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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem22 | Structured version Visualization version GIF version |
Description: Lemma for dath 37736. Show that lines 𝑐𝑑 and 𝑃𝑆 determine a plane. (Contributed by NM, 2-Aug-2012.) |
Ref | Expression |
---|---|
dalem.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))))) |
dalem.l | ⊢ ≤ = (le‘𝐾) |
dalem.j | ⊢ ∨ = (join‘𝐾) |
dalem.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dalem.ps | ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) |
dalem22.o | ⊢ 𝑂 = (LPlanes‘𝐾) |
dalem22.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
dalem22.z | ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
Ref | Expression |
---|---|
dalem22 | ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∨ (𝑃 ∨ 𝑆)) ∈ 𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalem.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 | dalem.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | dalem.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
4 | dalem.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | dalem.ps | . . 3 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) | |
6 | eqid 2738 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
7 | dalem22.o | . . 3 ⊢ 𝑂 = (LPlanes‘𝐾) | |
8 | dalem22.y | . . 3 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
9 | dalem22.z | . . 3 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | dalem21 37694 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑑)(meet‘𝐾)(𝑃 ∨ 𝑆)) ∈ 𝐴) |
11 | 1 | dalemkehl 37623 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL) |
12 | 11 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐾 ∈ HL) |
13 | 1, 2, 3, 4, 5 | dalemcjden 37692 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑐 ∨ 𝑑) ∈ (LLines‘𝐾)) |
14 | 1, 2, 3, 4, 7, 8 | dalempjsen 37653 | . . . . 5 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾)) |
15 | 14 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾)) |
16 | eqid 2738 | . . . . 5 ⊢ (LLines‘𝐾) = (LLines‘𝐾) | |
17 | 3, 6, 4, 16, 7 | 2llnmj 37560 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑐 ∨ 𝑑) ∈ (LLines‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾)) → (((𝑐 ∨ 𝑑)(meet‘𝐾)(𝑃 ∨ 𝑆)) ∈ 𝐴 ↔ ((𝑐 ∨ 𝑑) ∨ (𝑃 ∨ 𝑆)) ∈ 𝑂)) |
18 | 12, 13, 15, 17 | syl3anc 1370 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (((𝑐 ∨ 𝑑)(meet‘𝐾)(𝑃 ∨ 𝑆)) ∈ 𝐴 ↔ ((𝑐 ∨ 𝑑) ∨ (𝑃 ∨ 𝑆)) ∈ 𝑂)) |
19 | 18 | 3adant2 1130 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝑐 ∨ 𝑑)(meet‘𝐾)(𝑃 ∨ 𝑆)) ∈ 𝐴 ↔ ((𝑐 ∨ 𝑑) ∨ (𝑃 ∨ 𝑆)) ∈ 𝑂)) |
20 | 10, 19 | mpbid 231 | 1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∨ (𝑃 ∨ 𝑆)) ∈ 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5074 ‘cfv 6427 (class class class)co 7268 Basecbs 16900 lecple 16957 joincjn 18017 meetcmee 18018 Atomscatm 37263 HLchlt 37350 LLinesclln 37491 LPlanesclpl 37492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5485 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-proset 18001 df-poset 18019 df-plt 18036 df-lub 18052 df-glb 18053 df-join 18054 df-meet 18055 df-p0 18131 df-lat 18138 df-clat 18205 df-oposet 37176 df-ol 37178 df-oml 37179 df-covers 37266 df-ats 37267 df-atl 37298 df-cvlat 37322 df-hlat 37351 df-llines 37498 df-lplanes 37499 |
This theorem is referenced by: dalem23 37696 |
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