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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem8 | Structured version Visualization version GIF version |
Description: Lemma for dath 35543. Plane 𝑍 belongs to the 3-dimensional space. (Contributed by NM, 21-Jul-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))))) |
dalemc.l | ⊢ ≤ = (le‘𝐾) |
dalemc.j | ⊢ ∨ = (join‘𝐾) |
dalemc.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dalem6.o | ⊢ 𝑂 = (LPlanes‘𝐾) |
dalem6.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
dalem6.z | ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
dalem6.w | ⊢ 𝑊 = (𝑌 ∨ 𝐶) |
Ref | Expression |
---|---|
dalem8 | ⊢ (𝜑 → 𝑍 ≤ 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalem6.z | . 2 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
2 | dalema.ph | . . . . 5 ⊢ (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))))) | |
3 | dalemc.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
4 | dalemc.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
5 | dalemc.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | dalem6.o | . . . . 5 ⊢ 𝑂 = (LPlanes‘𝐾) | |
7 | dalem6.y | . . . . 5 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
8 | dalem6.w | . . . . 5 ⊢ 𝑊 = (𝑌 ∨ 𝐶) | |
9 | 2, 3, 4, 5, 6, 7, 1, 8 | dalem6 35475 | . . . 4 ⊢ (𝜑 → 𝑆 ≤ 𝑊) |
10 | 2, 3, 4, 5, 6, 7, 1, 8 | dalem7 35476 | . . . 4 ⊢ (𝜑 → 𝑇 ≤ 𝑊) |
11 | 2 | dalemkelat 35431 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat) |
12 | 2, 5 | dalemseb 35449 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾)) |
13 | 2, 5 | dalemteb 35450 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾)) |
14 | 2, 6 | dalemyeb 35456 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾)) |
15 | 2, 5 | dalemceb 35445 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾)) |
16 | eqid 2760 | . . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
17 | 16, 4 | latjcl 17272 | . . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → (𝑌 ∨ 𝐶) ∈ (Base‘𝐾)) |
18 | 11, 14, 15, 17 | syl3anc 1477 | . . . . . 6 ⊢ (𝜑 → (𝑌 ∨ 𝐶) ∈ (Base‘𝐾)) |
19 | 8, 18 | syl5eqel 2843 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾)) |
20 | 16, 3, 4 | latjle12 17283 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑆 ≤ 𝑊 ∧ 𝑇 ≤ 𝑊) ↔ (𝑆 ∨ 𝑇) ≤ 𝑊)) |
21 | 11, 12, 13, 19, 20 | syl13anc 1479 | . . . 4 ⊢ (𝜑 → ((𝑆 ≤ 𝑊 ∧ 𝑇 ≤ 𝑊) ↔ (𝑆 ∨ 𝑇) ≤ 𝑊)) |
22 | 9, 10, 21 | mpbi2and 994 | . . 3 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ≤ 𝑊) |
23 | 2, 3, 4, 5, 6, 7, 8 | dalem5 35474 | . . 3 ⊢ (𝜑 → 𝑈 ≤ 𝑊) |
24 | 2, 4, 5 | dalemsjteb 35453 | . . . 4 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾)) |
25 | 2, 5 | dalemueb 35451 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾)) |
26 | 16, 3, 4 | latjle12 17283 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ ((𝑆 ∨ 𝑇) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → (((𝑆 ∨ 𝑇) ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ 𝑊)) |
27 | 11, 24, 25, 19, 26 | syl13anc 1479 | . . 3 ⊢ (𝜑 → (((𝑆 ∨ 𝑇) ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ 𝑊)) |
28 | 22, 23, 27 | mpbi2and 994 | . 2 ⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ 𝑊) |
29 | 1, 28 | syl5eqbr 4839 | 1 ⊢ (𝜑 → 𝑍 ≤ 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 class class class wbr 4804 ‘cfv 6049 (class class class)co 6814 Basecbs 16079 lecple 16170 joincjn 17165 Latclat 17266 Atomscatm 35071 HLchlt 35158 LPlanesclpl 35299 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-preset 17149 df-poset 17167 df-plt 17179 df-lub 17195 df-glb 17196 df-join 17197 df-meet 17198 df-p0 17260 df-lat 17267 df-clat 17329 df-oposet 34984 df-ol 34986 df-oml 34987 df-covers 35074 df-ats 35075 df-atl 35106 df-cvlat 35130 df-hlat 35159 df-llines 35305 df-lplanes 35306 |
This theorem is referenced by: dalem13 35483 |
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