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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem2 | Structured version Visualization version GIF version |
Description: Lemma for dath 35890. Show the lines 𝑃𝑄 and 𝑆𝑇 form a plane. (Contributed by NM, 11-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))))) |
dalemc.l | ⊢ ≤ = (le‘𝐾) |
dalemc.j | ⊢ ∨ = (join‘𝐾) |
dalemc.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dalem1.o | ⊢ 𝑂 = (LPlanes‘𝐾) |
dalem1.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
Ref | Expression |
---|---|
dalem2 | ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ (𝑆 ∨ 𝑇)) ∈ 𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalema.ph | . . . 4 ⊢ (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 35777 | . . 3 ⊢ (𝜑 → 𝐾 ∈ HL) |
3 | 1 | dalempea 35780 | . . 3 ⊢ (𝜑 → 𝑃 ∈ 𝐴) |
4 | 1 | dalemqea 35781 | . . 3 ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
5 | 1 | dalemsea 35783 | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝐴) |
6 | 1 | dalemtea 35784 | . . 3 ⊢ (𝜑 → 𝑇 ∈ 𝐴) |
7 | dalemc.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
8 | dalemc.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | 7, 8 | hlatj4 35528 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ (𝑆 ∨ 𝑇)) = ((𝑃 ∨ 𝑆) ∨ (𝑄 ∨ 𝑇))) |
10 | 2, 3, 4, 5, 6, 9 | syl122anc 1447 | . 2 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ (𝑆 ∨ 𝑇)) = ((𝑃 ∨ 𝑆) ∨ (𝑄 ∨ 𝑇))) |
11 | dalemc.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
12 | dalem1.o | . . . . 5 ⊢ 𝑂 = (LPlanes‘𝐾) | |
13 | dalem1.y | . . . . 5 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
14 | 1, 11, 7, 8, 12, 13 | dalempjsen 35807 | . . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾)) |
15 | 1, 11, 7, 8, 12, 13 | dalemqnet 35806 | . . . . 5 ⊢ (𝜑 → 𝑄 ≠ 𝑇) |
16 | eqid 2778 | . . . . . 6 ⊢ (LLines‘𝐾) = (LLines‘𝐾) | |
17 | 7, 8, 16 | llni2 35666 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ 𝑄 ≠ 𝑇) → (𝑄 ∨ 𝑇) ∈ (LLines‘𝐾)) |
18 | 2, 4, 6, 15, 17 | syl31anc 1441 | . . . 4 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (LLines‘𝐾)) |
19 | 1, 11, 7, 8, 12, 13 | dalem1 35813 | . . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ≠ (𝑄 ∨ 𝑇)) |
20 | 1, 11, 7, 8, 12, 13 | dalemcea 35814 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
21 | 1 | dalemclpjs 35788 | . . . . 5 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆)) |
22 | 1 | dalemclqjt 35789 | . . . . 5 ⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇)) |
23 | eqid 2778 | . . . . . 6 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
24 | eqid 2778 | . . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
25 | 11, 23, 24, 8, 16 | 2llnm4 35724 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾) ∧ (𝑄 ∨ 𝑇) ∈ (LLines‘𝐾)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇))) → ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ≠ (0.‘𝐾)) |
26 | 2, 20, 14, 18, 21, 22, 25 | syl132anc 1456 | . . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ≠ (0.‘𝐾)) |
27 | 23, 24, 8, 16 | 2llnmat 35678 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾) ∧ (𝑄 ∨ 𝑇) ∈ (LLines‘𝐾)) ∧ ((𝑃 ∨ 𝑆) ≠ (𝑄 ∨ 𝑇) ∧ ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ≠ (0.‘𝐾))) → ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ∈ 𝐴) |
28 | 2, 14, 18, 19, 26, 27 | syl32anc 1446 | . . 3 ⊢ (𝜑 → ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ∈ 𝐴) |
29 | 7, 23, 8, 16, 12 | 2llnmj 35714 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾) ∧ (𝑄 ∨ 𝑇) ∈ (LLines‘𝐾)) → (((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ∈ 𝐴 ↔ ((𝑃 ∨ 𝑆) ∨ (𝑄 ∨ 𝑇)) ∈ 𝑂)) |
30 | 2, 14, 18, 29 | syl3anc 1439 | . . 3 ⊢ (𝜑 → (((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ∈ 𝐴 ↔ ((𝑃 ∨ 𝑆) ∨ (𝑄 ∨ 𝑇)) ∈ 𝑂)) |
31 | 28, 30 | mpbid 224 | . 2 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∨ (𝑄 ∨ 𝑇)) ∈ 𝑂) |
32 | 10, 31 | eqeltrd 2859 | 1 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ (𝑆 ∨ 𝑇)) ∈ 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 lecple 16345 joincjn 17330 meetcmee 17331 0.cp0 17423 Atomscatm 35417 HLchlt 35504 LLinesclln 35645 LPlanesclpl 35646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-proset 17314 df-poset 17332 df-plt 17344 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-lat 17432 df-clat 17494 df-oposet 35330 df-ol 35332 df-oml 35333 df-covers 35420 df-ats 35421 df-atl 35452 df-cvlat 35476 df-hlat 35505 df-llines 35652 df-lplanes 35653 |
This theorem is referenced by: dalemdea 35816 |
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