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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalempjqeb | Structured version Visualization version GIF version |
Description: Lemma for dath 35523. 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))))) |
dalemb.j | ⊢ ∨ = (join‘𝐾) |
dalemb.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
dalempjqeb | ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
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 35410 | . 2 ⊢ (𝜑 → 𝐾 ∈ HL) |
3 | 1 | dalempea 35413 | . 2 ⊢ (𝜑 → 𝑃 ∈ 𝐴) |
4 | 1 | dalemqea 35414 | . 2 ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
5 | eqid 2758 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | dalemb.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
7 | dalemb.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
8 | 5, 6, 7 | hlatjcl 35154 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
9 | 2, 3, 4, 8 | syl3anc 1477 | 1 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1072 = wceq 1630 ∈ wcel 2137 class class class wbr 4802 ‘cfv 6047 (class class class)co 6811 Basecbs 16057 joincjn 17143 Atomscatm 35051 HLchlt 35138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-8 2139 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 ax-rep 4921 ax-sep 4931 ax-nul 4939 ax-pow 4990 ax-pr 5053 ax-un 7112 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2045 df-eu 2609 df-mo 2610 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-ne 2931 df-ral 3053 df-rex 3054 df-reu 3055 df-rab 3057 df-v 3340 df-sbc 3575 df-csb 3673 df-dif 3716 df-un 3718 df-in 3720 df-ss 3727 df-nul 4057 df-if 4229 df-pw 4302 df-sn 4320 df-pr 4322 df-op 4326 df-uni 4587 df-iun 4672 df-br 4803 df-opab 4863 df-mpt 4880 df-id 5172 df-xp 5270 df-rel 5271 df-cnv 5272 df-co 5273 df-dm 5274 df-rn 5275 df-res 5276 df-ima 5277 df-iota 6010 df-fun 6049 df-fn 6050 df-f 6051 df-f1 6052 df-fo 6053 df-f1o 6054 df-fv 6055 df-riota 6772 df-ov 6814 df-oprab 6815 df-lub 17173 df-glb 17174 df-join 17175 df-meet 17176 df-lat 17245 df-ats 35055 df-atl 35086 df-cvlat 35110 df-hlat 35139 |
This theorem is referenced by: dalemcea 35447 dalem3 35451 dalem4 35452 dalem5 35454 dalem-cly 35458 dalem10 35460 dalem17 35467 dalem38 35497 dalem44 35503 dalem48 35507 dalem54 35513 dalem55 35514 dalem57 35516 |
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