-cly 39673 dalem10 39675 dalem13 39678 dalem16 39681 dalem17 39682 dalem21 39696 dalem25 39700 dalem27 39701 dalem38 39712 dalem39 39713 dalem43 39717 dalem44 39718 dalem45 39719 dalem48 39722 dalem54 39728 dalem55 39729 dalem56 39730 dalem57 39731 dalem60 39734 |
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