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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem61 | Structured version Visualization version GIF version |
Description: Lemma for dath 39719. Show that atoms 𝐷, 𝐸, and 𝐹 lie on the same line (axis of perspectivity). Eliminate hypotheses containing dummy atoms 𝑐 and 𝑑. (Contributed by NM, 11-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)))) |
dalem61.m | ⊢ ∧ = (meet‘𝐾) |
dalem61.o | ⊢ 𝑂 = (LPlanes‘𝐾) |
dalem61.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
dalem61.z | ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
dalem61.d | ⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) |
dalem61.e | ⊢ 𝐸 = ((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈)) |
dalem61.f | ⊢ 𝐹 = ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆)) |
Ref | Expression |
---|---|
dalem61 | ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐹 ≤ (𝐷 ∨ 𝐸)) |
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 | dalem61.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
7 | dalem61.o | . . 3 ⊢ 𝑂 = (LPlanes‘𝐾) | |
8 | dalem61.y | . . 3 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
9 | dalem61.z | . . 3 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
10 | dalem61.f | . . 3 ⊢ 𝐹 = ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆)) | |
11 | eqid 2735 | . . 3 ⊢ ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) | |
12 | eqid 2735 | . . 3 ⊢ ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇)) = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇)) | |
13 | eqid 2735 | . . 3 ⊢ ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈)) = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈)) | |
14 | eqid 2735 | . . 3 ⊢ (((((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ∨ ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))) ∨ ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))) ∧ 𝑌) = (((((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ∨ ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))) ∨ ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))) ∧ 𝑌) | |
15 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 | dalem59 39714 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐹 ≤ (((((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ∨ ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))) ∨ ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))) ∧ 𝑌)) |
16 | dalem61.d | . . 3 ⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) | |
17 | dalem61.e | . . 3 ⊢ 𝐸 = ((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈)) | |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17, 11, 12, 13, 14 | dalem60 39715 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐷 ∨ 𝐸) = (((((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ∨ ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))) ∨ ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))) ∧ 𝑌)) |
19 | 15, 18 | breqtrrd 5176 | 1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐹 ≤ (𝐷 ∨ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 lecple 17305 joincjn 18369 meetcmee 18370 Atomscatm 39245 HLchlt 39332 LPlanesclpl 39475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-lat 18490 df-clat 18557 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 |
This theorem is referenced by: dalem62 39717 |
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