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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem61 | Structured version Visualization version GIF version |
Description: Lemma for dath 36866. 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 2821 | . . 3 ⊢ ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) | |
12 | eqid 2821 | . . 3 ⊢ ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇)) = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇)) | |
13 | eqid 2821 | . . 3 ⊢ ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈)) = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈)) | |
14 | eqid 2821 | . . 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 36861 | . 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 36862 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐷 ∨ 𝐸) = (((((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ∨ ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))) ∨ ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))) ∧ 𝑌)) |
19 | 15, 18 | breqtrrd 5086 | 1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐹 ≤ (𝐷 ∨ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 lecple 16566 joincjn 17548 meetcmee 17549 Atomscatm 36393 HLchlt 36480 LPlanesclpl 36622 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-lat 17650 df-clat 17712 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-llines 36628 df-lplanes 36629 df-lvols 36630 |
This theorem is referenced by: dalem62 36864 |
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